To determine if the variances of two populations are equal, we can calculate the variance ratio **σ ^{2}_{1} / σ^{2}_{2}**, where σ

^{2}

_{1}is the variance of population 1 and σ

^{2}

_{2}is the variance of population 2.

To estimate the true population variance ratio, we typically take a simple random sample from each population and calculate the sample variance ratio, **s _{1}^{2} / s_{2}^{2}**, where s

_{1}

^{2}and s

_{2}

^{2}are the sample variances for sample 1 and sample 2, respectively.

This test assumes that both s_{1}^{2} and s_{2}^{2} are computed from independent samples of size n_{1} and n_{2}, both drawn from normally distributed populations.

The further this ratio is from one, the stronger the evidence for unequal population variances.

**The (1-α)100% confidence interval** for σ^{2}_{1} / σ^{2}_{2} is defined as:

(s_{1}^{2} / s_{2}^{2}) * F_{n1-1, n2-1, α/2 }≤ σ^{2}_{1} / σ^{2}_{2} ≤ (s_{1}^{2} / s_{2}^{2}) * F_{n2-1, n1-1, }_{α/2}

where F_{n2-1, n1-1, α/2 }and F_{n1-1, n2-1, }_{α/2}_{ }are the critical values from the F distribution for the chosen significance level α.

The following examples illustrate how to create a confidence interval for σ^{2}_{1} / σ^{2}_{2} using three different methods:

- By hand
- Using Microsoft Excel
- Using the statistical software
*R*

For each of the following examples, we will use the following information:

**α**= 0.05**n**= 16_{1}**n**= 11_{2}**s**=28.2_{1}^{2}**s**= 19.3_{2}^{2}

**Creating a Confidence Interval By Hand**

To calculate a confidence interval for **σ ^{2}_{1} / σ^{2}_{2}** by hand, we’ll simply plug in the numbers we have into the confidence interval formula:

(s_{1}^{2} / s_{2}^{2}) * F_{n1-1, n2-1,α/2 }≤ σ^{2}_{1} / σ^{2}_{2} ≤ (s_{1}^{2} / s_{2}^{2}) * F_{n2-1, n1-1, }_{α/2}

The only numbers we’re missing are the critical values. Luckily, we can locate these critical values in the F distribution table:

F_{n2-1, n1-1, α/2 }= F_{10, 15, 0.025 }= **3.0602**

F_{n1-1, n2-1, }_{α/2 }= 1/ F_{15, 10, 0.025} = 1 / 3.5217 = **0.2839**

*(Click to zoom in on the table)*

Now we can plug all of the numbers into the confidence interval formula:

(s_{1}^{2} / s_{2}^{2}) * F_{n1-1, n2-1,α/2 }≤ σ^{2}_{1} / σ^{2}_{2} ≤ (s_{1}^{2} / s_{2}^{2}) * F_{n2-1, n1-1, }_{α/2}

(28.2 / 19.3) * (0.2839) ≤ σ^{2}_{1} / σ^{2}_{2} ≤ (28.2 / 19.3) * (3.0602)

0.4148 ≤ σ^{2}_{1} / σ^{2}_{2} ≤ 4.4714

Thus, the 95% confidence interval for the ratio of the population variances is **(0.4148, 4.4714)**.

**Creating a Confidence Interval Using Excel**

The following image shows how to calculate a 95% confidence interval for the ratio of population variances in Excel. The lower and upper bounds of the confidence interval are displayed in column E and the formula used to find the lower and upper bounds are displayed in column F:

Thus, the 95% confidence interval for the ratio of the population variances is **(0.4148, 4.4714)**. This matches what we got when we calculated the confidence interval by hand.

**Creating a Confidence Interval Using R**

The following code illustrates how to calculate a 95% confidence interval for the ratio of population variances in R:

#define significance level, sample sizes, and sample variances alpha <- .05 n1 <- 16 n2 <- 11 var1 <- 28.2 var2 <- 19.3 #define F critical values upper_crit <- 1/qf(alpha/2, n1-1, n2-1) lower_crit <- qf(alpha/2, n2-1, n1-1) #find confidence interval lower_bound <- (var1/var2) * lower_crit upper_bound <- (var1/var2) * upper_crit #output confidence interval paste0("(", lower_bound, ", ", upper_bound, " )") #[1] "(0.414899337980266, 4.47137571035219 )"

Thus, the 95% confidence interval for the ratio of the population variances is **(0.4148, 4.4714)**. This matches what we got when we calculated the confidence interval by hand.

**Additional Resources**

**How to Read the F-Distribution Table
How to Find the F Critical Value in Excel**