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• Frequency Distribution | Tables, Types & Examples

Frequency Distribution | Tables, Types & Examples

Published on June 7, 2022 by Shaun Turney . Revised on June 21, 2023.

A frequency distribution describes the number of observations for each possible value of a variable . Frequency distributions are depicted using graphs and frequency tables.

Table of contents

What is a frequency distribution, how to make a frequency table, how to graph a frequency distribution, other interesting articles, frequently asked questions about frequency distributions.

The frequency of a value is the number of times it occurs in a dataset. A frequency distribution is the pattern of frequencies of a variable. It’s the number of times each possible value of a variable occurs in a dataset.

Types of frequency distributions

There are four types of frequency distributions:

• You can use this type of frequency distribution for categorical variables .
• You can use this type of frequency distribution for quantitative variables .
• You can use this type of frequency distribution for any type of variable when you’re more interested in comparing frequencies than the actual number of observations.
• You can use this type of frequency distribution for ordinal or quantitative variables when you want to understand how often observations fall below certain values .

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Frequency distributions are often displayed using frequency tables . A frequency table is an effective way to summarize or organize a dataset. It’s usually composed of two columns:

• The values or class intervals
• Their frequencies

The method for making a frequency table differs between the four types of frequency distributions. You can follow the guides below or use software such as Excel, SPSS, or R to make a frequency table.

How to make an ungrouped frequency table

• For ordinal variables , the values should be ordered from smallest to largest in the table rows.
• For nominal variables , the values can be in any order in the table. You may wish to order them alphabetically or in some other logical order.
• Especially if your dataset is large, it may help to count the frequencies by tallying . Add a third column called “Tally.” As you read the observations, make a tick mark in the appropriate row of the tally column for each observation. Count the tally marks to determine the frequency.

How to make a grouped frequency table

• Calculate the range . Subtract the lowest value in the dataset from the highest.

• Create a table with two columns and as many rows as there are class intervals. Label the first column using the variable name and label the second column “Frequency.” Enter the class intervals in the first column.
• Count the frequencies. The frequencies are the number of observations in each class interval. You can count by tallying if you find it helpful. Enter the frequencies in the second column of the table beside their corresponding class intervals.

Round the class interval width to 10.

The class intervals are 19 ≤ a < 29, 29 ≤ a < 39, 39 ≤ a < 49, 49 ≤ a < 59, and 59 ≤ a < 69.

How to make a relative frequency table

• Create an ungrouped or grouped frequency table .
• Add a third column to the table for the relative frequencies. To calculate the relative frequencies, divide each frequency by the sample size. The sample size is the sum of the frequencies.

How to make a cumulative frequency table

• Create an ungrouped or grouped frequency table for an ordinal or quantitative variable. Cumulative frequencies don’t make sense for nominal variables because the values have no order—one value isn’t more than or less than another value.
• Add a third column to the table for the cumulative frequencies. The cumulative frequency is the number of observations less than or equal to a certain value or class interval. To calculate the relative frequencies, add each frequency to the frequencies in the previous rows.
• Optional: If you want to calculate the cumulative relative frequency , add another column and divide each cumulative frequency by the sample size.

Pie charts, bar charts, and histograms are all ways of graphing frequency distributions. The best choice depends on the type of variable and what you’re trying to communicate.

A pie chart is a graph that shows the relative frequency distribution of a nominal variable .

A pie chart is a circle that’s divided into one slice for each value. The size of the slices shows their relative frequency.

This type of graph can be a good choice when you want to emphasize that one variable is especially frequent or infrequent, or you want to present the overall composition of a variable.

A disadvantage of pie charts is that it’s difficult to see small differences between frequencies. As a result, it’s also not a good option if you want to compare the frequencies of different values.

A bar chart is a graph that shows the frequency or relative frequency distribution of a categorical variable (nominal or ordinal).

The y -axis of the bars shows the frequencies or relative frequencies, and the x -axis shows the values. Each value is represented by a bar, and the length or height of the bar shows the frequency of the value.

A bar chart is a good choice when you want to compare the frequencies of different values. It’s much easier to compare the heights of bars than the angles of pie chart slices.

A histogram is a graph that shows the frequency or relative frequency distribution of a quantitative variable . It looks similar to a bar chart.

The continuous variable is grouped into interval classes , just like a grouped frequency table . The y -axis of the bars shows the frequencies or relative frequencies, and the x -axis shows the interval classes. Each interval class is represented by a bar, and the height of the bar shows the frequency or relative frequency of the interval class.

Although bar charts and histograms are similar, there are important differences:

A histogram is an effective visual summary of several important characteristics of a variable. At a glance, you can see a variable’s central tendency and variability , as well as what probability distribution it appears to follow, such as a normal , Poisson , or uniform distribution.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s t table
• Student’s t distribution
• Quartiles & Quantiles
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

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A histogram is an effective way to tell if a frequency distribution appears to have a normal distribution .

Plot a histogram and look at the shape of the bars. If the bars roughly follow a symmetrical bell or hill shape, like the example below, then the distribution is approximately normally distributed.

Categorical variables can be described by a frequency distribution. Quantitative variables can also be described by a frequency distribution, but first they need to be grouped into interval classes .

Probability is the relative frequency over an infinite number of trials.

For example, the probability of a coin landing on heads is .5, meaning that if you flip the coin an infinite number of times, it will land on heads half the time.

Since doing something an infinite number of times is impossible, relative frequency is often used as an estimate of probability. If you flip a coin 1000 times and get 507 heads, the relative frequency, .507, is a good estimate of the probability.

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Frequency Distribution

A frequency distribution shows the frequency of repeated items in a graphical form or tabular form. It gives a visual display of the frequency of items or shows the number of times they occurred. Let's learn about frequency distribution in this article in detail.

What is Frequency Distribution?

Frequency distribution is used to organize the collected data in table form. The data could be marks scored by students, temperatures of different towns, points scored in a volleyball match, etc. After data collection, we have to show data in a meaningful manner for better understanding. Organize the data in such a way that all its features are summarized in a table. This is known as frequency distribution.

Let's consider an example to understand this better. The following are the scores of 10 students in the G.K. quiz released by Mr. Chris 15, 17, 20, 15, 20, 17, 17, 14, 14, 20. Let's represent this data in frequency distribution and find out the number of students who got the same marks.

We can see that all the collected data is organized under the column quiz marks and the number of students. This makes it easier to understand the given information and we can see that the number of students who obtained the same marks. Thus, frequency distribution in statistics helps us to organize the data in an easy way to understand its features at a glance.

Frequency Distribution Graphs

There is another way to show data that is in the form of graphs and it can be done by using a frequency distribution graph. The graphs help us to understand the collected data in an easy way. The graphical representation of a frequency distribution can be shown using the following:

• Bar Graphs : Bar graphs represent data using rectangular bars of uniform width along with equal spacing between the rectangular bars.
• Histograms : A histogram is a graphical presentation of data using rectangular bars of different heights. In a histogram, there is no space between the rectangular bars.
• Pie Chart : A pie chart is a type of graph that visually displays data in a circular chart. It records data in a circular manner and then it is further divided into sectors that show a particular part of data out of the whole part.
• Frequency Polygon: A frequency polygon is drawn by joining the mid-points of the bars in a histogram.

Types of Frequency Distribution

There are four types of frequency distribution under statistics which are explained below:

• Ungrouped frequency distribution: It shows the frequency of an item in each separate data value rather than groups of data values.
• Grouped frequency distribution: In this type, the data is arranged and separated into groups called class intervals. The frequency of data belonging to each class interval is noted in a frequency distribution table. The grouped frequency table shows the distribution of frequencies in class intervals.
• Relative frequency distribution: It tells the proportion of the total number of observations associated with each category.
• Cumulative frequency distribution: It is the sum of the first frequency and all frequencies below it in a frequency distribution. You have to add a value with the next value then add the sum with the next value again and so on till the last. The last cumulative frequency will be the total sum of all frequencies.
• Frequency Distribution Table

A frequency distribution table is a chart that shows the frequency of each of the items in a data set. Let's consider an example to understand how to make a frequency distribution table using tally marks. A jar containing beads of different colors- red, green, blue, black, red, green, blue, yellow, red, red, green, green, green, yellow, red, green, yellow. To know the exact number of beads of each particular color, we need to classify the beads into categories. An easy way to find the number of beads of each color is to use tally marks . Pick the beads one by one and enter the tally marks in the respective row and column. Then, indicate the frequency for each item in the table.

Thus, the table so obtained is called a frequency distribution table .

Types of Frequency Distribution Table

There are two types of frequency distribution tables: Grouped and ungrouped frequency distribution tables.

Grouped Frequency Distribution Table: To arrange a large number of observations or data, we use grouped frequency distribution table. In this, we form class intervals to tally the frequency for the data that belongs to that particular class interval.

For example, Marks obtained by 20 students in the test are as follows. 5, 10, 20, 15, 5, 20, 20, 15, 15, 15, 10, 10, 10, 20, 15, 5, 18, 18, 18, 18. To arrange the data in grouped table we have to make class intervals. Thus, we will make class intervals of marks like 0 – 5, 6 – 10, and so on. Given below table shows two columns one is of class intervals (marks obtained in test) and the second is of frequency (no. of students). In this, we have not used tally marks as we counted the marks directly.

Ungrouped Frequency Distribution Table: In the ungrouped frequency distribution table, we don't make class intervals, we write the accurate frequency of individual data. Considering the above example, the ungrouped table will be like this. Given below table shows two columns: one is of marks obtained in the test and the second is of frequency (no. of students).

Important Notes:

Following are the important points related to frequency distribution.

• Figures or numbers collected for some definite purpose is called data.
• Frequency is the value in numbers that shows how often a particular item occurs in the given data set.
• There are two types of frequency table - Grouped Frequency Distribution and Ungrouped Frequency Distribution.
• Data can be shown using graphs like histograms, bar graphs, frequency polygons, and so on.

Related Articles on Frequency Distribution

To learn more about the frequency distribution, check the given articles.

• Data Handling

Frequency Distribution Examples

Example 1: There are 20 students in a class. The teacher, Ms. Jolly, asked the students to tell their favorite subject. The results are as follows - Mathematics, English, Science, Science, Mathematics, Science, English, Art, Mathematics, Mathematics, Science, Art, Art, Science, Mathematics, Art, Mathematics, English, English, Mathematics.

Represent this data in the form of frequency distribution and identify the most-liked subject?

Solution: 20 students have indicated their choices of preferred subjects. Let us represent this data using tally marks. The tally marks are showing the frequency of each subject.

According to the above frequency distribution, mathematics is the most liked subject.

Example 2: 100 schools decided to plant 100 tree saplings in their gardens on world environment day. Represent the given data in the form of frequency distribution and find the number of schools that are able to plant 50% of the plants or more? 95, 67, 28, 32, 65, 65, 69, 33, 98, 96, 76, 42, 32, 38, 42, 40, 40, 69, 95, 92, 75, 83, 76, 83, 85, 62, 37, 65, 63, 42, 89, 65, 73, 81, 49, 52, 64, 76, 83, 92, 93, 68, 52, 79, 81, 83, 59, 82, 75, 82, 86, 90, 44, 62, 31, 36, 38, 42, 39, 83, 87, 56, 58, 23, 35, 76, 83, 85, 30, 68, 69, 83, 86, 43, 45, 39, 83, 75, 66, 83, 92, 75, 89, 66, 91, 27, 88, 89, 93, 42, 53, 69, 90, 55, 66, 49, 52, 83, 34, 36

Solution: To include all the observations in groups, we will create various groups of equal intervals. These intervals are called class intervals. In the frequency distribution, the number of plants survived is showing the class intervals, tally marks are showing frequency, and the number of schools is the frequency in numbers.

So, according to class intervals starting from 50 – 59 to 90 – 99, the frequency of schools able to retain 50% or more plants are 8 + 18 + 10 + 23 + 12 = 71 schools. Thus, 71 schools are able to retain 50% or more plants in their garden.

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Frequency Distribution Practice Questions

Faqs on frequency distribution, what is a frequency distribution in math.

In statistics, the frequency distribution is a graph or data set organized to represent the frequency of occurrence of each possible outcome of an event that is observed a specific number of times. Frequency distribution is a tabular or graphical representation of the data that shows the frequency of all the observations.

What are the 2 Types of Frequency Distribution Table?

The 2 types of frequency distributions are:

• Ungrouped frequency distribution
• Grouped frequency distribution

Why are Frequency Distributions Important?

Frequency charts are the best way to organize data. Doctors use it to understand the frequency of diseases. Sports analysts use it to understand the performance of a sportsperson. Wherever you have a large amount of data, frequency distribution makes it easy to analyze the data.

How do you find Frequency Distribution?

