null hypothesis f statistic

Statistics Made Easy

A Simple Guide to Understanding the F-Test of Overall Significance in Regression

This tutorial explains how to identify the F-statistic in the output of a regression table as well as how to interpret this statistic and its corresponding p-value.

Understanding the F-Test of Overall Significance

The F-Test of overall significance in regression is a test of whether or not your linear regression model provides a better fit to a dataset than a model with no predictor variables. 

The F-Test of overall significance has the following two hypotheses:

Null hypothesis (H 0 ) : The model with no predictor variables (also known as an  intercept-only model ) fits the data as well as your regression model.

Alternative hypothesis (H A ) :  Your regression model fits the data better than the intercept-only model.

When you fit a regression model to a dataset, you will receive a regression table as output, which will tell you the F-statistic along with the corresponding p-value for that F-statistic.

If the p-value is less than the significance level you’ve chosen ( common choices are .01, .05, and .10 ), then you have sufficient evidence to conclude that your regression model fits the data better than the intercept-only model.

Example: F-Test in Regression

Suppose we have the following dataset that shows the total number of hours studied, total prep exams taken, and final exam score received for 12 different students:

To analyze the relationship between hours studied and prep exams taken with the final exam score that a student receives, we run a multiple linear regression using  hours studied and  prep  exams taken  as the predictor variables and  final exam score  as the response variable.

We receive the following output:

From these results, we will focus on the F-statistic given in the ANOVA table as well as the p-value of that F-statistic, which is labeled as  Significance F in the table. We will choose .05 as our significance level.

F-statistic: 5.090515

P-value: 0.0332

Technical note: The F-statistic is calculated as MS regression divided by MS residual. In this case MS regression / MS residual =273.2665 / 53.68151 = 5.090515 .

Since the p-value is less than the significance level, we can conclude that our regression model fits the data better than the intercept-only model.

In the context of this specific problem, it means that using our predictor variables Study Hours and  Prep Exams  in the model allows us to fit the data better than if we left them out and simply used the intercept-only model.

Notes on Interpreting the F-Test of Overall Significance

In general, if none of your predictor variables are statistically significant, the overall F-test will also not be statistically significant.

However, it’s possible on some occasions that this doesn’t hold because the F-test of overall significance tests whether all of the predictor variables are jointly  significant while the t-test of significance for each individual predictor variable merely tests whether each predictor variable is  individually  significant. 

Thus, the F-test determines whether or not  all  of the predictor variables are jointly significant.

It’s possible that each predictor variable is not significant and yet the F-test says that all of the predictor variables combined are jointly significant. 

Technical note:  In general, the more predictor variables you have in the model, the higher the likelihood that the The F-statistic and corresponding p-value will be statistically significant.

Another metric that you’ll likely see in the output of a regression is  R-squared , which measures the strength of the linear relationship between the predictor variables and the response variable is another.

Although R-squared can give you an idea of how strongly associated the predictor variables are with the response variable, it doesn’t provide a formal statistical test for this relationship.

This is why the F-Test is useful since it is a formal statistical test. In addition, if the overall F-test is significant, you can conclude that R-squared is not equal to zero and that the correlation between the predictor variable(s) and response variable is statistically significant.

Additional Resources

The following tutorials explain how to interpret other common values in regression models:

How to Read and Interpret a Regression Table Understanding the Standard Error of the Regression What is a Good R-squared Value?

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Statistics and probability

Course: statistics and probability   >   unit 16.

• ANOVA 1: Calculating SST (total sum of squares)
• ANOVA 2: Calculating SSW and SSB (total sum of squares within and between)

ANOVA 3: Hypothesis test with F-statistic

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Video transcript

F-statistic calculator

Table of contents

The F-statistic calculator (or F-test calculator) helps you compare the equality of the variances of two populations with normal distributions based on the ratio of the variances of a sample of observations drawn from them.

Read further, and learn the following:

• What is an F-statistic;
• What is the F-statistic formula; and
• How to interpret an F-statistic in regression.

What is F-statistic?

Broadly speaking, an F-statistic is a test procedure that compares variances of two given populations . While an F-test may appear in various statistical or econometric problems, we apply it most frequently to regression analysis containing multiple explanatory variables . In this vein, an F-statistic is comparable to a T-statistic , with the main difference of having a linear combination of multiple regression coefficients (F-test) instead of testing only an individual one (T-test).

In the following article, we introduce the F-test in its most basic form using the F-distribution table for better intuition. Then we show how to calculate F-statistic in linear regressions (see the calculator's Multiple regression mode) and explain how to interpret an F-statistic in regression analysis.

How to calculate the F-statistic using an F-statistic table?

The best way to grasp the essence of F-test statistics is to consider its most basic form . Let's consider two populations, from which we each draw an equal number of observation samples. If we want to test whether the two populations are likely to have the same variance (denoted by S i 2 S^2_i S i 2 ​ , i = 1 , 2 i = 1, 2 i = 1 , 2 ), we need to follow these steps:

• Specify the null hypothesis H 0 H_0 H 0 ​ (which in our simple case is that the two variances are equal) and the alternative hypothesis H 1 H_1 H 1 ​ (which supposes that the two variances are different).

Determine the variance of the samples (here you may find our variance calculator useful).

Calculate the F-test statistic by dividing the two variances.

