Fstatistic calculator
Table of contents
What is Fstatistic?How to calculate the Fstatistic using an Fstatistic table?How to calculate the Fstatistic in linear regression?FAQsThe Fstatistic calculator (or Ftest 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 Fstatistic;
 What is the Fstatistic formula; and
 How to interpret an Fstatistic in regression.
What is Fstatistic?
Broadly speaking, an Fstatistic is a test procedure that compares variances of two given populations. While an Ftest may appear in various statistical or econometric problems, we apply it most frequently to regression analysis containing multiple explanatory variables. In this vein, an Fstatistic is comparable to a Tstatistic, with the main difference of having a linear combination of multiple regression coefficients (Ftest) instead of testing only an individual one (Ttest).
In the following article, we introduce the Ftest in its most basic form using the Fdistribution table for better intuition. Then we show how to calculate Fstatistic in linear regressions (see the calculator's Multiple regression
mode) and explain how to interpret an Fstatistic in regression analysis.
How to calculate the Fstatistic using an Fstatistic table?
The best way to grasp the essence of Ftest 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 $S^2_i$, $i = 1, 2$), we need to follow these steps:
 Specify the null hypothesis $H_0$ (which in our simple case is that the two variances are equal) and the alternative hypothesis $H_1$ (which supposes that the two variances are different).

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

Calculate the Ftest statistic by dividing the two variances.
 Determine the degrees of freedom $(\text{df}_i)$ of the two samples, with $n$ being the number of observations taken from the two populations in each case.

Choose the significance level of the Fstatistic $(\alpha)$ — for example, $\alpha = 0.05$ corresponds to a 95 percent confidence interval.

Check the critical value of the Fstatistic in the
as follows: Look for the appropriate Fstatistic table with the given significance level $(\alpha)$.
 Find the right column at the top of the Ftable 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 Fdistribution graph below.
 Compare the Fstatistic critical value to the previously obtained Fvalue – check our critical value calculator to learn more about the concept. If the Fvalue is larger than the critical value collected from the Ftable statistic $(F > F_\text{critical})$, 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 Fstatistic in linear regression?
Analysts mainly apply Fstatistic on multiple regressions models (and so can you, with our Ftest 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 nonsignificant (equal to zero) in the population:
where:
 $\beta_0$ – Constant or intercept;
 $y$ – Dependent variable (also called the regressand, response variable, explained variable, or output variable);
 $x_i\, , \ i = 1, 2, 3$ is the independent variable (also called the regressor, explanatory variable, controlled variable, or input variable);
 $\beta_i\, ,\ i = 1, 2, 3$ are the coefficients; and
 $\hat{u}$ is the residual (or error term).
To conduct the Ftest and obtain the Fstatistic (or Fvalue), we need to take the following steps:

State the hypothesis we want to test.
In our case, the null hypothesis $(H_0)$ 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 $(H_1)$ is that at least one of these coefficients is not equal to zero.
 where:
 $J$ is the number of restrictions (in the present case, $J=2$); and
 $K$ is the total number of coefficients (in the present case, $K = 3$).
 Now, to gain information on which model fits better, we need to obtain the sum square of residuals ($\text{SSR}$), where we expect that the sum square of residuals of the restricted model is larger than that of the full model (i.e. $\text{SSR}_R > \text{SSR}_F$).
 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. $\text{SSR}_R \gg \text{SSR}_F$). To do so, we need to apply the following Fstatistic formula to estimate the Fratio.
 where:
 $F$ – Fstatistic;
 $\text{SSR}_F$ – Sum square of residuals of the full model;
 $\text{SSR}_R$ – Sum square of residuals of the restricted model;
 $J$ – Number of restrictions;
 $K$ – Total number of coefficients; and
 $N$ – Number of observations representing the population.

Naturally, the larger the Fstatistic, the more evidence we have to reject the null hypothesis (note that the Fstatistic increases when the difference between the two variances gets larger). However, to be more precise, we need to find a critical value of the Fstatistic to decide on the rejection. In other words, if $F$ 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 Fvalue $(F^J_{NK;\alpha})$ in the F distribution table with a specified significance level Fstatistic $(\alpha)$ and looking for the intercept corresponding to the degrees of freedom, where $\text{df}_1 = J$ is at the top and $\text{df}_1 = NK$ is at the side of the table (we can also say that $F$ has an Fdistribution with $J$ and $N − K$ degrees of freedom). If $F$ is larger than its critical value, we can reject the null hypothesis.
So how to interpret Fstatistic in regression?
The Ftest 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 $\alpha$ is the significance level of the test. For example, if $N − K = 40$ and $J = 4$, the critical value at the 5% level is $F^J_{NK; \alpha} = 2.606$.
What is the difference between Ftest vs Ttest?
There are some differences between the Ftest vs a Ttest.

The Ttest is applied to test the significance of one explanatory variable, but the Ftest studies the whole model.

While the Ttest is used to compare the means of two populations, Ftest is applied for comparing two population variances.

The Tstatistic is based on the student tdistribution, while the Fstatistic follows the Fdistribution under the null hypothesis.

While the Ttest is a univariate hypothesis test where the standard deviation is unknown, the Ftest is applied to determine the equality of the two normal populations.
Can an Fstatistic be negative?
No. Since variances always take a positive value (squared values), both the numerator and the denominator of the Fstatistic formula must always be positive, resulting in a positive Fvalue.
What is a high Fstatistic?
While a large Fvalue tends to indicate that the null hypothesis can be rejected, you can confidently reject the null if the Tvalue is larger than its critical value.
Is the Fdistribution symmetric?
No. The curve of the Fdistribution 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 Fstatistic?
To calculate Fstatistic, in general, you need to follow the below steps.

State the null hypothesis and the alternate hypothesis.

Determine the Fvalue 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 Fstatistic as determined by Fstatistic = variance of the group means / mean of the withingroup variances.

Find the Fstatistic in the Ftable.

Support or reject the null hypothesis.
What is the Fstatistic of two populations with variances of 10 and 5?
The Fstatistic of two populations with variances of 10 and 5 is 2. To get this result, it suffices to divide the two variances.