Follow the steps to find frequency distribution:

• Step 1: To make a frequency chart, first, write the categories in the first column.
• Step 2: In the next step, tally the score in the second column.
• Step 3: And finally, count the tally to write the frequency of each category in the third column.

Thus, in this way, we can find the frequency distribution of an event.

What is the Difference Between Frequency Table and Frequency Distribution?

The frequency table is a tabular method where the frequency is assigned to its respective category. Whereas, a frequency distribution is known as the graphical representation of the frequency table.

What is Grouped Frequency Distribution?

A grouped frequency distribution shows the scores by grouping the observations into intervals and then lists these intervals in the frequency distribution table. The intervals in grouped frequency distribution are called class limits.

What is Ungrouped Frequency Distribution?

The ungrouped frequency distribution is a type of frequency distribution that displays the frequency of each individual data value instead of groups of data values. In this type of frequency distribution, we can directly see how often different values occurred in the table.

What are the Components of Frequency Distribution?

The components of the frequency distribution are as follows:

• Class interval
• Types of class interval
• Class boundaries
• Midpoint or classmark
• Width or size of class interval
• Class frequency
• Frequency class width

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What Is a Frequency Distribution?

• How It Works
• Visual Representation

Frequency Distribution in Trading

• Frequency Distribution FAQs

The Bottom Line

• Corporate Finance
• Financial Analysis

Frequency Distribution: Definition in Statistics and Trading

Investopedia / Sydney Saporito

A frequency distribution is a representation, either in a graphical or tabular format, that displays the number of observations within a given interval. The frequency is how often a value occurs in an interval while the distribution is the pattern of frequency of the variable.

The interval size depends on the data being analyzed and the goals of the analyst. The intervals must be mutually exclusive and exhaustive. Frequency distributions are typically used within a statistical context. Generally, frequency distributions can be associated with the charting of a normal distribution .

Key Takeaways

• A frequency distribution in statistics is a representation that displays the number of observations within a given interval.
• The representation of a frequency distribution can be graphical or tabular so that it is easier to understand.
• Frequency distributions are particularly useful for normal distributions, which show the observations of probabilities divided among standard deviations.
• In finance, traders use frequency distributions to take note of price action and identify trends.

Understanding a Frequency Distribution

As a statistical tool, a frequency distribution provides a visual representation of the distribution of observations within a particular test. Analysts often use a frequency distribution to visualize or illustrate the data collected in a sample. For example, the height of children can be split into several different categories or ranges.

In measuring the height of 50 children, some are tall and some are short, but there is a high probability of a higher frequency or concentration in the middle range. The most important factors for gathering data are that the intervals used must not overlap and must contain all of the possible observations.

Visual Representation of a Frequency Distribution

Both histograms and bar charts provide a visual display using columns, with the y-axis representing the frequency count, and the x-axis representing the variable to be measured. In the height of children, for example, the y-axis is the number of children, and the x-axis is the height. The columns represent the number of children observed with heights measured in each interval.

In general, a histogram chart will typically show a normal distribution, which means that the majority of occurrences will fall in the middle columns. Frequency distributions can be a key aspect of charting normal distributions which show observation probabilities divided among standard deviations .

Frequency distributions can be presented as a frequency table, a histogram, or a bar chart. Below is an example of a frequency distribution as a table.

Frequency distributions are not commonly used in the world of investments; however, traders who follow Richard D. Wyckoff , a pioneering early 20th-century trader, use an approach to trading that involves frequency distribution.

Investment houses still use the approach, which requires considerable practice, to teach traders. The frequency chart is referred to as a point-and-figure chart and was created out of a need for  floor traders  to take note of price action and to identify trends.

The y-axis is the variable measured, and the x-axis is the frequency count. Each change in price action is denoted in Xs and Os. Traders interpret it as an uptrend when three X's emerge; in this case, demand has overcome supply. In the reverse situation, when the chart shows three O's, it indicates that supply has overcome demand.

What Are the Types of Frequency Distribution?

The types of frequency distribution are grouped frequency distribution, ungrouped frequency distribution, cumulative frequency distribution, relative frequency distribution, and relative cumulative frequency distribution.

What Is the Importance of a Frequency Distribution?

A frequency distribution is a means to organize a large amount of data. It takes data from a population based on certain characteristics and organizes the data in a way that is comprehensible to an individual that wants to make assumptions about a given population.

How Can I Construct a Frequency Distribution?

To construct a frequency distribution, first, note the specific classes determined by intervals in one column then sum the numbers in each isolated category based on how many times it shows up. The frequency can then be noted in the second column.

A frequency distribution is used to display the number of observations within a particular interval. This method, while not always commonly used in investing, is still used by some traders. In this case, the frequency chart is called a point-and-figure chart and is used to identify trends through the observation of price action.

National Association of Securities Dealers Automated Quotations. " Point & Figure Basics ," Page 3.

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StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

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Exploratory data analysis: frequencies, descriptive statistics, histograms, and boxplots.

Jacob Shreffler ; Martin R. Huecker .

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Last Update: November 3, 2023 .

• Definition/Introduction

Researchers must utilize exploratory data techniques to present findings to a target audience and create appropriate graphs and figures. Researchers can determine if outliers exist, data are missing, and statistical assumptions will be upheld by understanding data. Additionally, it is essential to comprehend these data when describing them in conclusions of a paper, in a meeting with colleagues invested in the findings, or while reading others’ work.

• Issues of Concern

This comprehension begins with exploring these data through the outputs discussed in this article. Individuals who do not conduct research must still comprehend new studies, and knowledge of fundamentals in analyzing data and interpretation of histograms and boxplots facilitates the ability to appraise recent publications accurately. Without this familiarity, decisions could be implemented based on inaccurate delivery or interpretation of medical studies.

Frequencies and Descriptive Statistics

Effective presentation of study results, in presentation or manuscript form, typically starts with frequencies and descriptive statistics (ie, mean, medians, standard deviations). One can get a better sense of the variables by examining these data to determine whether a balanced and sufficient research design exists. Frequencies also inform on missing data and give a sense of outliers (will be discussed below).

Luckily, software programs are available to conduct exploratory data analysis. For this chapter, we will be examining the following research question.

RQ: Are there differences in drug life (length of effect) for Drug 23 based on the administration site?

A more precise hypothesis could be: Is drug 23 longer-lasting when administered via site A compared to site B?

To address this research question, exploratory data analysis is conducted. First, it is essential to start with the frequencies of the variables. To keep things simple, only variables of minutes (drug life effect) and administration site (A vs B) are included. See Image. Figure 1 for outputs for frequencies.

Figure 1 shows that the administration site appears to be a balanced design with 50 individuals in each group. The excerpt for minutes frequencies is the bottom portion of Figure 1 and shows how many cases fell into each time frame with the cumulative percent on the right-hand side. In examining Figure 1, one suspiciously low measurement (135) was observed, considering time variables. If a data point seems inaccurate, a researcher should find this case and confirm if this was an entry error. For the sake of this review, the authors state that this was an entry error and should have been entered 535 and not 135. Had the analysis occurred without checking this, the data analysis, results, and conclusions would have been invalid. When finding any entry errors and determining how groups are balanced, potential missing data is explored. If not responsibly evaluated, missing values can nullify results.  

After replacing the incorrect 135 with 535, descriptive statistics, including the mean, median, mode, minimum/maximum scores, and standard deviation were examined. Output for the research example for the variable of minutes can be seen in Figure 2. Observe each variable to ensure that the mean seems reasonable and that the minimum and maximum are within an appropriate range based on medical competence or an available codebook. One assumption common in statistical analyses is a normal distribution. Image . Figure 2 shows that the mode differs from the mean and the median. We have visualization tools such as histograms to examine these scores for normality and outliers before making decisions.

Histograms are useful in assessing normality, as many statistical tests (eg, ANOVA and regression) assume the data have a normal distribution. When data deviate from a normal distribution, it is quantified using skewness and kurtosis. [1]  Skewness occurs when one tail of the curve is longer. If the tail is lengthier on the left side of the curve (more cases on the higher values), this would be negatively skewed, whereas if the tail is longer on the right side, it would be positively skewed. Kurtosis is another facet of normality. Positive kurtosis occurs when the center has many values falling in the middle, whereas negative kurtosis occurs when there are very heavy tails. [2]

Additionally, histograms reveal outliers: data points either entered incorrectly or truly very different from the rest of the sample. When there are outliers, one must determine accuracy based on random chance or the error in the experiment and provide strong justification if the decision is to exclude them. [3]  Outliers require attention to ensure the data analysis accurately reflects the majority of the data and is not influenced by extreme values; cleaning these outliers can result in better quality decision-making in clinical practice. [4]  A common approach to determining if a variable is approximately normally distributed is converting values to z scores and determining if any scores are less than -3 or greater than 3. For a normal distribution, about 99% of scores should lie within three standard deviations of the mean. [5]  Importantly, one should not automatically throw out any values outside of this range but consider it in corroboration with the other factors aforementioned. Outliers are relatively common, so when these are prevalent, one must assess the risks and benefits of exclusion. [6]

Image . Figure 3 provides examples of histograms. In Figure 3A, 2 possible outliers causing kurtosis are observed. If values within 3 standard deviations are used, the result in Figure 3B are observed. This histogram appears much closer to an approximately normal distribution with the kurtosis being treated. Remember, all evidence should be considered before eliminating outliers. When reporting outliers in scientific paper outputs, account for the number of outliers excluded and justify why they were excluded.

Boxplots can examine for outliers, assess the range of data, and show differences among groups. Boxplots provide a visual representation of ranges and medians, illustrating differences amongst groups, and are useful in various outlets, including evidence-based medicine. [7]  Boxplots provide a picture of data distribution when there are numerous values, and all values cannot be displayed (ie, a scatterplot). [8]  Figure 4 illustrates the differences between drug site administration and the length of drug life from the above example.

Image . Figure 4 shows differences with potential clinical impact. Had any outliers existed (data from the histogram were cleaned), they would appear outside the line endpoint. The red boxes represent the middle 50% of scores. The lines within each red box represent the median number of minutes within each administration site. The horizontal lines at the top and bottom of each line connected to the red box represent the 25th and 75th percentiles. In examining the difference boxplots, an overlap in minutes between 2 administration sites were observed: the approximate top 25 percent from site B had the same time noted as the bottom 25 percent at site A. Site B had a median minute amount under 525, whereas administration site A had a length greater than 550. If there were no differences in adverse reactions at site A, analysis of this figure provides evidence that healthcare providers should administer the drug via site A. Researchers could follow by testing a third administration site, site C. Image . Figure 5 shows what would happen if site C led to a longer drug life compared to site A.

Figure 5 displays the same site A data as Figure 4, but something looks different. The significant variance at site C makes site A’s variance appear smaller. In order words, patients who were administered the drug via site C had a larger range of scores. Thus, some patients experience a longer half-life when the drug is administered via site C than the median of site A; however, the broad range (lack of accuracy) and lower median should be the focus. The precision of minutes is much more compacted in site A. Therefore, the median is higher, and the range is more precise. One may conclude that this makes site A a more desirable site.

• Clinical Significance

Ultimately, by understanding basic exploratory data methods, medical researchers and consumers of research can make quality and data-informed decisions. These data-informed decisions will result in the ability to appraise the clinical significance of research outputs. By overlooking these fundamentals in statistics, critical errors in judgment can occur.

• Nursing, Allied Health, and Interprofessional Team Interventions

All interprofessional healthcare team members need to be at least familiar with, if not well-versed in, these statistical analyses so they can read and interpret study data and apply the data implications in their everyday practice. This approach allows all practitioners to remain abreast of the latest developments and provides valuable data for evidence-based medicine, ultimately leading to improved patient outcomes.

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Exploratory Data Analysis Figure 1 Contributed by Martin Huecker, MD and Jacob Shreffler, PhD

Exploratory Data Analysis Figure 2 Contributed by Martin Huecker, MD and Jacob Shreffler, PhD

Exploratory Data Analysis Figure 3 Contributed by Martin Huecker, MD and Jacob Shreffler, PhD

Exploratory Data Analysis Figure 4 Contributed by Martin Huecker, MD and Jacob Shreffler, PhD

Exploratory Data Analysis Figure 5 Contributed by Martin Huecker, MD and Jacob Shreffler, PhD

Disclosure: Jacob Shreffler declares no relevant financial relationships with ineligible companies.

Disclosure: Martin Huecker declares no relevant financial relationships with ineligible companies.

This book is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ), which permits others to distribute the work, provided that the article is not altered or used commercially. You are not required to obtain permission to distribute this article, provided that you credit the author and journal.

• Cite this Page Shreffler J, Huecker MR. Exploratory Data Analysis: Frequencies, Descriptive Statistics, Histograms, and Boxplots. [Updated 2023 Nov 3]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

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Goodness of Fit and Related Chi-Square Tests

13 Frequency Distributions

13.1 analyzing distributions of data.