• Determine the degrees of freedom ( df i ) (\text{df}_i) ( df i ​ ) of the two samples, with n n n being the number of observations taken from the two populations in each case.

Choose the significance level of the F-statistic ( α ) (\alpha) ( α ) — for example, α = 0.05 \alpha = 0.05 α = 0.05 corresponds to a 95 percent confidence interval .

Check the critical value of the F-statistic in the F-distribution table as follows:

• Look for the appropriate F-statistic table with the given significance level ( α ) (\alpha) ( α ) .
• Find the right column at the top of the F-table statistics that correspond to the degree of freedom of your first sample (nominator).
• Check the row on the side that corresponds to the degree of freedom of your second sample (denominator).
• Read the F critical value at the intersection , which represents the shaded area on the F-distribution graph below.

• Compare the F-statistic critical value to the previously obtained F-value – check our critical value calculator to learn more about the concept. If the F-value is larger than the critical value collected from the F-table statistic ( F > F critical ) (F > F_\text{critical}) ( F > F critical ​ ) , you can reject the null hypothesis . That is, we can state with high confidence that the variances in the two observation samples are not equal.

How to calculate the F-statistic in linear regression?

Analysts mainly apply F-statistic on multiple regressions models (and so can you, with our F-test statistic calculator in Multiple regression mode). It's therefore a good idea that we step further in this direction from the previous basic analysis.

Let's assume we have the following regression model ( full model , or unrestricted model ), where we would like to know if it is more significant than its reduced form ( restricted model ). In other words, we are testing whether the restricted coefficients (or the effects of the restricted variables) are jointly non-significant (equal to zero) in the population:

• β 0 \beta_0 β 0 ​ – Constant or intercept;
• y y y – Dependent variable (also called the regressand , response variable , explained variable , or output variable );
• x i   ,   i = 1 , 2 , 3 x_i\, , \ i = 1, 2, 3 x i ​ ,   i = 1 , 2 , 3 is the independent variable (also called the regressor , explanatory variable , controlled variable , or input variable );
• β i   ,   i = 1 , 2 , 3 \beta_i\, ,\ i = 1, 2, 3 β i ​ ,   i = 1 , 2 , 3 are the coefficients; and
• u ^ \hat{u} u ^ is the residual (or error term ).

To conduct the F-test and obtain the F-statistic (or F-value), we need to take the following steps :

State the hypothesis we want to test.

In our case, the null hypothesis ( H 0 ) (H_0) ( H 0 ​ ) is that the last two coefficients are jointly equal to zero in the unrestricted model. Or, stating the same differently, the joint effect of the related independent variables is insignificant.

In turn, the alternative hypothesis ( H 1 ) (H_1) ( H 1 ​ ) is that at least one of these coefficients is not equal to zero.

• J J J is the number of restrictions (in the present case, J = 2 J=2 J = 2 ); and
• K K K is the total number of coefficients (in the present case, K = 3 K = 3 K = 3 ).
• Now, to gain information on which model fits better, we need to obtain the sum square of residuals ( SSR \text{SSR} SSR ), where we expect that the sum square of residuals of the restricted model is larger than that of the full model (i.e. SSR R > SSR F \text{SSR}_R > \text{SSR}_F SSR R ​ > SSR F ​ ).
• However, the real question is to determine whether the sum square of residuals of the restricted model is significantly larger than the one in the full model (i.e. SSR R ≫ SSR F \text{SSR}_R \gg \text{SSR}_F SSR R ​ ≫ SSR F ​ ). To do so, we need to apply the following F-statistic formula to estimate the F-ratio.
• F F F – F-statistic;
• SSR F \text{SSR}_F SSR F ​ – Sum square of residuals of the full model;
• SSR R \text{SSR}_R SSR R ​ – Sum square of residuals of the restricted model;
• J J J – Number of restrictions;
• K K K – Total number of coefficients; and
• N N N – Number of observations representing the population.

Naturally, the larger the F-statistic , the more evidence we have to reject the null hypothesis (note that the F-statistic increases when the difference between the two variances gets larger). However, to be more precise, we need to find a critical value of the F-statistic to decide on the rejection. In other words, if F F F is larger than its critical value , we can reject the null hypothesis .

Now, we can proceed in the way we described in the previous section by finding the critical F-value ( F N − K ; α J ) (F^J_{N-K;\alpha}) ( F N − K ; α J ​ ) in the F distribution table with a specified significance level F-statistic ( α ) (\alpha) ( α ) and looking for the intercept corresponding to the degrees of freedom, where df 1 = J \text{df}_1 = J df 1 ​ = J is at the top and df 1 = N − K \text{df}_1 = N-K df 1 ​ = N − K is at the side of the table (we can also say that F F F has an F-distribution with J J J and N − K N − K N − K degrees of freedom). If F F F is larger than its critical value, we can reject the null hypothesis.

So how to interpret F-statistic in regression?

The F-test can be interpreted as testing whether the increase in variance moving from the restricted model to the more general model is significant. We may write it formally in the following way:

where α \alpha α is the significance level of the test. For example, if N − K = 40 N − K = 40 N − K = 40 and J = 4 J = 4 J = 4 , the critical value at the 5% level is F N − K ; α J = 2.606 F^J_{N-K; \alpha} = 2.606 F N − K ; α J ​ = 2.606 .

What is the difference between F-test vs T-test?

There are some differences between the F-test vs a T-test.

