# How to Perform an F-Test in R

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)

To perform an F-test in R, we can use the function var.test() with one of the following syntaxes:

• Method 1: var.test(x, y, alternative = “two.sided”)
• Method 2: var.test(values ~ groups, data, alternative = “two.sided”)

Note that alternative indicates the alternative hypothesis to use. The default is “two.sided” but you can specify it to be “left” or “right” instead.

This tutorial explains how to perform an F-test in R using both methods.

## Method 1: F-Test in R

The following code shows how to perform an F-test using the first method:

```#define the two groups
x <- c(18, 19, 22, 25, 27, 28, 41, 45, 51, 55)
y <- c(14, 15, 15, 17, 18, 22, 25, 25, 27, 34)

#perform an F-test to determine if the variances are equal
var.test(x, y)

F test to compare two variances

data:  x and y
F = 4.3871, num df = 9, denom df = 9, p-value = 0.03825
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
1.089699 17.662528
sample estimates:
ratio of variances
4.387122
```

The F test statistic is 4.3871 and the corresponding p-value is 0.03825. Since this p-value is less than .05, we would reject the null hypothesis.

This means we have sufficient evidence to say that the two population variances are not equal.

## Method 2: F-Test in R

The following code shows how to perform an F-test using the second method:

```#define the two groups
data <- data.frame(values=c(18, 19, 22, 25, 27, 28, 41, 45, 51, 55,
14, 15, 15, 17, 18, 22, 25, 25, 27, 34),
group=rep(c('A', 'B'), each=10))

#perform an F-test to determine in the variances are equal
var.test(values~group, data=data)

F test to compare two variances

data:  x and y
F = 4.3871, num df = 9, denom df = 9, p-value = 0.03825
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
1.089699 17.662528
sample estimates:
ratio of variances
4.387122
```

Once again the F test statistic is 4.3871 and the corresponding p-value is 0.03825. Since this p-value is less than .05, we would reject the null hypothesis.

This means we have sufficient evidence to say that the two population variances are not equal.

Related: Perform an F-test using this free F-Test for Equal Variances Calculator.

## When to Use the F-Test

The F-test is typically used to answer one of the following questions:

1. Do two samples come from populations with equal variances?

2. Does a new treatment or process reduce the variability of some current treatment or process?