Throughout this text, we will focus on using frequency analysis and descriptive statistics. These simple but powerful analyses enable you to examine your data and identify patterns including the shapes and distributions of data, missing values, and outliers. Frequencies and distributions are important concepts in the quantitative analysis of data that underlie the overall statistical approach covered in this book. In fact, this approach is often referred to as “frequentist statistics” because it relies on frequencies to make inferences about the data. An alternative approach is called the Bayesian statistical analysis which relies on probabilities. While we won’t go into detail about the differences between frequentist and Bayesian statistical approaches, it is important to recognize that frequencies play a key role in the approach that we are demonstrating here but that a frequentist approach is not the only way to analyze your data.

A frequency is simply the number of times something happens. It could be, for example, the number of people with brown hair, the number of children in a family, the number of deaths in a hospital. It could also be the number of times an electrical signal with a given level of energy-intensity is recorded.

A distribution shows the relative frequencies of each possible value or category for a variable. Distributions are used to describe the organization or shape of a set of scores or values for a particular variable. If you studied statistics previously you are most likely familiar with the normal distribution or bell curve. What you may not realize is that distributions other than the normal distribution are also used in statistic analyses and that datasets can take the shape of these other distributions. For example, datasets that include only discrete scores ranging from 1 to 5 would not be expected to fit a normal distribution curve but would rather be compared to a categorical distribution curve – like the chi-square distribution or the Poisson distribution.

Distributions can be obtained by counting the number of events that occur or how many participants in a sample have a specific score on a questionnaire or measure (i.e., counting frequencies). For example, you might look at the number of patients presenting to the Emergency Department for different reasons: cardiovascular concerns, accidents, infections, reported symptoms. You might also consider responses to an anxiety questionnaire scored on a Likert scale (using discrete scaled scores) ranging from 1 to 5. You may consider reviewing the number of respondents in your sample had each possible score (i.e. 1, 2, 3, 4, or 5) – in other words, how frequent each score appeared within the total set of scores.

The following is an example of a table showing a frequency distribution for a set of responses to a categorical variable ranging from 1 to 5, and a graphical representation of the frequency of responses in each category. The Category Label is presented on the x-axis, and the number of responses—frequencies for each response are presented on the y-axis.

Table 13.1 Frequency Distribution For Categorical Responses

The PROC FREQ Procedure

SAS Code to produce the Frequency Distribution and Corresponding Figure

Figure 13.1 Frequency Distribution For a Categorical Response Variable

Frequency distributions are useful in describing variables, helping to identify errors (impossible values) and outliers, assessing how well a continuous variable fits the normal distribution, or to test hypotheses using specific statistical tests such as using a chi-square test to evaluate categorical variables.

Frequency & Distribution of a Count Variable

Count variables refer to those that simply tally the number of items or events that occur. For example, you might want to count the number of adverse events that occur when people take a medication, the number of times nurses wash their hands during their shift or the number of babies born in each month of the year. In health research, there are many items or events that can be counted!

Note that for a count variable, the values are arithmetically meaningful and represent the number of events or items for a specific variable– the count variable is quite literally storing the count of items of interest. Therefore, values differ by a magnitude and are meaningful. For example, 4 adverse events are twice as many as 2 adverse events.

Count variables are different than categorical variables.

Categorical variables are used when the researcher wishes to use numbers to represent different kinds of items or events. In the categorical variable the numbers are arbitrary. For example, hair colour could be coded as 1 = blonde, 2 = brown, 3 = gray, 4 = red, and 5 = other but it could also be coded as 11 = blonde, 22 = brown, 33 = gray, 44 = red, and 55 = other. The numbers representing a category label are not mathematically meaningful and do not represent the number of people with a specific hair colour. Of course, you can analyze the frequency of people with each response which we will cover later when we talk about categorical variables in more detail.

Working example to process a “count” variable

Let’s say we would like to take a sample of 50 families from a population of 1000 households in a small town and record the number of children in each household. Here we will create two variables, the first we will call “ NKIDS ” and the second we will call “ HOUSEHOLDS ”. The variable NKIDS is the categorical variable for the number of children in each household that we sampled, while the variable HOUSEHOLDS represents the number of response houses that report having a given number of children.

The Scenario

We arrive at the small town and knock on the front door of the first house. Below is the dialogue between the researchers and the respondents.

“Good day, we are Biostatisticians and we are conducting a study of the number of children in your family.”

“Oh we don’t have any children.”

“Okay, thank-you.”

We note that for Household #1 there are 0 children. We then knock on the front door of the second house.

“We have 7 children. Would you like some?”

“No thank-you, but have a nice day.”

We note that for Household #2 there are 7 children. We then knock on the front door of the third house, and continue our process for each of 50 houses in the town.

In this example, the categorical variable is NKIDS and is considered the independent variable, while the continuous-discrete variable is HOUSEHOLDS and is considered the dependent variable – aka the measure of interest.

Since there can only be whole numbers for the variable NKIDS (i.e., you can’t actually have 1.2 children), the variable NKIDS is a discrete categorical variable, and likewise, because we are counting families on a whole number line (i.e. not partial families) then the variable NFAMILIES is a discrete random variable.

The frequency distribution recording sheet for this example is shown below. Notice that as a rule, we want to keep our variable labels at or near 8 characters so that HOUSEHOLDS is shortened to HSEHLD.

Table 13.2 Tally Sheet to produce the Frequency Distribution for Number of Children in Each Household Sampled  

Counting events such as the number of children in a family, the number of needles found on the ground near a safe injection site, or the number of patients readmitted to the hospital after discharge, typically follow the whole number line. Frequency tables are often used to show how many times an event has occurred.

In our example, we can say that the variable HOUSEHOLDS is a discrete random variable because in a given sample of 50 families the variable can take on (contain) any value between 0 and 50 (the total sample) on the whole number line.

Table 13.1 shows how we can determine the frequency and relative frequency (percentage out of 100) for the number of children in each of the families in our sample.  Of the 50 families in our sample, nine families did not have children, 7 families had 1 child, 12 families had 2 children, 9 families had 3 children, 5 families had 4 children, 6 families had 5 children, no families had 6 children, and 2 families had 7 children. Notice here that the variable of interest is the number of families reporting each of the possible number of children.

Relative frequency refers to the proportion of the entire sample that had a particular value. In this example, the relative frequency tells us what percentage of the sample had a specific number of children. To calculate the relative frequency, simply divide each frequency by the total number of families and then multiply the result by 100 to calculate the percentage value. For example, from the data in Table 4.1 we see that in this sample, 24% of the families had 2 children while only 4% had 7 children.

Creating the SAS Program to compute a frequency distribution for a discrete random variable

Below are the SAS commands to produce the frequency distribution table of the data recorded for the number of children in our sample of 50 families.

SAS Code to produce a Frequency Distribution Table For Number of Children in Each Household sampled

In this SAS program, we are using the PROC FREQ statistical processing command with the keyword TABLES to produce a frequency distribution for the data recorded for our sample of 50 households. Notice in the PROC FREQ command sequence we included the statement WEIGHT HSEHLD. In this example, the independent or categorical variable is NKIDS and the dependent discrete random variable is HSEHLD.  The WEIGHT command enables us to enter the summary data for the dependent variable HSEHLD as the count related to the categorical variable NKIDS.

Notice the table indicates that 9 households reported no children, while no households reported having 6 children. The table also indicates that most households reported having 2 children.

TABLE 13.3 Frequency distribution for number of children in each household

The FREQ Procedure

The following is a SAS program to compute elements of PROC FREQ for frequency distributions. The data are fictitious and are used here to enable you to work through the various options and features of the PROC FREQ command with relevant options.

As you work through the SAS program take note of the specific features that are identified, therein. The scenario is based on a public health study in which a group of researchers intended to determine the number of discarded needles left on the ground within a 100-metre radius of safe injection sites. We begin the program first by reading the data set and then using the essential SAS statistical processing commands with relevant options for PROC FREQ.

The program begins by labeling the working SAS program as DATA FREQ13_4; – which simply creates a label for the SAS program in the present SAS work session;

The second line is the listing of variables to be read within the sample data set. The SAS command begins with the SAS keyword INPUT which is followed by the names of each variable. Notice that the variable names are kept to eight characters and each variable name begins with an alphabetic character rather than a number or a special character. In this example the variables SITE, NDLCNT and INCREG are used to indicate that we have a variable to list the various sites from which the data were collected (SITE), the number of needles found on the ground within a 100-metre radius of the exit door of the safe injection site (NDLCNT), and the estimated average household income reported in thousands of dollars for the region in which the injection site is located (INCREG).

We also use simple IF-THEN logic commands to create summary groups for both the variable NDLCNT – number of needles recorded at each site, as well as to group the average household income – INCGRP.

SAS Code to Demonstrate Features of PROC FREQ

F REQUENCY DISTRIBUTION FOR THE NUMBER NEEDLES FOUND ACROSS ALL SITES

The SAS commands to sort the data and run the PROC FREQ using the * between variables helps to summarize the data into 2-way frequency distribution tables. In this way, we can see at a glance, a summary of the dataset.

In the sequence of SAS processing commands, we first sort the data using PROC SORT, followed by the SAS commands PROC FREQ with the keyword TABLES and then the two variables that we wish to include in the 2-way table – NDLGRP * INCGRP.

Code Snippet for SAS Code to Demonstrate PROC SORT code added to PROC FREQ

PROC SORT; BY INCGRP; PROC FREQ; TABLES NDLGRP*INCGRP; TITLE1 ‘FREQ DIST FOR GROUP NEEDLES BY INCOME GROUP’; RUN;

The result of this sequence of commands enables us to produce the 2-way SAS table of the groups of needles found arranged by income groups.  The problem with this table is that the delivery of information is not optimized for the reader if the reader does not know what an income group of 5, or an NDLGRP of 3 refers.

Using the PROC FORMAT command enables us to explain the categories within each variable. The code to explain the levels of each category uses the following two-step approach.

1.) At the start of the program add the PROC FORMAT statement and the VALUE for each categorical variable.

In our example, we have two categorical variables: INCGRP and NDLGRP. The variable INCGRP has 5 levels, while the variable NDLGRP has three groups.

Later in the program, after we call a SAS procedure, like in this case we call PROC FREQ, we then call the FORMAT function and assign the predefined format to each variable used by the SAS procedure.

Notice we first call the variable – in this example the variable of interest is NDLGRP and this is followed by the PROC FORMAT VALUE name NDL. Notice also that when we include the VALUE name we follow it with a period(.). This command will place the full text for the variable category in the frequency distribution.

The results of this analysis demonstrate that the highest number of needles found near the areas of safe injection sites tended to be higher among low-income neighborhoods than the number of needles found near the safe injection sites located in more affluent areas.

Figure 13.5 Features of Proc Freq: Adding Proc Format to the Frequency Procedure for a block chart 

In the following output the SAS syntax is shown here.

At the top of the program add:

13.2 Distribution for a categorical variable

As previously discussed, categorical variables involve grouping items, persons, or attributes, whereby the assignment of numbers to each group is arbitrary. For example, you might be interested in looking at the employment status of nursing home workers. The variable: employment status would be a categorical or grouping variable and might contain the following categories: full-time, part-time, casual, and temporary . You could assign any number you wish to represent the group label because the number is merely a label when applied to represent the category and doesn’t hold any mathematical significance – the number simply enables you to group persons based on that variable (in this case, employment status).

It is important to remember that with categorical data our interest is not to compute measures of centrality or variance like means and standard deviations, and therefore we won’t compare the distribution of items of persons to a normal distribution (i.e., the bell curve). Rather, the data that is held in the categories are counts and so our evaluation approach is to use statistical methods based on frequencies and ranks.

In the following steps, we calculate frequencies, relative frequencies, proportions, and percentages for categorical variables. Consider this simple data set.

We add up how many participants are in each employment status group and transfer the information to our chart:

Better yet, here we will use SAS to produce a frequency distribution table.

SAS code to produce a frequency distribution for employment status

This program produces the basic frequency distribution table for a set of categorical data and since we included the PROC FORMAT commands we can explain the data output clearly.

Features of Proc Freq: Distribution for a Categorical Variable

FREQUENCY DISTRIBUTION OF EMPLOYMENT STATUS

The FREQ Procedure for Employment Status

Distribution for a continuous variable

Now let’s talk about analyzing data for continuous variables.

Suppose we recorded the heights (in inches) of 200 students.  In this example, height is a continuous variable since the possible values include decimals (not just whole numbers), there are equal intervals between each line on a tape measure, and there is a meaningful 0.

While we can examine frequencies and create a histogram for a continuous variable, it is likely that we will have many different values in our dataset because each student will have a slightly different height. For example, Tom might be 61.5 inches tall while Cara is 61.6 inches tall. As a result, few students will record the exact same height. It may be, therefore, more meaningful to group these data and create categories. In other words, you can transform a continuous variable into a categorical variable simply by grouping the data with the IF-THEN logic statements.