The T-test is applied to test the significance of one explanatory variable, but the F-test studies the whole model.

While the T-test is used to compare the means of two populations, F-test is applied for comparing two population variances.

The T-statistic is based on the student t-distribution, while the F-statistic follows the F-distribution under the null hypothesis.

While the T-test is a univariate hypothesis test where the standard deviation is unknown, the F-test is applied to determine the equality of the two normal populations.

Can an F-statistic be negative?

No. Since variances always take a positive value (squared values), both the numerator and the denominator of the F-statistic formula must always be positive, resulting in a positive F-value.

What is a high F-statistic?

While a large F-value tends to indicate that the null hypothesis can be rejected, you can confidently reject the null if the T-value is larger than its critical value.

Is the F-distribution symmetric?

No. The curve of the F-distribution is not symmetrical but skewed to the right (the curve has a long tail on its right side), where the shape of the curve depends on the degrees of freedom.

How to calculate F-statistic?

To calculate F-statistic, in general, you need to follow the below steps.

State the null hypothesis and the alternate hypothesis.

Determine the F-value by the formula of F = [(SSE₁ – SSE₂) / m] / [SSE₂ / (n−k)] , where SSE is the residual sum of squares, m is the number of restrictions and k is the number of independent variables.

Find the critical value for the F-statistic as determined by F-statistic = variance of the group means / mean of the within-group variances .

Find the F-statistic in the F-table.

Support or reject the null hypothesis.

What is the F-statistic of two populations with variances of 10 and 5?

The F-statistic of two populations with variances of 10 and 5 is 2. To get this result, it suffices to divide the two variances.

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Sum square of residuals — full model (SSR F )

Sum square of residuals — restricted model (SSR R )

Number of excluded coefficients (J)

Total number of coefficients (K)

Sample size (N)

F-statistic (F)

• 5.6 - The General Linear F-Test

The " general linear F-test " involves three basic steps, namely:

• Define a larger full model . (By "larger," we mean one with more parameters.)
• Define a smaller reduced model . (By "smaller," we mean one with fewer parameters.)
• Use an F- statistic to decide whether or not to reject the smaller reduced model in favor of the larger full model.

As you can see by the wording of the third step, the null hypothesis always pertains to the reduced model, while the alternative hypothesis always pertains to the full model.

The easiest way to learn about the general linear F-test is to first go back to what we know, namely the simple linear regression model. Once we understand the general linear F-test for the simple case, we then see that it can be easily extended to the multiple case. We take that approach here.

The full model

The " full model ", which is also sometimes referred to as the " unrestricted model ," is the model thought to be most appropriate for the data. For simple linear regression, the full model is:

\[y_i=(\beta_0+\beta_1x_{i1})+\epsilon_i\]

Here's a plot of a hypothesized full model for a set of data that we worked with previously in this course (student heights and grade point averages):

And, here's another plot of a hypothesized full model that we previously encountered (state latitudes and skin cancer mortalities):

In each plot, the solid line represents what the hypothesized population regression line might look like for the full model. The question we have to answer in each case is "does the full model describe the data well?" Here, we might think that the full model does well in summarizing the trend in the second plot but not the first.

The reduced model

The " reduced model ," which is sometimes also referred to as the " restricted model ," is the model described by the null hypothesis H 0 . For simple linear regression, a common null hypothesis is H 0 : β 1 = 0. In this case, the reduced model is obtained by "zeroing-out" the slope β 1 that appears in the full model. That is, the reduced model is:

\[y_i=\beta_0+\epsilon_i\]

This reduced model suggests that each response y i is a function only of some overall mean, β 0 , and some error ε i .

Let's take another look at the plot of student grade point average against height, but this time with a line representing what the hypothesized population regression line might look like for the reduced model:

Not bad — there (fortunately?!) doesn't appear to be a relationship between height and grade point average. And, it appears as if the reduced model might be appropriate in describing the lack of a relationship between heights and grade point averages. How does the reduced model do for the skin cancer mortality example?

It doesn't appear as if the reduced model would do a very good job of summarizing the trend in the population.

How do we decide if the reduced model or the full model does a better job of describing the trend in the data when it can't be determined by simply looking at a plot? What we need to do is to quantify how much error remains after fitting each of the two models to our data. That is, we take the general linear F-test approach:

• Obtain the least squares estimates of β 0 and β 1 .
• Determine the error sum of squares, which we denote " SSE ( F )."
• Obtain the least squares estimate of β 0 .
• Determine the error sum of squares, which we denote " SSE ( R )."

Recall that, in general, the error sum of squares is obtained by summing the squared distances between the observed and fitted (estimated) responses:

\[\sum(\text{observed } - \text{ fitted})^2\]

Therefore, since \(y_i\) is the observed response and \(\hat{y}_i\) is the fitted response for the full model :

\[SSE(F)=\sum(y_i-\hat{y}_i)^2\]

And, since \(y_i\) is the observed response and \(\bar{y}\) is the fitted response for the reduced model :

\[SSE(R)=\sum(y_i-\bar{y})^2\]

Let's get a better feel for the general linear F-test approach by applying it to two different two datasets. First, let's look at the heightgpa data . The following plot of grade point averages against heights contains two estimated regression lines — the solid line is the estimated line for the full model, and the dashed line is the estimated line for the reduced model:

As you can see, the estimated lines are almost identical. Calculating the error sum of squares for each model, we obtain:

\[SSE(F)=\sum(y_i-\hat{y}_i)^2=9.7055\]

\[SSE(R)=\sum(y_i-\bar{y})^2=9.7331\]

The two quantities are almost identical. Adding height to the reduced model to obtain the full model reduces the amount of error by only 0.0276 (from 9.7331 to 9.7055). That is, adding height to the model does very little in reducing the variability in grade point averages. In this case, there appears to be no advantage in using the larger full model over the simpler reduced model.

