# F-Test vs. T-Test: What’s the Difference?

Two statistical tests that students often get mixed up are the F-Test and the T-Test. This tutorial explains the difference between the two tests.

### F-Test: The Basics

An F-test is used to test whether two population variances are equal. The null and alternative hypotheses for the test are as follows:

H0: σ12 = σ22 (the population variances are equal)

H1: σ12 ≠ σ22 (the population variances are not equal)

The F test statistic is calculated as s12 / s22.

If the p-value of the test statistic is less than some significance level (common choices are 0.10, 0.05, and 0.01), then the null hypothesis is rejected.

Example: F-Test for Equal Variances

A researcher wants to know if the variance in height between two species of plants is the same. To test this, she collects a random sample of 20 plants from each population and calculates the sample variance for each sample.

The F test statistic turns out to be 4.38712 and the corresponding p-value is 0.0191. Since this p-value is less than .05, she rejects the null hypothesis of the F-Test. This means she has sufficient evidence to say that the variance in height between the two plant species is not equal.

### T-Test: The Basics

two sample t-test is used to test whether or not the means of two populations are equal.

A two-sample t-test always uses the following null hypothesis:

• H0: μ1 = μ2 (the two population means are equal)

The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed:

• H1 (two-tailed): μ1 ≠ μ2 (the two population means are not equal)
• H1 (left-tailed): μ1 < μ2 (population 1 mean is less than population 2 mean)
• H1 (right-tailed): μ1> μ2 (population 1 mean is greater than population 2 mean)

The test statistic is calculated as:

Test statistic: (x1 – x2)  /  sp(√1/n1 + 1/n2)

where x1 and x2 are the sample means, nand nare the sample sizes, and where sp is calculated as:

sp = √ (n1-1)s12 +  (n2-1)s22 /  (n1+n2-2)

where s12 and s22 are the sample variances.

If the p-value that corresponds to the test statistic t with (n1+n2-1) degrees of freedom is less than your chosen significance level (common choices are 0.10, 0.05, and 0.01) then you can reject the null hypothesis.

Example: Two Sample t-test

A researcher wants to know if the mean height between two species of plants is equal. To test this, she collects a random sample of 20 plants from each population and calculates the sample mean for each sample.

The t test statistic turns out to be 1.251 and the corresponding p-value is 0.2148. Since this p-value is not less than .05, she fails to reject the null hypothesis of the T-Test. This means she does not have sufficient evidence to say that the mean heights between these two plant species is different.

## F-Test vs. T-Test: When to Use Each

We typically use an F-test to answer the following questions:

• Do two samples come from populations with equal variances?
• Does a new treatment or process reduce the variability of some current treatment or process?

And we typically use a T-test to answer the following questions:

• Are two population means equal? (We use a two sample t-test to answer this)
• Is one population mean equal to a certain value? (We use a one sample t-test to answer this)