In the following example, we will use the grouping approach so that we can create a more comprehensive frequency distribution.

Let’s start with a dataset that includes two variables for our sample of 200 students (ID and HEIGHT).  For each participant, we assign an ID and then record the height in inches for each of our participants. (note: despite that in Canada we use the metric scale for most of our measurements, we continue to refer to our heights in inches and feet – old habits die hard!)

Here we can use SAS to produce the frequency distribution table based on our grouping strategy for the data. We start by naming the working file and then include the appropriate SYNTAX to describe the variables and add the simple logic statements.

Raw Dataset 13.1 Two-hundred Height Measurements (inches)

Below is the SAS code required for the frequency analysis for the dataset above:

PROC FORMAT; VALUE HT 1=’LESS THAN 66.0′ 2=’66.1 TO 68.0′ 3=’68.1 TO 70.0′ 4=’70.1 TO 72.0′ 5=’MORE THAN 72.0′; DATA HEIGHTS; LABEL ID = ‘PARTICIPANT ID’ HEIGHT = ‘PARTICIPANT HEIGHT’ HTGRP=’HEIGHT GROUP’; INPUT ID HEIGHT @@;

Notice some specific features of the INPUT statement above. Here we list two variables: ID and HEIGHT, followed by two @ symbols at the end of the list of variables. When two  @ symbols are presented together SAS does not skip to a new line after reading the list of variables (in this case ID and HEIGHT), but rather reads across the page. This format enables us to read the data as a constant stream across the page for as many rows as is required to present the entire dataset. The computer reads the data in the order of the variables listed. That is, the computer reads through the dataset assigning the first value as the ID and the second value as the HEIGHT until all data are read.

Below is the paragraph of simple logic statements that follow the INPUT format statement. With these simple IF-THEN logic statements we organize the large unwieldy data set into six manageable groups.

IF HEIGHT <=66.0 THEN HTGRP=1; IF HEIGHT >66.0 AND HEIGHT <=68.0 THEN HTGRP=2; IF HEIGHT >68.0 AND HEIGHT <=70.0 THEN HTGRP=3; IF HEIGHT >70.0 AND HEIGHT <=72.0 THEN HTGRP=4; IF HEIGHT >72.0  THEN HTGRP=5; DATALINES; 001 58.5 002 58.8 003 60.1 004 61.3 005 61.75 006 61.96 . . .

199 79.40 200 79.47 ; PROC FREQ; TABLES HEIGHT; RUN; PROC FREQ; TABLES HTGRP; FORMAT HTGRP HT. ; RUN;

As you see in the partial output presented in Figure 13.4 below, when we run the SAS command: PROC FREQ; TABLES HEIGHT; RUN; most of the values occur only once because height is a continuous variable which allows greater variation than categorical or count type variables. When reading this output, make sure that you screen for outliers by looking at the high and low values for the variable. SAS will also indicate the number of missing values which is also important when you are cleaning and screening your data.

Table 13.4 The output from PROC FREQ Applied to Continuous Data –Participant Height.

13.3  Creating a Histogram in SAS

Producing graphs in SAS enables us to examine the distribution of the data visually rather than in a table. The SAS code shown here includes the option to produce a histogram. A histogram is more than a vertical bar chart. Histograms use rectangles to illustrate the frequency and interval, whereby the height of the rectangle is relative to the frequency (y axis) and the width of the rectangle is relative to the interval (x axis).

The SAS code used here provides the analysis for our sample of 200 measures of height within a cohort of children.  In order to establish the appropriate number of intervals in our sample we calculate the range of our set of scores. The range refers to the spread of scores between the lowest estimate from our sample, and the highest estimate from our sample.  We can estimate the range apriori by running the PROC UNIVARIATE command. When we include the command MIDPOINTS= we can customize the output. Here we include the command HISTOGRAM /MIDPOINTS = 55 TO 85 BY 0.5, to produce the expected RANGE of highest and lowest values and then plot the midpoints for all categories within the range.

PROC UNIVARIATE; VAR HEIGHT; HISTOGRAM / MIDPOINTS=55 TO 85 BY 0.5 NORMAL; RUN;

Notice in Figure 13.6 that the x-axis is a continuous variable. An overlay of the shape of the distribution is represented by the blue BELL-SHAPED normal curve.

Figure 13.6 Histogram for Heights of Students in Sample of 200 Participants

Use the following SAS Code to group the data into categories and add a representation of the shape of the distribution (CTEXT = BLUE) when plotting the histogram.

proc univariate; var HEIGHT; histogram HEIGHT / normal midpoints = 55 60 65 70 75 80 85 90 CTEXT = BLUE;

Of you can use: proc sgplot; histogram HEIGHT; density HEIGHT;

Figure 13.7 Histogram for Heights of Students N= 200 Grouped Data

Dividing a continuous variable into categories

In the example shown above, it is easy to see how helpful it is to arrange these continuous data into categories that represent a range of values rather than individual values on a continuum.

In our height example, we can optimize these data by creating height categories rather than exact heights because there is so much variance in exact heights. However, in this process of arbitrarily categorizing our response variable by grouping the data together we recognize that there will be a loss of information. When grouping data we are essentially saying that the responses are exactly the same even though differences are observed. For example, when you group a student who is 61.5 inches tall with a student who is 64 inches tall, the difference between the two individuals will be ignored within the category. There is absolutely nothing wrong with using categories that represent a range of continuous values but if you are planning to collect your own data, it is usually best to collect continuous data from the source and group the data later. Generally speaking, you can always convert data from continuous data to categories but without the original estimates, you cannot go the other way!

Once you deem it helpful to transform a continuous variable into categories, next you need to decide how to chop up your data. Ideally each you should have an equal range of values in each group. In our example, which you can see the would-be participants are quite tall, we decided to transform the continuous height data into categories that are each 2 inches wide. Group 1 includes students with a height of less than 66 inches, Group 2 starts at 66.1 and tops out at 68 inches, Group 3 starts at 68.1 and tops out at 70 inches … and so on.

Alternatively, we could have decided to group students in intervals of 5 inches or even 1 inch. As the researcher, you decide how to group the data based on what will make meaningful groupings. This might be based on past literature, clinical reference values, logic, or a combination of these factors. Once we create our categories we can sort students into each group based on their height.  Then we can count how many students fall into each group and create a frequency distribution table (Table 13.5) and a histogram. Notice that the shape (distribution) of this histogram is the one we created using the continuous version of the height variable (Figure 13.6).  This is because in the first histogram exact values are included and the data is divided into quintiles based on the mean. On the other hand, in the second histogram students are grouped together in 2-inch height intervals – depending on the cut-off values that we chose the distribution would be different.

Using the grouping routines with simple logic statements helps to simplify the organization of the data. The output for the grouped data is invoked with the SAS command: PROC FREQ; TABLES HTGRP;  FORMAT HTGRP HT. ; RUN;

Table 13.5 Grouping PROC FREQ output for –Participant Height.

Figure 13.7 Histogram for Heights of Students using Height Group (N=200)

In addition to the histogram, SAS also includes a number of tables in the output with the PROC UNIVARIATE command. The data in these tables provide important information about our variable (height, in this example). As you can see in Table 13.8 the moments’ table provides descriptive statistics about our variable including the mean, standard deviation, and standard error, as well as the overall variance. Skewness and kurtosis are also provided which provide valuable information about how well the variable fits the normal distribution. Keep your critical thinking hat on when looking at these data because some of this information is not relevant for categorical variables. For example, you cannot have an average for a categorical variable and the normal distribution doesn’t make sense.

  Table 13.8 Descriptive Statistics for the Dataset of Heights

The next table presents the Basic Statistical Measures (Figure 13.9). In addition to the mean, standard deviation, and variance, this table also provides the median, mode, range, and interquartile range for the variable.

Table 13.9 Basic Statistical Measures Table

Finally, Table 13.10 is the Extreme Observations Table which identifies the highest and lowest values of the variable. Here the data do not differ from what we would expect but when datasets contain outliers, this table is one way to identify the outlier data points easily. Notice that the table provides the case number along with the data point value, making it easy to revisit the original dataset and verify original values or consider adjustments to extreme values if needed.

Figure 13.10 Extreme Observations Table 

Below is a SAS program to produce a graphical presentation of the data while also creating an organized frequency distribution table.

OPTIONS PAGESIZE=55 LINESIZE=120 CENTER DATE; DATA RPRT1; INPUT DIVISION $ 1-12 CASES 14-16; LABEL DIVISION=’CATEGORIES’; DATALINES; 58.5-61.5   4 61.5-64.5  12 64.5-67.5  44 67.5-70.5  64 70.5-73.5  56 73.5-76.5  16 76.5-79.5   4 ; RUN; PROC GCHART DATA=REPORTS.RPRT1; VBAR DIVISION/SUMVAR=CASES; RUN;

Figure 13.8. Vertical Bar Chart of height categories produced in SAS.

PROC FREQ DATA=REPORTS.RPRT1; WEIGHT CASES; TABLES DIVISION; RUN;

 Table 13.11 Frequency table of height categories produced in SAS

13.4 Outliers

As depicted in Table 13.12. below, outliers can be defined as   Cases with extreme values on one (univariate) or more variables (multivariate) (Tabachnick & Fidell, 2013).  Outliers can be either error outliers (incorrect values) or  “interesting” outliers (correct but unusual) (Orr, Sackett, & Dubois, 1991). Error outliers need to be checked against the original data for verification and then either corrected or removed from the data set. Interesting outliers are less easy to deal with (see Tabachnick and Fidell, 2013 for recommended strategies) but they are important to think about because they pull the mean towards them and have a stronger influence on the data than other values. The bottom line is that before you move on to further analysis or data transformation, it is essential to run a frequency analysis and screen your data for outliers.

Error outliers can be detected using PROC FREQ and checking for values that don’t make sense. For example, if you had 976 as someone’s age, that would be a red flag and you would investigate further.

Interesting outliers, while unusual, are still within the realm of possibility. They can be identified using the PROC GPLOT procedure outlined here This command produces a table with the five highest and five lowest values for a particular variable.

In the graph below, one person’s data doesn’t follow the same pattern as the rest of the sample. This is an example of an outlier.

Figure 13.12. Example of an Outlier within a Distribution

Applied Statistics in Healthcare Research Copyright © 2020 by William J. Montelpare, Ph.D., Emily Read, Ph.D., Teri McComber, Alyson Mahar, Ph.D., and Krista Ritchie, Ph.D. is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License , except where otherwise noted.

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Frequency Distribution

• Ricardo A. Olea 7  
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Part of the book series: Encyclopedia of Earth Sciences Series ((EESS))

Given a numerical dataset, a frequency distribution is a summary displaying fluctuations of an attribute within the range of values. In contrast to an analytical probability distribution, a frequency distribution always deals with empirically observed values (Everitt and Skondall 2010 ). In general, the larger the number of values, the more useful is the frequency distribution relative to listing all values. Today, multiple software packages allow easy display of a frequency distribution.

The Merriam-Webster Dictionary ( 2020 ) states that the first use of the term with the meaning observed here was in 1895. However, it provides no reference.

The dataset may be univariate or multivariate. Here, for simplicity, the entry deals with univariate datasets.

Different Forms

In the broadest sense, a frequency distribution may be tabular or graphical. In either case, the most common practice is to classify the data into classes with the same interval length. Table 1 and Fig. 1 provide...

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Olea RA (2017) Resampling of spatially correlated data with preferential sampling for the estimation of frequency distributions and semivariograms. Stoch Env Res Risk A 31(2):481–491

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Quickonomics

Frequency Distribution

Definition of frequency distribution.

A frequency distribution is a representation of how often different values or intervals occur in a dataset. It organizes data into groups or intervals, along with a count of how many times each value or interval appears. Frequency distributions are commonly used to analyze and understand the distribution of data.

Let’s consider a dataset of students’ test scores in a class. The scores range from 60 to 100. We want to create a frequency distribution to understand how many students scored within certain ranges. We decide to group the scores into intervals of 10 (e.g., 60-69, 70-79, etc.).

After analyzing the dataset, we find the following frequency distribution:

– 60-69: 4 students – 70-79: 8 students – 80-89: 10 students – 90-99: 5 students – 100: 2 students

This frequency distribution tells us that there are 4 students who scored between 60 and 69, 8 students who scored between 70 and 79, and so on. We can use this information to identify the most common score range and understand the distribution of scores in the class.

Why Frequency Distribution Matters

Frequency distributions are important for analyzing and summarizing data. They provide an organized and concise representation of how values are distributed within a dataset. By visualizing the frequency distribution, patterns and trends in the data can be identified, such as the presence of outliers or the shape of the distribution (e.g., normal, skewed, etc.). This helps in making informed decisions, identifying areas of improvement, and understanding the characteristics of the data. Frequency distributions are widely used in statistics, research, and data analysis across various fields and industries.