Look what happens when we fit the full and reduced models to the skin cancer mortality and latitude dataset :

Here, there is quite a big difference in the estimated equation for the reduced model (solid line) and the estimated equation for the full model (dashed line). The error sums of squares quantify the substantial difference in the two estimated equations:

\[SSE(F)=\sum(y_i-\hat{y}_i)^2=17173\]

\[SSE(R)=\sum(y_i-\bar{y})^2=53637\]

Adding latitude to the reduced model to obtain the full model reduces the amount of error by 36464 (from 53637 to 17173). That is, adding latitude to the model substantially reduces the variability in skin cancer mortality. In this case, there appears to be a big advantage in using the larger full model over the simpler reduced model.

Where are we going with this general linear F-test approach? In short:

• The general linear F-test involves a comparison between SSE ( R ) and SSE ( F ).
• If SSE ( F ) is close to SSE ( R ), then the variation around the estimated full model regression function is almost as large as the variation around the estimated reduced model regression function. If that's the case, it makes sense to use the simpler reduced model.
• On the other hand, if SSE ( F ) and SSE ( R ) differ greatly, then the additional parameter(s) in the full model substantially reduce the variation around the estimated regression function. In this case, it makes sense to go with the larger full model.

How different does SSE ( R ) have to be from SSE ( F ) in order to justify using the larger full model? The general linear F -statistic:

\[F^*=\left( \frac{SSE(R)-SSE(F)}{df_R-df_F}\right)\div\left( \frac{SSE(F)}{df_F}\right)\]

helps answer this question. The F -statistic intuitively makes sense — it is a function of SSE ( R )- SSE ( F ), the difference in the error between the two models. The degrees of freedom — denoted df R and df F — are those associated with the reduced and full model error sum of squares, respectively.

We use the general linear F -statistic to decide whether or not:

• to reject the null hypothesis H 0 : the reduced model,
• in favor of the alternative hypothesis H A : the full model.

In general, we reject H 0 if F * is large — or equivalently if its associated P -value is small.

The test applied to the simple linear regression model

For simple linear regression, it turns out that the general linear F -test is just the same ANOVA F -test that we learned before. As noted earlier for the simple linear regression case, the full model is:

and the reduced model is:

Therefore, the appropriate null and alternative hypotheses are specified either as:

• H 0 : y i = β 0 + ε i
• H A : y i = β 0 + β 1 x i + ε i
• H 0 : β 1 = 0
• H A : β 1 ≠ 0

The degrees of freedom associated with the error sum of squares for the reduced model is n -1, and:

\[SSE(R)=\sum(y_i-\bar{y})^2=SSTO\]

The degrees of freedom associated with the error sum of squares for the full model is n -2, and:

\[SSE(F)=\sum(y_i-\hat{y}_i)^2=SSE\]

Now, we can see how the general linear F -statistic just reduces algebraically to the ANOVA F -test that we know:

That is, the general linear F -statistic reduces to the ANOVA F -statistic:

\[F^*=\frac{MSR}{MSE}\]

For the student height and grade point average example:

\[F^*=\frac{MSR}{MSE}=\frac{0.0276/1}{9.7055/33}=\frac{0.0276}{0.2941}=0.094\]

For the skin cancer mortality example:

\[F^*=\frac{MSR}{MSE}=\frac{36464/1}{17173/47}=\frac{36464}{365.4}=99.8\]

The P -value is calculated as usual. The P -value answers the question: "what is the probability that we’d get an F* statistic as large as we did, if the null hypothesis were true?" The P -value is determined by comparing F * to an F distribution with 1 numerator degree of freedom and n -2 denominator degrees of freedom. For the student height and grade point average example, the P -value is 0.761 (so we fail to reject H 0 and we favor the reduced model), while for the skin cancer mortality example, the P -value is 0.000 (so we reject H 0 and we favor the full model).

Does alcoholism have an effect on muscle strength? Some researchers (Urbano-Marquez, et al , 1989) who were interested in answering this question collected the following data ( alcoholarm.txt ) on a sample of 50 alcoholic men:

• x = the total lifetime dose of alcohol ( kg per kg of body weight) consumed
• y = the strength of the deltoid muscle in the man's non-dominant arm

The full model is the model that would summarize a linear relationship between alcohol consumption and arm strength. The reduced model, on the other hand, is the model that claims there is no relationship between alcohol consumption and arm strength.

Upon fitting the reduced model to the data, we obtain:

\[SSE(R)=\sum(y_i-\bar{y})^2=1224.32\]

Note that the reduced model does not appear to summarize the trend in the data very well.

Upon fitting the full model to the data, we obtain:

\[SSE(F)=\sum(y_i-\hat{y}_i)^2=720.27\]

The full model appears to decribe the trend in the data better than the reduced model.

