# How to Conduct an F-Test in R to Compare Two Variances An F-Test is used to test if the variances of two populations are equal.

The two-tailed version tests whether or not the variances are equal, while the one-tailed version tests in one direction, that is the variance from the first population is either greater or less than (but not both) the variance fro the second population.

An F-Test is commonly used to answer the following questions:

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

2. Does a new process/treatment have lower variation than the current process/treatment?

## Defining the F Test

The two hypotheses for the F-Test are as follows:

H0 (null hypothesis):
σ12 = σ2(the two population variances are equal)

HA (alternative hypothesis):
σ12 < σ22      (lower one-tailed test)
σ12 > σ22      (upper one-tailed test)
σ12 ≠ σ22      (two-tailed test)

The test statistic for the F-Test is defined as follows:

F-statistic = s12 / s22

where s12  and s22 are the sample variances. The further this ratio is from one, the stronger the evidence for unequal population variances.

The critical value for the F-Test is defined as follows:

F Critical Value = Fα, N1-1, N1-1 from the F-distribution table with N1-1 and N2-1 degrees of freedom and a significance level of α.

## Conducting an F-Test in R

The built-in R function var.test() can be used to compare two variances using the following syntax:

var.test(x, y, ratio = 1, alternative = c(“two.sided”, “less”, “greater”), conf.level = 0.95, …)

• x, y – numeric vectors
• ratio – hypothesized ratio of the population variances of x and y (default is 1)
• alternative – the alternative hypothesis of the test (default is “two.sided”)
• conf.level – optional confidence level for the test (default is 0.95)

The following code illustrates how to conduct an F-Test for two samples and y:

```#create two vectors x and y
x <- rnorm(n = 100, mean = 1, sd = 2.7)
y <- rnorm(n = 100, mean = 1, sd = 2)

#conduct two.sided F-test to test for equality of variances
var.test(x, y)

#	F test to compare two variances
#
#data:  x and y
#F = 1.6294, num df = 99, denom df = 99, p-value = 0.01592
#alternative hypothesis: true ratio of variances is not equal to 1
#95 percent confidence interval:
# 1.096316 2.421641
#sample estimates:
#ratio of variances
#          1.629381

```

We can see that the F-statistic for the test is 1.6294 and the corresponding p-value is 0.01592.

Since the p-value is less than our significance level of 0.05, we have sufficient evidence to reject the null hypothesis and say that the difference between the two variances is statistically significant.