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11 Frequency Distributions

Jenna Lehmann

In statistics, a lot of tests are run using many different points of data and it’s important to understand how those data are spread out and what their individual values are in comparison with other data points. A frequency distribution is just that–an outline of what the data look like as a unit. A frequency table is one way to go about this. It’s an organized tabulation showing the number of individuals located in each category on the scale of measurement. When used in a table, you are given each score from highest to lowest (X) and next to it the number of times that score appears in the data (f). A table in which one is able to read the scores that appear in a data set and how often those particular scores appear in the data set. Here’s a Khan Academy video we found to be helpful in explaining this concept:

Organizing Data into a Frequency Distribution

• Find the range
• Order the table from highest score to lowest score, not skipping scores that might not have shown up in the data set
• In the next column, document how many times this score shows up in the data set

Organizing data into a group frequency table

• The grouped frequency table should have about 10 intervals. A good strategy is to come up with some widths according to Guideline 2 and divide the total range of numbers by that width to see if there are close to 10 intervals.
• The width of the interval should be a relatively simple number (like 2, 5, or 10)
• The bottom score in each class interval should be a multiple of the width (0-9, 10-19, 20-19, etc.)
• All intervals should be the same width.

Proportions and Percentages

Proportions measure the fraction of the total group that is associated with each score (they’re called relative frequencies because they describe the frequency in relation to the total number of scores). For example, if I have 10 pieces of fruit and 3 of them are oranges, 3/10 is the proportion of oranges. On the other hand, percentages express relative frequency out of 100, but essentially report the same values. Keeping in line with our fruit example, 30% of my fruit is oranges. Here’s a YouTube video which might be helpful:

Real Limits

Real limits are continuous variables require a calculation of a real limit. They can be calculated by taking the apparent limit and subtracting and then separately adding half the value of the smallest digit available or presented. For example, I have a value of 50 and I want the real limits. To make it easier to see, I make the number 50.0. The smallest digit shown is the 1 digit, so I subtract half of one (49.5) and add half of one (50.5). Sometimes one isn’t the smallest digit. If I have a value of 34.5, I add another digit to the end to make 34.50, and the smallest value is the 0.5, so we divide by 2 to get 0.25. So the limits are 34.75 and 34.25. Finally, sometimes the smallest value of measurement is given. If the smallest unit a scale can measure is 0.2 pounds, and you have a value of 80 pounds, you add and subtract half of 0.2 pounds and get 80.1 and 79.9. This can be a difficult concept two grasp, so here are two YouTube videos we found helpful.

Frequency Distribution Graphs

A frequency distribution is often best grasped conceptually though the use of graphs. These graphs are like the tables in that they represent the same data, but graphs show it in a different way. This can be done with bar graphs (discrete), histograms (continuous), or polygons (continuous). Here are two Khan Academy videos we found helpful.

These graphs can come in a multitude of shapes, but here are just a few important descriptive words generally used in statistics:

• Symmetrical : When the shape of the distribution is, at least for the most part, mirrored on both sides if you were to view the mean as the flipping point.
• Asymmetrical : When the shape of the distribution is not mirrored on both sides for whatever reason (usually because of skew).
• Positively Skewed : This is when there is what looks like a tail of data trailing off to the right. I like to remember this is as the P in Positive having fallen on its back.
• Negatively Skewed : This is when there is what looks like a tail of data trailing off to the left.
• Unimodal : This literally means having a buildup of data around what looks to be one number, so one mode. Your typical bell curve is unimodal.
• Bimodal : This is when there is data clustering around two different numbers or spots on the distribution, so having two modes. This can often look like camel humps.
• Multimodal : When a distribution has two or more “humps” in the graph.

Here’s a video which may be helpful in teaching you how to interpret data presented in a table and organizing data into a frequency distribution graph.

This chapter was originally posted to the Math Support Center blog at the University of Baltimore on on June 4, 2019. 

Math and Statistics Guides from UB's Math & Statistics Center Copyright © by Jenna Lehmann is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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1.4: Frequency Distributions

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Frequency Distributions

[latexpage]

Table 1.9 lists the different data values in ascending order and their frequencies.

A frequency is the number of times a value of the data occurs. According to Table 1.9 , there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.

A frequency distribution (also known as a grouped frequency distribution table or GFDT for short) is the table that lists the class (this is the “Data Value” column in Table 1.9) and the frequency. Each class can be a single value like in Table 1.9, or a range of values like in Table 1.12 below.

A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample–in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.

Cumulative frequency is the accumulation of the previous frequencies. To find the cumulative frequencies, add all the previous frequencies to the frequency for the current row, as shown in Table 1.11a.

The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.

Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row, as shown in Table 1.11b.

Because of rounding, the relative frequency column may not always sum to one, and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one.

Example – Professional Soccer Players

Table 1.12 represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.

The data in this table have been grouped into the following intervals, which are the classes :

• 60 to 61 inches
• 62 to 63 inches
• 64 to 65 inches
• 66 to 67 inches
• 68 to 69 inches
• 70 to 71 inches
• 72 to 73 inches
• 74 to 75 inches

The  lower class limit for each class is the starting point for each class, and the upper class limit for each class is the upper end of the range. So for example, from Table 1.12, the set of lower class limits are 60, 62, 64, 66, 68, 70, 72, & 74 and the set of upper class limits are 61, 63, 65, 67, 69, 71, 73, & 75.

The class boundaries are the values in the middle between the upper class limit of one class, and the lower class limit of the next class. For example, the class boundary between the first class and the second class in Table 1.12 is 61.5; we can get the class boundary mathematically by adding the upper class limit and the next lower class limit, and dividing the result by two (find the average or mean of the two numbers) $\frac{61+62}{2}=61.5$. Doing this for each class, we see the set of class boundaries is 61.5, 63.5, 65.5, 67.5, 69.5, 71.5, & 73.5

But wait , the class boundaries also include the boundaries before the first class, and after the last class. So if we continue the pattern from the class boundaries above, we get the actual list of class boundaries 59.5 , 61.5, 63.5, 65.5, 67.5, 69.5, 71.5, 73.5, & 75.5 (the two added numbers are highlighted for emphasis).

The class width is the difference between one lower class limit and the next class’ lower class limit. In table 1.12, we can simply take the difference of the first two class’ lower class limits and find the class width is 62 – 60 = 2. The class width is the same for all the classes.

The class midpoint is the value in the middle of the class, which can be found by “averaging” a class’ lower class limit and it’s upper class limit. For example, for the frequency distribution that we’ve been working with here, we would find the first class midpoint as $\frac{60+61}{2}=60.5$. Similarly, the second class midpoint would be 62.5, and so on.

Building a Frequency Distribution

To build a Frequency Distribution, you are generally given two things: 1) a set of data which you are going to group into the frequency distribution, and 2) the number of classes two use. We will illustrate the process with an example.

Suppose you are working for an advertising agency and a client wants you to summarize how long potential customers on the internet watch their commercial before hitting the skip button. You collect the below data from 30 viewers which shows the number of seconds they watched the advertisement, and you figure with 30 data points, you should have 7 classes. Note: the number of classes will be given to you in any problems you work on in this class.

Table 1.13 Data from 30 viewers which shows the number of seconds of watching an advertisement

Calculate the Class Width

The first thing to do is to calculate the class width. To do this, we simply subtract the smallest number in our data set from the largest number in our dataset, and divide that by the number of classes.

$$\frac{\text{max – min}}{\text{num. classes}} = \text{class width (rounded up)}$$

Our class width is calculated as $\frac{31-1}{7} = 4.429$. We always round this number up, so in this example, the class width is $5$.

Set the Lower Class Limits

Now we can start building our frequency distribution. The lower class limit of the first class is always the smallest number in the original dataset, which is 1 second in this example. From there, we add our class width to get each of the next lower class limits.

Table 1.13a A frequency distribution with just the lower class limits

Set the Upper Class Limits

Now we can determine the upper class limits. Since the original data in Table 1.13 has no decimal points, our upper class limits are simply 1 less than the next class’ lower class limit. So the upper class limit of the first class is $6 – 1 = 5$. From there we can keep adding the class width of 5 to get each of the subsequent upper class limits.

Table 1.13b A frequency distribution with just the upper and lower class limits

Count the Frequencies

Now we simply count up how many data points from the original dataset fit into each class. So I see that there are 15 values between 1 & 5, so 15 is my frequency for the first class. We can fill out the rest of the table.

Table 1.13c A frequency distribution, complete with the upper and lower class limits and frequencies.

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What is a Frequency Distribution?

Hey guys! It’s Chip, and I’m here to talk about frequency distributions and frequency tables! 📊

📈 Frequency Distribution : A frequency distribution is a way to show how often values occur in a dataset. It tells us the number of times each value appears and can help us identify any patterns or trends in the data.

• For example, if we look at the number of pets that people own, a frequency distribution could tell us how many people own one pet, two pets, three pets, and so on.

📈 Frequency Table : A frequency table is a way of organizing and summarizing data. It shows how many times each value appears in a dataset. The table is divided into categories, or intervals, that represent a range of values.

• For example, if we look at the height of a group of people, we could create a frequency table with intervals that represent different ranges of heights, such as 5 feet to 5 feet 2 inches, 5 feet 2 inches to 5 feet 4 inches, and so on.

So, how does a frequency table help us understand the data?

Well, a frequency table can help us see how common certain values are, which values are extreme, and what the range of values in a distribution is. We can also use a frequency table to explore percentiles, percentages (relative frequency), and the frequency distribution.

📈 Percentiles Percentiles : are a way to divide the data into 100 equal parts. They can help us to understand where a certain value falls in the distribution.

• For example, if we look at the score of a student on a test, we can use percentiles to see how their score compares to the scores of other students.

📈 Relative Frequency : Relative frequency is the percentage of times that a value appears in the dataset.

• For example, if we look at the frequency table for the number of pets that people own, we can calculate the relative frequency by dividing the frequency of each category by the total number of pets and multiplying by 100.

📈 Percentages : Percentages are another way to show the relative frequency of values in a dataset. They tell us what proportion of the data falls into each category.

• For example, if we look at the frequency table for the height of a group of people, we can calculate the percentage of people who fall into each interval by dividing the frequency of each interval by the total number of people and multiplying by 100.

So, in summary, a frequency distribution tells us how often values occur in a dataset, and a frequency table is a way of organizing and summarizing data. A frequency table can help us to understand how common certain values are, which values are extreme, and the range of values in a distribution. We can also use a frequency table to explore percentiles, percentages (relative frequency), and the frequency distribution.

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28 Frequency Distribution

S. Gandhimathi

Introduction

Formation of frequency distribution is the process of forming frequency table. It involves the counting of items, the number of times a particular value is repeated which is called the frequency of that class. The formation of frequency distribution involves discrete and continuous frequency distribution. In the discrete frequency distribution, the value of X will be certain. For certain value of x value, the value of frequency is identified through tally marks. In the continuous frequency distribution, the value of X will be a range of values. For each value of x value, the frequency is identified through tally mark

In order to facilitate counting, prepare a column of “tallies”. In another column, place all possible values of variable from the lowest to the highest. Then put a bar (vertical line) opposite the particular value to which it relates. To facilitate counting, blocks of five bars are prepared and some space is left in between each block. We finally count the number of bars and get frequency.

1.To constructunivariatevariate frequency distribution

2.To construct bi variate frequency distribution

1. Univariatevariate frequency distribution

Univariate frequency distribution is explained though the following examples.

Example: In a survey of 35 families in a village, the number of children per family was recorded and the following data obtained.

Represent the data in the form of a discrete frequency distribution.

FREQUENCY DISTRIBUTION OF THE NUMBER OF CHILDREN

It is clear from the table that the number of children varied from 0 to 12. There were 1 family with no child, 5 families with 4 children each and only one family with 12 children.

FORMATION OF CONTINUOUS FREQUENCY DISTRIBUTION

This type of classification is most popular in practice. The following technical terms are important when a continuous frequency distribution is formed or data are classified according to class intervals

1. Class Limits : The lowest value of a class is the lower class limit and the upper value of a class is the upper class limit. For example, take the class 20-40. The lowest value of  the class is 20 and the highest is 40. The two boundaries of class are known as the lower limit and the upper limit of the class. The lower limit of a class is the value below which there can be no item in the class. The upper limit of a class is the value above which no item can belong to that class. Of the class 70-89, 70 is the lower limit and 89 upper limit, ie., in this class there can be no value which is less than 70 or more than 89. Similarly, if we take the class 90-109, there can be no value in that class which is less than 90 or more than 109. The way in which class limits are stated depends upon the nature of the data.