The good news is that in the simple linear regression case, we don't have to bother with calculating the general linear F -statistic. Statistical software does it for us in the ANOVA table:

As you can see, the output reports both SSE ( F ) — the amount of error associated with the full model — and SSE ( R ) — the amount of error associated with the reduced model. The F -statistic is:

\[F^*=\frac{MSR}{MSE}=\frac{504.04/1}{720.27/48}=\frac{504.04}{15.006}=33.59\]

and its associated P -value is < 0.001 (so we reject H 0 and we favor the full model). We can conclude that there is a statistically significant linear association between lifetime alcohol consumption and arm strength.

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Understanding Analysis of Variance (ANOVA) and the F-test

Topics: ANOVA , Hypothesis Testing , Data Analysis

Analysis of variance (ANOVA) can determine whether the means of three or more groups are different. ANOVA uses F-tests to statistically test the equality of means. In this post, I’ll show you how ANOVA and F-tests work using a one-way ANOVA example.

But wait a minute...have you ever stopped to wonder why you’d use an analysis of variance to determine whether means are different? I'll also show how variances provide information about means.

As in my posts about understanding t-tests , I’ll focus on concepts and graphs rather than equations to explain ANOVA F-tests.

What are F-statistics and the F-test?

F-tests are named after its test statistic, F, which was named in honor of Sir Ronald Fisher. The F-statistic is simply a ratio of two variances. Variances are a measure of dispersion, or how far the data are scattered from the mean. Larger values represent greater dispersion.

Variance is the square of the standard deviation. For us humans, standard deviations are easier to understand than variances because they’re in the same units as the data rather than squared units. However, many analyses actually use variances in the calculations.

F-statistics are based on the ratio of mean squares. The term “ mean squares ” may sound confusing but it is simply an estimate of population variance that accounts for the degrees of freedom (DF) used to calculate that estimate.

Despite being a ratio of variances, you can use F-tests in a wide variety of situations. Unsurprisingly, the F-test can assess the equality of variances. However, by changing the variances that are included in the ratio, the F-test becomes a very flexible test. For example, you can use F-statistics and F-tests to test the overall significance for a regression model , to compare the fits of different models, to test specific regression terms, and to test the equality of means.

Using the F-test in One-Way ANOVA

To use the F-test to determine whether group means are equal, it’s just a matter of including the correct variances in the ratio. In one-way ANOVA, the F-statistic is this ratio:

F = variation between sample means / variation within the samples

The best way to understand this ratio is to walk through a one-way ANOVA example.

We’ll analyze four samples of plastic to determine whether they have different mean strengths. You can download the sample data  if you want to follow along. (If you don't have Minitab, you can download a free 30-day trial .) I'll refer back to the one-way ANOVA output as I explain the concepts.

In Minitab, choose Stat > ANOVA > One-Way ANOVA...  In the dialog box, choose "Strength" as the response, and "Sample" as the factor. Press OK, and Minitab's Session Window displays the following output: 

Numerator: Variation Between Sample Means

One-way ANOVA has calculated a mean for each of the four samples of plastic. The group means are: 11.203, 8.938, 10.683, and 8.838. These group means are distributed around the overall mean for all 40 observations, which is 9.915. If the group means are clustered close to the overall mean, their variance is low. However, if the group means are spread out further from the overall mean, their variance is higher.

Clearly, if we want to show that the group means are different, it helps if the means are further apart from each other. In other words, we want higher variability among the means.

Imagine that we perform two different one-way ANOVAs where each analysis has four groups. The graph below shows the spread of the means. Each dot represents the mean of an entire group. The further the dots are spread out, the higher the value of the variability in the numerator of the F-statistic.

What value do we use to measure the variance between sample means for the plastic strength example? In the one-way ANOVA output, we’ll use the adjusted mean square (Adj MS) for Factor, which is 14.540. Don’t try to interpret this number because it won’t make sense. It’s the sum of the squared deviations divided by the factor DF. Just keep in mind that the further apart the group means are, the larger this number becomes.

Denominator: Variation Within the Samples

We also need an estimate of the variability within each sample. To calculate this variance, we need to calculate how far each observation is from its group mean for all 40 observations. Technically, it is the sum of the squared deviations of each observation from its group mean divided by the error DF.

If the observations for each group are close to the group mean, the variance within the samples is low. However, if the observations for each group are further from the group mean, the variance within the samples is higher.

In the graph, the panel on the left shows low variation in the samples while the panel on the right shows high variation. The more spread out the observations are from their group mean, the higher the value in the denominator of the F-statistic.

If we’re hoping to show that the means are different, it's good when the within-group variance is low. You can think of the within-group variance as the background noise that can obscure a difference between means.

For this one-way ANOVA example, the value that we’ll use for the variance within samples is the Adj MS for Error, which is 4.402. It is considered “error” because it is the variability that is not explained by the factor.

Ready for a demo of Minitab Statistical Software? Just ask! 

The F-Statistic: Variation Between Sample Means / Variation Within the Samples

The F-statistic is the test statistic for F-tests. In general, an F-statistic is a ratio of two quantities that are expected to be roughly equal under the null hypothesis, which produces an F-statistic of approximately 1.

The F-statistic incorporates both measures of variability discussed above. Let's take a look at how these measures can work together to produce low and high F-values. Look at the graphs below and compare the width of the spread of the group means to the width of the spread within each group.

The low F-value graph shows a case where the group means are close together (low variability) relative to the variability within each group. The high F-value graph shows a case where the variability of group means is large relative to the within group variability. In order to reject the null hypothesis that the group means are equal, we need a high F-value.

For our plastic strength example, we'll use the Factor Adj MS for the numerator (14.540) and the Error Adj MS for the denominator (4.402), which gives us an F-value of 3.30.

Is our F-value high enough? A single F-value is hard to interpret on its own. We need to place our F-value into a larger context before we can interpret it. To do that, we’ll use the F-distribution to calculate probabilities.

F-distributions and Hypothesis Testing

For one-way ANOVA, the ratio of the between-group variability to the within-group variability follows an F-distribution when the null hypothesis is true.

When you perform a one-way ANOVA for a single study, you obtain a single F-value. However, if we drew multiple random samples of the same size from the same population and performed the same one-way ANOVA, we would obtain many F-values and we could plot a distribution of all of them. This type of distribution is known as a sampling distribution .

Because the F-distribution assumes that the null hypothesis is true, we can place the F-value from our study in the F-distribution to determine how consistent our results are with the null hypothesis and to calculate probabilities.

The probability that we want to calculate is the probability of observing an F-statistic that is at least as high as the value that our study obtained. That probability allows us to determine how common or rare our F-value is under the assumption that the null hypothesis is true. If the probability is low enough, we can conclude that our data is inconsistent with the null hypothesis. The evidence in the sample data is strong enough to reject the null hypothesis for the entire population.

This probability that we’re calculating is also known as the p-value!

To plot the F-distribution for our plastic strength example, I’ll use Minitab’s probability distribution plots . In order to graph the F-distribution that is appropriate for our specific design and sample size, we'll need to specify the correct number of DF. Looking at our one-way ANOVA output, we can see that we have 3 DF for the numerator and 36 DF for the denominator.

The graph displays the distribution of F-values that we'd obtain if the null hypothesis is true and we repeat our study many times. The shaded area represents the probability of observing an F-value that is at least as large as the F-value our study obtained. F-values fall within this shaded region about 3.1% of the time when the null hypothesis is true. This probability is low enough to reject the null hypothesis using the common significance level of 0.05. We can conclude that not all the group means are equal.

Learn how to correctly interpret the p-value.

Assessing Means by Analyzing Variation

ANOVA uses the F-test to determine whether the variability between group means is larger than the variability of the observations within the groups. If that ratio is sufficiently large, you can conclude that not all the means are equal.

This brings us back to why we analyze variation to make judgments about means.  Think about the question: "Are the group means different?" You are implicitly asking about the variability of the means. After all, if the group means don't vary, or don't vary by more than random chance allows, then you can't say the means are different. And that's why you use analysis of variance to test the means.

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F test is a statistical test that is used in hypothesis testing to check whether the variances of two populations or two samples are equal or not. In an f test, the data follows an f distribution. This test uses the f statistic to compare two variances by dividing them. An f test can either be one-tailed or two-tailed depending upon the parameters of the problem.

The f value obtained after conducting an f test is used to perform the one-way ANOVA (analysis of variance) test. In this article, we will learn more about an f test, the f statistic, its critical value, formula and how to conduct an f test for hypothesis testing.

What is F Test in Statistics?

F test is statistics is a test that is performed on an f distribution. A two-tailed f test is used to check whether the variances of the two given samples (or populations) are equal or not. However, if an f test checks whether one population variance is either greater than or lesser than the other, it becomes a one-tailed hypothesis f test.

F Test Definition

F test can be defined as a test that uses the f test statistic to check whether the variances of two samples (or populations) are equal to the same value. To conduct an f test, the population should follow an f distribution and the samples must be independent events. On conducting the hypothesis test, if the results of the f test are statistically significant then the null hypothesis can be rejected otherwise it cannot be rejected.

F Test Formula

The f test is used to check the equality of variances using hypothesis testing . The f test formula for different hypothesis tests is given as follows:

Left Tailed Test:

• Null Hypothesis: \(H_{0}\) : \(\sigma_{1}^{2} = \sigma_{2}^{2}\)

Alternate Hypothesis: \(H_{1}\) : \(\sigma_{1}^{2} < \sigma_{2}^{2}\)

Decision Criteria: If the f statistic < f critical value then reject the null hypothesis

Right Tailed test:

• Alternate Hypothesis: \(H_{1}\) : \(\sigma_{1}^{2} > \sigma_{2}^{2}\)

Decision Criteria: If the f test statistic > f test critical value then reject the null hypothesis

Two Tailed test:

Alternate Hypothesis: \(H_{1}\) : \(\sigma_{1}^{2} ≠ \sigma_{2}^{2}\)

Decision Criteria: If the f test statistic > f test critical value then the null hypothesis is rejected

F Statistic

The f test statistic or simply the f statistic is a value that is compared with the critical value to check if the null hypothesis should be rejected or not. The f test statistic formula is given below:

F statistic for large samples: F = \(\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\), where \(\sigma_{1}^{2}\) is the variance of the first population and \(\sigma_{2}^{2}\) is the variance of the second population.

F statistic for small samples: F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\), where \(s_{1}^{2}\) is the variance of the first sample and \(s_{2}^{2}\) is the variance of the second sample.