2.  Class intervals: The difference between the Upper and Lower Limit of a class is known as class interval of that class. For example, in the class 100-200, the class interval is 100 (i.e., 200 minus 100). An important decision while constructing a frequency distribution is about the width of the class interval, ie., whether it should be 10, 20, 50, 100, 500 etc., The decision would depend upon a number of factors such as the range in the data, i.e., the difference between the smallest and largest item, the details required and number of classes to be formed, etc., A simple formula to obtain the estimate the appropriate class interval, ie., I is –

For example, if the salary of 100 employees in a commercial undertaking varied between Rs.500 and Rs.5,500 and we want to form 10 classes, then the class interval would be

The starting class would be 500 – 1000, the next 1000 – 1500, and so on.

The question now is how to fix the number of classes, i.e., k. The number can be either fixed arbitrarily keeping in view the nature of problem under study or it can be decided with the formula:

    k= 1 + 3.322 log N

where N = total number of observations

and log = logarithms of the number

Thus, if 10 observations are being studied, the number of classes shall be:

k = 1 + (3.322 x 1) = 4.322 or 4 and if 100 observations are being studied, the number of classes shall be:

k = 1 + (3.322×3) = 1 + 6.644 = 7.644 or 8.

It should be noted that since log is used in the formula, the number of classes will generally be between 4 and 20 – it cannot be less than 4 even if N is less than 10 .

Sturges suggested the following formula for determining the magnitude of class interval:

where range is the difference between the largest and smallest items.

For example, if in the above illustration we apply this formula the magnitude of class interval shall be

If we take a class interval of 650, the number of classes formed would be 5000/650, ie., 7.69 or 8.

It may be noted that the application of above formula may give a value involving fractions and odd intervals. For example, we have I = 6.541. In such cases suitable approximation should be made.

3.  Class Frequency : The number of observations corresponding to a particular class is known as the frequency of that class or the class frequency. In the following example,  the frequency of the class 1000-1100 is 50 which implies that there are 50 persons having income between Rs.1000 and Rs.1100. If we add together the frequencies of all individual classes, we obtain the total frequency. Thus, in the same problem, the total frequency of the six classes is 550 which means that in all there are 550 persons whose income has been studied.

Class Mid-point or Class Mark: It is the value lying half-way between the lower and upper class limits of a class interval. Mid-point of a class is ascertained as follows

Mid-point of a class = Upper limit of the class+Lower limit of the class 2

For the purpose of further calculation in statistical work the mid-point of each class is taken to represent that class.

There are two methods of classifying the data according to class-intervals, namely (i) ‘exclusive’ method and (ii) ‘inclusive’ method.

(i)  ‘Exclusive Method : When the class intervals are so fixed that the upper limit of one class is the lower limit of the next class it is known as the exclusive method of classification. The following data are classified on this basis.

Thus, in the above example, there are 50 persons whose income is between Rs.1000 and Rs.1099.99. A person whose income is Rs.1100 would be included in the class 1100-1200. This method is widely followed in practice.

(ii)  ‘Inclusive’ method: Under the ‘inclusive’ method of classification, the upper limit of one class is included in that class itself. The following example illustrates the method.

In the class 1000 – 1099 we include persons whose income is between Rs.1000 and Rs.1099. If the income of person is exactly Rs,1100 he is included in the next class. The above example makes it clear that there is no confusion here of the type we find under the exclusive method. We may have classes like 100 – 1099.5 or 1000 – 1099.99 and so on.

Considerations in the Construction of Frequency Distributions

It is difficult to lay down and hard and fast rules for constructing a frequency distribution, as much depends on the nature of the given data and the object of classification.

However, the following general considerations may be kept in mind for ensuring meaningful classification of data:

The number of classes should preferably be between 5 and 20. However, there is no rigidity about it. The classes can be more than 20 depending upon the total number of items in the series and the details required, but they should not be less  than five because in that case the classification may not reveal the essential characteristics. The choice of number of classes basically depends upon:

    (a)   the number of figures to be classified

(b)   the magnitude of the figures

(c)   the details required and

(d)   ease of calculation for further statistical work.

As far as possible one should avoid values of class-intervals, as 3, 7, 11, 26, 39, etc., Preferably, one should have class intervals of either five or multiples of 5 like 10, 20, 25, 100 etc.

The starting point, ie., the lower limit of the first class, should either be zero or 5 or multiple of 5. For example, if the lowest value of the data is 63 and we have taken a class interval of 10. Then the first class should be 60-70, instead of 63-73. Similarly, if the lowest value of the data is 75 to 80 rather than 76 to 81.

To ensure continuity and to get correct class interval we should adopt ‘exclusive’ method of classification. However, where ‘inclusive’ method has been adopted it is necessary to make an adjustment consists of finding the difference between the lower limit of the second class and the upper limit of the first class, dividing the difference by two, subtracting the value so obtained from all lower limits and adding this value to all upper limits. This can be expressed in the form of a formula as follows:

Correction factor = Lower Limit of the 2nd class−Upper limit of the 1st class 2

How the adjustment is made when data are given by inclusive method can be seen from the following examples:

To adjust the class limits, we take here the difference between 900 and 899, which is one. By dividing it by two we get ½ or 0.5. ‘This (0.5) is called the correction factor. Deduct 0.5 from the lower limits of all classes and add 0.5 to upper limits. The adjusted classes would then be as follows:

It should be noted that before adjustment the class interval was 99 but after adjustment, it is 100.

Open-end distribution presents problems of graphing and further analysis. When the frequency distribution is being employed as the only technique of presentation, open-end classes do not seriously reduce its usefulness as long as only a few items fall in these classes. However, use of the distribution for purposes of further mathematical computation is difficult because a mid-point value, which can be used to present the class, cannot be determined for an open-end class.

In any frequency distribution the size of items or the values are indicated on the left-hand side and the number of times the items in thise sizes or values have repeated are indicated by frequencies on the right-hand side corresponding to the respective size of value.

Example: Prepare a frequency table for the following data with width of each class interval as 10. Use exclusive method of classification:

Relative Frequency Distribution

At times it may be desirable to convert class frequencies to relative class frequencies to show the percentage of the total number of observations in each class. For example, we may be interested in knowing the percentage of employees earning less than Rs.2000 per month.

In order to convert a frequency distribution to a relative frequency distribution, each of the class frequencies is divided by the total number of frequencies so that the relative frequencies would always total 1.

The relative frequency distribution would be as follows:

Bivariate or Two-way Frequency Distribution

In the previous few pages of this module, we described frequency distributions involving one variable only. Such frequency distributions are called univariate frequency distributions. In many situations simultaneous study of two variables becomes necessary. For example, we want to classify data relating to age of husbands and age of wives or data relating to marks in statistics and marks in accountancy or height and weight of students. The data so classified on the basis of two variables is called a bivariate frequency distribution. While preparing a bivariate frequency distribution, the same considerations of classification apply as for univariate frequency distribution, i.e., the values of each variable. If the data corresponding to one variable, say X, is grouped into m classes and the data corresponding to the other variable, say Y, is grouped into n classes then the bivariate table will consist of m x n cells. By going through the different pairs of the values (X, Y) of the variables and using tally marks we can find the frequency of each cell and thus from bivariate frequency distribution.

The frequency distribution of the values of the variable X together with their frequency distribution of the values of variable Y together with the total frequencies is known as the marginal frequency distribution of Y.

Let us summarise, the statistical inferences can not be drawn based on the raw data. Hence, the raw data are to be summarised. One way of summarizing the data is the  formation of frequency distribution. There are two types of frequency distribution. One is the univariate frequency distribution and the other is the bivariate frequency distribution. The univariate frequency distribution is based on only one variable. The bi variate frequency distribution is based on the two variables. The above discussion may give an idea to group your statistical data and to construct frequency table.

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4.2 Frequency Distributions for Qualitative Data

4.2: frequency distributions for qualitative data, 4.2.1: describing qualitative data.

Qualitative data is a categorical measurement expressed not in terms of numbers, but rather by means of a natural language description.

Learning Objectives

Summarize the processes available to researchers that allow qualitative data to be analyzed similarly to quantitative data.

Key Takeaways

• Observer impression is when expert or bystander observers examine the data, interpret it via forming an impression and report their impression in a structured and sometimes quantitative form.
• To discover patterns in qualitative data, one must try to find frequencies, magnitudes, structures, processes, causes, and consequences.
• The Ground Theory Method (GTM) is an inductive approach to research in which theories are generated solely from an examination of data rather than being derived deductively.
• Coding is an interpretive technique that both organizes the data and provides a means to introduce the interpretations of it into certain quantitative methods.
• Most coding requires the analyst to read the data and demarcate segments within it.

Qualitative data is a categorical measurement expressed not in terms of numbers, but rather by means of a natural language description. In statistics, it is often used interchangeably with “categorical” data. When there is not a natural ordering of the categories, we call these nominal categories. Examples might be gender, race, religion, or sport.

When the categories may be ordered, these are called ordinal variables. Categorical variables that judge size (small, medium, large, etc.) are ordinal variables. Attitudes (strongly disagree, disagree, neutral, agree, strongly agree) are also ordinal variables; however, we may not know which value is the best or worst of these issues. Note that the distance between these categories is not something we can measure.

Qualitative Analysis

Qualitative Analysis is the numerical examination and interpretation of observations for the purpose of discovering underlying meanings and patterns of relationships. The most common form of qualitative qualitative analysis is observer impression—when an expert or bystander observers examine the data, interpret it via forming an impression and report their impression in a structured and sometimes quantitative form.

An important first step in qualitative analysis and observer impression is to discover patterns. One must try to find frequencies, magnitudes, structures, processes, causes, and consequences. One method of this is through cross-case analysis, which is analysis that involves an examination of more than one case. Cross-case analysis can be further broken down into variable-oriented analysis and case-oriented analysis . Variable-oriented analysis is that which describes and/or explains a particular variable, while case-oriented analysis aims to understand a particular case or several cases by looking closely at the details of each.

The Ground Theory Method (GTM) is an inductive approach to research, introduced by Barney Glaser and Anselm Strauss, in which theories are generated solely from an examination of data rather than being derived deductively. A component of the Grounded Theory Method is the constant comparative method , in which observations are compared with one another and with the evolving inductive theory.

Four Stages of the Constant Comparative Method

• comparing incident application to each category
• integrating categories and their properties
• delimiting the theory
• writing theory

Other methods of discovering patterns include semiotics and conversation analysis. Semiotics is the study of signs and the meanings associated with them. It is commonly associated with content analysis. Conversation analysis is a meticulous analysis of the details of conversation, based on a complete transcript that includes pauses and other non-verbal communication.

Conceptualization and Coding

In quantitative analysis, it is usually obvious what the variables to be analyzed are, for example, race, gender, income, education, etc. Deciding what is a variable, and how to code each subject on each variable, is more difficult in qualitative data analysis.

Concept formation is the creation of variables (usually called themes ) out of raw qualitative data. It is more sophisticated in qualitative data analysis. Casing is an important part of concept formation. It is the process of determining what represents a case. Coding is the actual transformation of qualitative data into themes.

More specifically, coding is an interpretive technique that both organizes the data and provides a means to introduce the interpretations of it into certain quantitative methods. Most coding requires the analyst to read the data and demarcate segments within it, which may be done at different times throughout the process. Each segment is labeled with a “code” – usually a word or short phrase that suggests how the associated data segments inform the research objectives. When coding is complete, the analyst prepares reports via a mix of: summarizing the prevalence of codes, discussing similarities and differences in related codes across distinct original sources/contexts, or comparing the relationship between one or more codes.

Some qualitative data that is highly structured (e.g., close-end responses from surveys or tightly defined interview questions) is typically coded without additional segmenting of the content. In these cases, codes are often applied as a layer on top of the data. Quantitative analysis of these codes is typically the capstone analytical step for this type of qualitative data.

A frequent criticism of coding method is that it seeks to transform qualitative data into empirically valid data that contain actual value range, structural proportion, contrast ratios, and scientific objective properties. This can tend to drain the data of its variety, richness, and individual character. Analysts respond to this criticism by thoroughly expositing their definitions of codes and linking those codes soundly to the underlying data, therein bringing back some of the richness that might be absent from a mere list of codes.

Alternatives to Coding

Alternatives to coding include recursive abstraction and mechanical techniques. Recursive abstraction involves the summarizing of datasets. Those summaries are then further summarized and so on. The end result is a more compact summary that would have been difficult to accurately discern without the preceding steps of distillation.

Mechanical techniques rely on leveraging computers to scan and reduce large sets of qualitative data. At their most basic level, mechanical techniques rely on counting words, phrases, or coincidences of tokens within the data. Often referred to as content analysis, the output from these techniques is amenable to many advanced statistical analyses.

4.2.2: Interpreting Distributions Constructed by Others

Graphs of distributions created by others can be misleading, either intentionally or unintentionally.