The selection criteria for the \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\) for an f statistic is given below:

• For a right-tailed and a two-tailed f test, the variance with the greater value will be in the numerator. Thus, the sample corresponding to \(\sigma_{1}^{2}\) will become the first sample. The smaller value variance will be the denominator and belongs to the second sample.
• For a left-tailed test, the smallest variance becomes the numerator (sample 1) and the highest variance goes in the denominator (sample 2).

F Test Critical Value

A critical value is a point that a test statistic is compared to in order to decide whether to reject or not to reject the null hypothesis. Graphically, the critical value divides a distribution into the acceptance and rejection regions. If the test statistic falls in the rejection region then the null hypothesis can be rejected otherwise it cannot be rejected. The steps to find the f test critical value at a specific alpha level (or significance level), \(\alpha\), are as follows:

• Find the degrees of freedom of the first sample. This is done by subtracting 1 from the first sample size. Thus, x = \(n_{1} - 1\).
• Determine the degrees of freedom of the second sample by subtracting 1 from the sample size. This given y = \(n_{2} - 1\).
• If it is a right-tailed test then \(\alpha\) is the significance level. For a left-tailed test 1 - \(\alpha\) is the alpha level. However, if it is a two-tailed test then the significance level is given by \(\alpha\) / 2.
• The F table is used to find the critical value at the required alpha level.
• The intersection of the x column and the y row in the f table will give the f test critical value.

ANOVA F Test

The one-way ANOVA is an example of an f test. ANOVA stands for analysis of variance. It is used to check the variability of group means and the associated variability in observations within that group. The F test statistic is used to conduct the ANOVA test. The hypothesis is given as follows:

\(H_{0}\): The means of all groups are equal.

\(H_{1}\): The means of all groups are not equal.

Test Statistic: F = explained variance / unexplained variance

Decision rule: If F > F critical value then reject the null hypothesis.

To determine the critical value of an ANOVA f test the degrees of freedom are given by \(df_{1}\) = K - 1 and \(df_{1}\) = N - K, where N is the overall sample size and K is the number of groups.

F Test vs T-Test

F test and t-test are different types of statistical tests used for hypothesis testing depending on the distribution followed by the population data. The table given below outlines the differences between the F test and the t-test.

Related Articles:

• Probability and Statistics
• Data Handling
• Summary Statistics

Important Notes on F Test

• The f test is a statistical test that is conducted on an F distribution in order to check the equality of variances of two populations.
• The f test formula for the test statistic is given by F = \(\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\).
• The f critical value is a cut-off value that is used to check whether the null hypothesis can be rejected or not.
• A one-way ANOVA is an example of an f test that is used to check the variability of group means and the associated variability in the group observations.

Examples on F Test

• Example 2: Pizza delivery times of two cities are given below City 1: Number of delivery times observed = 28, Variance = 38 City 2: Number of delivery times observed = 25, Variance = 83 Check if the delivery times of city 1 are lesser than city 2 at a 0.05 alpha level. Solution: This is an example of a left-tailed F test. Thus, the alpha level is 1 - 0.05 = 0.95 \(H_{0}\) : \(s_{1}^{2} = s_{2}^{2}\) \(H_{1}\) : \(s_{1}^{2} < s_{2}^{2}\) As 38 < 83 thus, city 1 will be sample 1 and city 2 is sample 2. \(n_{1}\) = 28, \(n_{2}\) = 25 \(df_{1}\) = 28 - 1 = 27 \(df_{2}\) = 25 - 1 = 24 \(s_{1}^{2}\) = 38, \(s_{2}^{2}\) = 83 F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\) = 38 / 83 F = 0.4578 As an F table for 0.95 alpha level is not available, the critical value is determined as follows: F(0.95, 27, 24) = 1 / F(0.05, 24, 27) F(0.05, 24, 27) = 1.93 F(0.95, 27, 24) = 1 / 1.93 = 0.5181 As 0.4578 < 0.5181, thus, the null hypothesis can be rejected and it can be concluded that there is enough evidence to support the claim that the delivery times in city 1 are less than in city 2. Answer: Reject the null hypothesis
• Example 3: A toy manufacturer wants to get batteries for toys. A team collected 41 samples from supplier A and the variance was 110 hours. The team also collected 21 samples from supplier B with a variance of 65 hours. At a 0.05 alpha level determine if there is a difference in the variances. Solution: This is an example of a two-tailed F test. Thus, the alpha level is 0.05 / 2 = 0.025 \(H_{0}\) : \(s_{1}^{2} = s_{2}^{2}\) \(H_{1}\) : \(s_{1}^{2} \neq s_{2}^{2}\) \(n_{1}\) = 41, \(n_{2}\) = 21 \(df_{1}\) = 41 - 1 = 40 \(df_{2}\) = 21 - 1 = 20 \(s_{1}^{2}\) = 110, \(s_{2}^{2}\) = 65 F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\) = 110 / 65 F = 1.69 Using the F table F(0.025, 40, 20) = 2.287 As 1.69 < 2.287 thus, the null hypothesis cannot be rejected, Answer: Fail to reject the null hypothesis.

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FAQs on F Test

What is the f test.

The f test in statistics is used to find whether the variances of two populations are equal or not by using a one-tailed or two-tailed hypothesis test.

What is the F Test Formula?

The f test formula can be used to find the f statistic. The f test formula is given as follows:

• F statistic for large samples: F = \(\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\)
• F statistic for small samples: F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\)

What is the Decision Criterion for a Right Tailed F Test?

The algorithm to set up an right tailed f test hypothesis along with the decision criteria are given as follows:

• Decision Criteria: Reject \(H_{0}\) if the f test statistic > f test critical value.

What is the Critical Value for an F Test?

The F critical value for an f test can be defined as the cut-off value that is compared with the test statistic to decide if the null hypothesis should be rejected or not.

Why is an F Test Used in ANOVA?

A one-way ANOVA test uses the f test to compare if there is a difference between the variability of group means and the associated variability of observations of those groups.

Can the F statistic in an F Test be Negative?

As the f test statistic is the ratio of variances thus, it cannot be negative. This is because the square of a number will always be positive.

What is the Difference Between F Test and T-Test?

An F test is conducted on an f distribution to determine the equality of variances of two samples. The t-test is performed on a student t distribution when the number of samples is less and the population standard deviation is not known. It is used to compare means.

Stats: F-Test

• The F-values are all non-negative
• The distribution is non-symmetric
• The mean is approximately 1
• There are two independent degrees of freedom, one for the numerator, and one for the denominator.
• There are many different F distributions, one for each pair of degrees of freedom.

Avoiding Left Critical Values

Assumptions / notes.

• The larger variance should always be placed in the numerator
• The test statistic is F = s1^2 / s2^2 where s1^2 > s2^2
• Divide alpha by 2 for a two tail test and then find the right critical value
• If standard deviations are given instead of variances, they must be squared
• When the degrees of freedom aren't given in the table, go with the value with the larger critical value (this happens to be the smaller degrees of freedom). This is so that you are less likely to reject in error (type I error)
• The populations from which the samples were obtained must be normal.
• The samples must be independent

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9.1: Null and Alternative Hypotheses

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The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

\(H_0\): The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

\(H_a\): The alternative hypothesis: It is a claim about the population that is contradictory to \(H_0\) and what we conclude when we reject \(H_0\). This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject \(H_0\)" if the sample information favors the alternative hypothesis or "do not reject \(H_0\)" or "decline to reject \(H_0\)" if the sample information is insufficient to reject the null hypothesis.

\(H_{0}\) always has a symbol with an equal in it. \(H_{a}\) never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example \(\PageIndex{1}\)

• \(H_{0}\): No more than 30% of the registered voters in Santa Clara County voted in the primary election. \(p \leq 30\)
• \(H_{a}\): More than 30% of the registered voters in Santa Clara County voted in the primary election. \(p > 30\)

Exercise \(\PageIndex{1}\)

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

• \(H_{0}\): The drug reduces cholesterol by 25%. \(p = 0.25\)
• \(H_{a}\): The drug does not reduce cholesterol by 25%. \(p \neq 0.25\)

Example \(\PageIndex{2}\)

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

• \(H_{0}: \mu = 2.0\)
• \(H_{a}: \mu \neq 2.0\)

Exercise \(\PageIndex{2}\)

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.

• \(H_{0}: \mu \_ 66\)
• \(H_{a}: \mu \_ 66\)
• \(H_{0}: \mu = 66\)
• \(H_{a}: \mu \neq 66\)

Example \(\PageIndex{3}\)

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

• \(H_{0}: \mu \geq 5\)
• \(H_{a}: \mu < 5\)

Exercise \(\PageIndex{3}\)

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• \(H_{0}: \mu \_ 45\)
• \(H_{a}: \mu \_ 45\)
• \(H_{0}: \mu \geq 45\)
• \(H_{a}: \mu < 45\)

Example \(\PageIndex{4}\)

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

• \(H_{0}: p \leq 0.066\)
• \(H_{a}: p > 0.066\)

Exercise \(\PageIndex{4}\)

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (\(=, \neq, \geq, <, \leq, >\)) for the null and alternative hypotheses.

• \(H_{0}: p \_ 0.40\)
• \(H_{a}: p \_ 0.40\)
• \(H_{0}: p = 0.40\)
• \(H_{a}: p > 0.40\)

COLLABORATIVE EXERCISE

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:

• Evaluate the null hypothesis , typically denoted with \(H_{0}\). The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality \((=, \leq \text{or} \geq)\)
• Always write the alternative hypothesis , typically denoted with \(H_{a}\) or \(H_{1}\), using less than, greater than, or not equals symbols, i.e., \((\neq, >, \text{or} <)\).
• If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
• Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

\(H_{0}\) and \(H_{a}\) are contradictory.

• If \(\alpha \leq p\)-value, then do not reject \(H_{0}\).
• If\(\alpha > p\)-value, then reject \(H_{0}\).

\(\alpha\) is preconceived. Its value is set before the hypothesis test starts. The \(p\)-value is calculated from the data.References

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm .

9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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• Null and Alternative Hypotheses | Definitions & Examples

Null & Alternative Hypotheses | Definitions, Templates & Examples

Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis ( H 0 ): There’s no effect in the population .
• Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:

• The null hypothesis ( H 0 ) answers “No, there’s no effect in the population.”
• The alternative hypothesis ( H a ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses .

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

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The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept . Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect,” “no difference,” or “no relationship.” When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis ( H a ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes < or >). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question.
• They both make claims about the population.
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

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To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

General template sentences

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis ( H 0 ): Independent variable does not affect dependent variable.
• Alternative hypothesis ( H a ): Independent variable affects dependent variable.

Test-specific template sentences

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

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.

• Normal distribution
• Descriptive statistics
• 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

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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