Learning Objective

Demonstrate how distributions constructed by others may be misleading, either intentionally or unintentionally

• Misleading graphs will misrepresent data, constituting a misuse of statistics that may result in an incorrect conclusion being derived from them.
• Graphs can be misleading if they’re used excessively, if they use the third dimensions where it is unnecessary, if they are improperly scaled, or if they’re truncated.

The use of biased or loaded words in the graph’s title, axis labels, or caption may inappropriately prime the reader.

Distributions Constructed by Others

Unless you are constructing a graph of a distribution on your own, you need to be very careful about how you read and interpret graphs. Graphs are made in order to display data; however, some people may intentionally try to mislead the reader in order to convey certain information.

In statistics, these types of graphs are called misleading graphs (or distorted graphs). They misrepresent data, constituting a misuse of statistics that may result in an incorrect conclusion being derived from them. Graphs may be misleading through being excessively complex or poorly constructed. Even when well-constructed to accurately display the characteristics of their data, graphs can be subject to different interpretation.

Misleading graphs may be created intentionally to hinder the proper interpretation of data, but can also be created accidentally by users for a variety of reasons including unfamiliarity with the graphing software, the misinterpretation of the data, or because the data cannot be accurately conveyed. Misleading graphs are often used in false advertising.

Types of Misleading Graphs

The use of graphs where they are not needed can lead to unnecessary confusion/interpretation. Generally, the more explanation a graph needs, the less the graph itself is needed. Graphs do not always convey information better than tables. This is often called excessive usage.

Pie charts can be especially misleading. Comparing pie charts of different sizes could be misleading as people cannot accurately read the comparative area of circles. The usage of thin slices which are hard to discern may be difficult to interpret. The usage of percentages as labels on a pie chart can be misleading when the sample size is small. A perspective (3D) pie chart is used to give the chart a 3D look. Often used for aesthetic reasons, the third dimension does not improve the reading of the data; on the contrary, these plots are difficult to interpret because of the distorted effect of perspective associated with the third dimension. In a 3D pie chart, the slices that are closer to the reader appear to be larger than those in the back due to the angle at which they’re presented .

3-D Pie Chart

In the misleading pie chart, Item C appears to be at least as large as Item A, whereas in actuality, it is less than half as large.

When using pictogram in bar graphs, they should not be scaled uniformly as this creates a perceptually misleading comparison. The area of the pictogram is interpreted instead of only its height or width. This causes the scaling to make the difference appear to be squared .

Improper Scaling

Note how in the improperly scaled pictogram bar graph, the image for B is actually 9 times larger than A.

A truncated graph has a y-axis that does not start at 0. These graphs can create the impression of important change where there is relatively little change .

Truncated Bar Graph

Note that both of these graphs display identical data; however, in the truncated bar graph on the left, the data appear to show significant differences, whereas in the regular bar graph on the right, these differences are hardly visible.

Usage in the Real World

Graphs are useful in the summary and interpretation of financial data. Graphs allow for trends in large data sets to be seen while also allowing the data to be interpreted by non-specialists. Graphs are often used in corporate annual reports as a form of impression management. In the United States, graphs do not have to be audited as they fall under AU Section 550 Other Information in Documents Containing Audited Financial Statements. Several published studies have looked at the usage of graphs in corporate reports for different corporations in different countries and have found frequent usage of improper design, selectivity, and measurement distortion within these reports. The presence of misleading graphs in annual reports have led to requests for standards to be set. Research has found that while readers with poor levels of financial understanding have a greater chance of being misinformed by misleading graphs, even those with financial understanding, such as loan officers, may be misled.

4.2.3: Graphs of Qualitative Data

Qualitative data can be graphed in various ways, including using pie charts and bar charts.

Create a pie chart and bar chart representing qualitative data.

• Since qualitative data represent individual categories, calculating descriptive statistics is limited. Mean, median, and measures of spread cannot be calculated; however, the mode can be calculated.
• One way in which we can graphically represent qualitative data is in a pie chart. Categories are represented by slices of the pie, whose areas are proportional to the percentage of items in that category.
• The key point about the qualitative data is that they do not come with a pre-established ordering (the way numbers are ordered).
• Bar charts can also be used to graph qualitative data. The Y axis displays the frequencies and the X axis displays the categories.

Qualitative Data

Recall the difference between quantitative and qualitative data. Quantitative data are data about numeric values. Qualitative data are measures of types and may be represented as a name or symbol. Statistics that describe or summarize can be produced for quantitative data and to a lesser extent for qualitative data. As quantitative data are always numeric they can be ordered, added together, and the frequency of an observation can be counted. Therefore, all descriptive statistics can be calculated using quantitative data. As qualitative data represent individual (mutually exclusive) categories, the descriptive statistics that can be calculated are limited, as many of these techniques require numeric values which can be logically ordered from lowest to highest and which express a count. Mode can be calculated, as it it the most frequency observed value. Median, measures of shape, measures of spread such as the range and interquartile range, require an ordered data set with a logical low-end value and high-end value. Variance and standard deviation require the mean to be calculated, which is not appropriate for categorical variables as they have no numerical value.

Graphing Qualitative Data

There are a number of ways in which qualitative data can be displayed. A good way to demonstrate the different types of graphs is by looking at the following example:

When Apple Computer introduced the iMac computer in August 1998, the company wanted to learn whether the iMac was expanding Apple’s market share. Was the iMac just attracting previous Macintosh owners? Or was it purchased by newcomers to the computer market, and by previous Windows users who were switching over? To find out, 500 iMac customers were interviewed. Each customer was categorized as a previous Macintosh owners, a previous Windows owner, or a new computer purchaser. The qualitative data results were displayed in a frequency table.

Frequency Table for Mac Data

The frequency table shows how many people in the study were previous Mac owners, previous Windows owners, or neither.

The key point about the qualitative data is that they do not come with a pre-established ordering (the way numbers are ordered). For example, there is no natural sense in which the category of previous Windows users comes before or after the category of previous iMac users. This situation may be contrasted with quantitative data, such as a person’s weight. People of one weight are naturally ordered with respect to people of a different weight.

One way in which we can graphically represent this qualitative data is in a pie chart. In a pie chart, each category is represented by a slice of the pie. The area of the slice is proportional to the percentage of responses in the category. This is simply the relative frequency multiplied by 100. Although most iMac purchasers were Macintosh owners, Apple was encouraged by the 12% of purchasers who were former Windows users, and by the 17% of purchasers who were buying a computer for the first time .

Pie Chart for Mac Data

The pie chart shows how many people in the study were previous Mac owners, previous Windows owners, or neither.

Pie charts are effective for displaying the relative frequencies of a small number of categories. They are not recommended, however, when you have a large number of categories. Pie charts can also be confusing when they are used to compare the outcomes of two different surveys or experiments.

Here is another important point about pie charts. If they are based on a small number of observations, it can be misleading to label the pie slices with percentages. For example, if just 5 people had been interviewed by Apple Computers, and 3 were former Windows users, it would be misleading to display a pie chart with the Windows slice showing 60%. With so few people interviewed, such a large percentage of Windows users might easily have accord since chance can cause large errors with small samples. In this case, it is better to alert the user of the pie chart to the actual numbers involved. The slices should therefore be labeled with the actual frequencies observed (e.g., 3) instead of with percentages.

Bar Chart for Mac Data

The bar chart shows how many people in the study were previous Mac owners, previous Windows owners, or neither.

Bar charts can also be used to represent frequencies of different categories . Frequencies are shown on the Y axis and the type of computer previously owned is shown on the X axis. Typically the Y-axis shows the number of observations rather than the percentage of observations in each category as is typical in pie charts.

4.2.4: Misleading Graphs

A misleading graph misrepresents data and may result in incorrectly derived conclusions.

• Misleading graphs may be created intentionally to hinder the proper interpretation of data, but can be also created accidentally by users for a variety of reasons.
• The use of graphs where they are not needed can lead to unnecessary confusion/interpretation. This is referred to as excessive usage.
• The use of biased or loaded words in the graph’s title, axis labels, or caption may inappropriately sway the reader. This is called biased labeling.
• Graphs can also be misleading if they are improperly labeled, if they are truncated, if there is an axis change, if they lack a scale, or if they are unnecessarily displayed in the third dimension.

What is a Misleading Graph?

In statistics, a misleading graph, also known as a distorted graph, is a graph which misrepresents data, constituting a misuse of statistics and with the result that an incorrect conclusion may be derived from it. Graphs may be misleading through being excessively complex or poorly constructed. Even when well-constructed to accurately display the characteristics of their data, graphs can be subject to different interpretation.

Misleading graphs may be created intentionally to hinder the proper interpretation of data, but can be also created accidentally by users for a variety of reasons including unfamiliarity with the graphing software, the misinterpretation of the data, or because the data cannot be accurately conveyed. Misleading graphs are often used in false advertising. One of the first authors to write about misleading graphs was Darrell Huff, who published the best-selling book How to Lie With Statistics in 1954. It is still in print.

Excessive Usage

There are numerous ways in which a misleading graph may be constructed. The use of graphs where they are not needed can lead to unnecessary confusion/interpretation. Generally, the more explanation a graph needs, the less the graph itself is needed. Graphs do not always convey information better than tables.

Biased Labeling

The use of biased or loaded words in the graph’s title, axis labels, or caption may inappropriately sway the reader.

When using pictogram in bar graphs, they should not be scaled uniformly as this creates a perceptually misleading comparison. The area of the pictogram is interpreted instead of only its height or width. This causes the scaling to make the difference appear to be squared.

In the improperly scaled pictogram bar graph, the image for B is actually 9 times larger than A.

Truncated Graphs

A truncated graph has a y-axis that does not start at zero. These graphs can create the impression of important change where there is relatively little change.Truncated graphs are useful in illustrating small differences. Graphs may also be truncated to save space. Commercial software such as MS Excel will tend to truncate graphs by default if the values are all within a narrow range.

Both of these graphs display identical data; however, in the truncated bar graph on the left, the data appear to show significant differences, whereas in the regular bar graph on the right, these differences are hardly visible.

Misleading 3D Pie Charts

A perspective (3D) pie chart is used to give the chart a 3D look. Often used for aesthetic reasons, the third dimension does not improve the reading of the data; on the contrary, these plots are difficult to interpret because of the distorted effect of perspective associated with the third dimension. The use of superfluous dimensions not used to display the data of interest is discouraged for charts in general, not only for pie charts. In a 3D pie chart, the slices that are closer to the reader appear to be larger than those in the back due to the angle at which they’re presented .

Misleading 3D Pie Chart

Other Misleading Graphs

Graphs can also be misleading for a variety of other reasons. An axis change affects how the graph appears in terms of its growth and volatility. A graph with no scale can be easily manipulated to make the difference between bars look larger or smaller than they actually are. Improper intervals can affect the appearance of a graph, as well as omitting data. Finally, graphs can also be misleading if they are overly complex or poorly constructed.

Graphs in Finance and Corporate Reports

Graphs are useful in the summary and interpretation of financial data. Graphs allow for trends in large data sets to be seen while also allowing the data to be interpreted by non-specialists. Graphs are often used in corporate annual reports as a form of impression management. In the United States, graphs do not have to be audited. Several published studies have looked at the usage of graphs in corporate reports for different corporations in different countries and have found frequent usage of improper design, selectivity, and measurement distortion within these reports. The presence of misleading graphs in annual reports have led to requests for standards to be set. Research has found that while readers with poor levels of financial understanding have a greater chance of being misinformed by misleading graphs, even those with financial understanding, such as loan officers, may be misled.

4.2.5: Do It Yourself: Plotting Qualitative Frequency Distributions

Qualitative frequency distributions can be displayed in bar charts, Pareto charts, and pie charts.

• The first step to plotting a qualitative frequency distributions is to create a frequency table.
• If drawing a bar graph or Pareto chart, first draw two axes. The y-axis is labeled with the frequency (or relative frequency) and the x-axis is labeled with the category.
• In bar graphs and Pareto graphs, draw rectangles of equal width and heights that correspond to their frequencies/relative frequencies.
• A pie chart shows the distribution in a different way, where each percentage is a slice of the pie.

Ways to Organize Data

When data is collected from a survey or an experiment, they must be organized into a manageable form. Data that is not organized is referred to as raw data. A few different ways to organize data include tables, graphs, and numerical summaries.

One common way to organize qualitative, or categorical, data is in a frequency distribution. A frequency distribution lists the number of occurrences for each category of data.

Step-by-Step Guide to Plotting Qualitative Frequency Distributions

The first step towards plotting a qualitative frequency distribution is to create a table of the given or collected data. For example, let’s say you want to determine the distribution of colors in a bag of Skittles. You open up a bag, and you find that there are 15 red, 7 orange, 7 yellow, 13 green, and 8 purple. Create a two column chart, with the titles of Color and Frequency, and fill in the corresponding data.

To construct a frequency distribution in the form of a bar graph, you must first draw two axes. The y-axis (vertical axis) should be labeled with the frequencies and the x-axis (horizontal axis) should be labeled with each category (in this case, Skittle color). The graph is completed by drawing rectangles of equal width for each color, each as tall as their frequency .

This graph shows the frequency distribution of a bag of Skittles.

Sometimes a relative frequency distribution is desired. If this is the case, simply add a third column in the table called Relative Frequency. This is found by dividing the frequency of each color by the total number of Skittles (50, in this case). This number can be written as a decimal, a percentage, or as a fraction. If we decided to use decimals, the relative frequencies for the red, orange, yellow, green, and purple Skittles are respectively 0.3, 0.14, 0.14, 0.26, and 0.16. The decimals should add up to 1 (or very close to it due to rounding). Bar graphs for relative frequency distributions are very similar to bar graphs for regular frequency distributions, except this time, the y-axis will be labeled with the relative frequency rather than just simply the frequency. A special type of bar graph where the bars are drawn in decreasing order of relative frequency is called a Pareto chart .

Pareto Chart

This graph shows the relative frequency distribution of a bag of Skittles.

The distribution can also be displayed in a pie chart, where the percentages of the colors are broken down into slices of the pie. This may be done by hand, or by using a computer program such as Microsoft Excel . If done by hand, you must find out how many degrees each piece of the pie corresponds to. Since a circle has 360 degrees, this is found out by multiplying the relative frequencies by 360. The respective degrees for red, orange, yellow, green, and purple in this case are 108, 50.4, 50.4, 93.6, and 57.6. Then, use a protractor to properly draw in each slice of the pie.

This pie chart shows the frequency distribution of a bag of Skittles.

4.2.6: Summation Notation

In statistical formulas that involve summing numbers, the Greek letter sigma is used as the summation notation.

• There is no special notation for the summation of explicit sequences (such as 1+2+4+2), as the corresponding repeated addition expression will do.
• If the terms of the sequence are given by a regular pattern, possibly of variable length, then the summation notation may be useful or even essential.

Many statistical formulas involve summing numbers. Fortunately there is a convenient notation for expressing summation. This section covers the basics of this summation notation.

Summation is the operation of adding a sequence of numbers, the result being their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group. For finite sequences of such elements, summation always produces a well-defined sum.

The summation of the sequence [1, 2, 4, 2] is an expression whose value is the sum of each of the members of the sequence. In the example, 1+2+4+2=9. Since addition is associative, the value does not depend on how the additions are grouped. For instance (1+2) + (4+2) and 1 + ((2+4) + 2) both have the value 9; therefore, parentheses are usually omitted in repeated additions. Addition is also commutative, so changing the order of the terms of a finite sequence does not change its sum.

There is no special notation for the summation of such explicit sequences as the example above, as the corresponding repeated addition expression will do. If, however, the terms of the sequence are given by a regular pattern, possibly of variable length, then a summation operator may be useful or even essential.

For the summation of the sequence of consecutive integers from 1 to 100 one could use an addition expression involving an ellipsis to indicate the missing terms: 1 + 2 + 3 + 4 + ⋯ + 99 + 100 . In this case the reader easily guesses the pattern; however, for more complicated patterns, one needs to be precise about the rule used to find successive terms. This can be achieved by using the summation notation “ Σ ” Using this sigma notation, the above summation is written as:

In this notation, i represents the index of summation, a i is an indexed variable representing each successive term in the series, m is the lower bound of summation, and n is the upper bound of summation. The “ i = m ” under the summation symbol means that the index i starts out equal to m . The index, i , is incremented by 1 for each successive term, stopping when i = n .

Here is an example showing the summation of exponential terms (terms to the power of 2):

Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context, as in:

4.2.7: Graphing Bivariate Relationships

We can learn much more by displaying bivariate data in a graphical form that maintains the pairing of variables.

Compare the strengths and weaknesses of the various methods used to graph bivariate data.

• When one variable increases with the second variable, we say that x and y have a positive association.
• Conversely, when y decreases as x increases, we say that they have a negative association.
• The presence of qualitative data leads to challenges in graphing bivariate relationships.
• If both variables are qualitative, we would be able to graph them in a contingency table.

Introduction to Bivariate Data

Measures of central tendency, variability, and spread summarize a single variable by providing important information about its distribution. Often, more than one variable is collected on each individual. For example, in large health studies of populations it is common to obtain variables such as age, sex, height, weight, blood pressure, and total cholesterol on each individual. Economic studies may be interested in, among other things, personal income and years of education. As a third example, most university admissions committees ask for an applicant’s high school grade point average and standardized admission test scores (e.g., SAT). In the following text, we consider bivariate data, which for now consists of two quantitative variables for each individual. Our first interest is in summarizing such data in a way that is analogous to summarizing univariate (single variable) data.

By way of illustration, let’s consider something with which we are all familiar: age. More specifically, let’s consider if people tend to marry other people of about the same age. One way to address the question is to look at pairs of ages for a sample of married couples. Bivariate Sample 1 shows the ages of 10 married couples. Going across the columns we see that husbands and wives tend to be of about the same age, with men having a tendency to be slightly older than their wives.

Bivariate Sample 1

Sample of spousal ages of 10 white American couples.

These pairs are from a dataset consisting of 282 pairs of spousal ages (too many to make sense of from a table). What we need is a way to graphically summarize the 282 pairs of ages, such as a histogram. as in .

Bivariate Histogram

Histogram of spousal ages.

Each distribution is fairly skewed with a long right tail. From the first figure we see that not all husbands are older than their wives. It is important to see that this fact is lost when we separate the variables. That is, even though we provide summary statistics on each variable, the pairing within couples is lost by separating the variables. Only by maintaining the pairing can meaningful answers be found about couples, per se.

Therefore, we can learn much more by displaying the bivariate data in a graphical form that maintains the pairing. shows a scatter plot of the paired ages. The x-axis represents the age of the husband and the y-axis the age of the wife.

Bivariate Scatterplot

Scatterplot showing wife age as a function of husband age.

There are two important characteristics of the data revealed by this figure. First, it is clear that there is a strong relationship between the husband’s age and the wife’s age: the older the husband, the older the wife. When one variable increases with the second variable, we say that x and y have a positive association. Conversely, when y decreases as x increases, we say that they have a negative association. Second, the points cluster along a straight line. When this occurs, the relationship is called a linear relationship.

Bivariate Relationships in Qualitative Data

The presence of qualitative data leads to challenges in graphing bivariate relationships. We could have one qualitative variable and one quantitative variable, such as SAT subject and score. However, making a scatter plot would not be possible as only one variable is numerical. A bar graph would be possible.

If both variables are qualitative, we would be able to graph them in a contingency table. We can then use this to find whatever information we may want. In , this could include what percentage of the group are female and right-handed or what percentage of the males are left-handed.

Contingency Table

Contingency tables are useful for graphically representing qualitative bivariate relationships.

Attributions

• “Boundless.” http://www.boundless.com/ . Boundless Learning CC BY-SA 3.0 .
• “qualitative analysis.” http://en.wikipedia.org/wiki/qualitative%20analysis . Wikipedia CC BY-SA 3.0 .
• “Qualitative research.” http://en.wikipedia.org/wiki/Qualitative_research . Wikipedia CC BY-SA 3.0 .
• “ordinal.” http://en.wiktionary.org/wiki/ordinal . Wiktionary CC BY-SA 3.0 .
• “nominal.” http://en.wiktionary.org/wiki/nominal . Wiktionary CC BY-SA 3.0 .
• “Statistics/Different Types of Data/Quantitative and Qualitative Data.” http://en.wikibooks.org/wiki/Statistics/Different_Types_of_Data/Quantitative_and_Qualitative_Data%23Qualitative_data . Wikibooks CC BY-SA 3.0 .
• “Social Research Methods/Qualitative Research.” http://en.wikibooks.org/wiki/Social_Research_Methods/Qualitative_Research . Wikibooks CC BY-SA 3.0 .
• “Misleading graph.” http://en.wikipedia.org/wiki/Misleading_graph . Wikipedia CC BY-SA 3.0 .
• “distribution.” http://en.wiktionary.org/wiki/distribution . Wiktionary CC BY-SA 3.0 .
• “truncate.” http://en.wiktionary.org/wiki/truncate . Wiktionary CC BY-SA 3.0 .
• “bias.” http://en.wiktionary.org/wiki/bias . Wiktionary CC BY-SA 3.0 .
• “Misleading graph.” http://en.wikipedia.org/wiki/Misleading_graph . Wikipedia GNU FDL .
• “descriptive statistics.” http://en.wiktionary.org/wiki/descriptive_statistics . Wiktionary CC BY-SA 3.0 .
• “David Lane, Graphing Qualitative Variables. September 17, 2013.” http://cnx.org/content/m10927/latest/ . OpenStax CNX CC BY 3.0 .
• “Error 404.” http://www.abs.gov.au/websitedbs/a3121120.nsf/89a5f3d8684682b6ca256de4002c809b/e200e8e572a2ae52ca25794900127f4f!OpenDocument . Austrailian Bureau of Statistics CC BY .
• “David Lane, Graphing Qualitative Variables. April 22, 2013.” http://cnx.org/content/m10927/latest/ . OpenStax CNX CC BY 3.0 .
• “volatility.” http://en.wiktionary.org/wiki/volatility . Wiktionary CC BY-SA 3.0 .
• “pictogram.” http://en.wiktionary.org/wiki/pictogram . Wiktionary CC BY-SA 3.0 .
• “Misleading Graph.” http://en.wikipedia.org/wiki/Misleading_graph . Wikipedia GNU FDL .
• “Frequency distribution.” http://en.wikipedia.org/wiki/Frequency_distribution . Wikipedia CC BY-SA 3.0 .
• “Microsoft Excel &#8211; spreadsheet software – Office.com.” http://office.microsoft.com/en-us/excel/ . Microsoft License: Other .
• “summation notation.” http://en.wikipedia.org/wiki/summation%20notation . Wikipedia CC BY-SA 3.0 .
• “Summation.” http://en.wikipedia.org/wiki/Summation%23Capital-sigma_notation . Wikipedia CC BY-SA 3.0 .
• “ellipsis.” http://en.wiktionary.org/wiki/ellipsis . Wiktionary CC BY-SA 3.0 .
• “Bivariate Data Tutorial | Sophia Learning.” http://www.sophia.org/bivariate-data-tutorial . Sophia Learning Online CC BY .
• “bivariate.” http://en.wiktionary.org/wiki/bivariate . Wiktionary CC BY-SA 3.0 .
• “contingency table.” http://en.wiktionary.org/wiki/contingency_table . Wiktionary CC BY-SA 3.0 .
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• “David Lane, Introduction to Bivariate Data. September 17, 2013.” http://cnx.org/content/m10949/latest/ . OpenStax CNX CC BY 3.0 .
• “David Lane, Introduction to Bivariate Data. May 6, 2013.” http://cnx.org/content/m10949/latest/ . OpenStax CNX CC BY 3.0 .
• “Contingency table.” http://en.wikipedia.org/wiki/Contingency_table . Wikipedia GNU FDL .

Boundless Statistics for Organizations Copyright © 2021 by Brad Griffith and Lisa Friesen is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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Chapter 2: Summarizing and Visualizing Data

Chapter 3: measure of central tendency, chapter 4: measures of variation, chapter 5: measures of relative standing, chapter 6: probability distributions, chapter 7: estimates, chapter 8: distributions, chapter 9: hypothesis testing, chapter 10: analysis of variance, chapter 11: correlation and regression, chapter 12: statistics in practice.

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Consider a relative frequency distribution table of hockey players with different heights. This table provides information about the fraction, or proportion, of data values under each class.

If this relative frequency is expressed in terms of percentage, it is called the percentage frequency distribution.

Suppose one is interested in the percentage of players with heights between 152 and 157 centimeters. To find out, multiply the corresponding relative frequency with 100 to get the percentage frequency. This indicates that 5 percent of players fall within the required height range.

Repeat the similar calculation for all the other relative frequencies to obtain the percentage frequencies under each class. Generally, the sum of all the percentage frequencies is equal to 100.

2.5: Percentage Frequency Distribution

A percentage frequency distribution, in general, is a display of data that indicates the percentage of observations for each data point or grouping of data points. It is a commonly used method for expressing the relative frequency of survey responses and other data. The percentage frequency distributions are often displayed as bar graphs, pie charts, or tables.

The process of making a percentage frequency distribution involves the following few steps: note the total number of observations; count the total number of observations within each data point or grouping of data points; and finally, divide the total observations within each data point or grouping of data points by the total number of observations. However, it is to be noted that whenever the percentage frequencies are used in the relative frequency distribution, it is also sometimes termed as percentage frequency distribution.

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