# Two Sample Hypothesis Testing to Compare Variances

Theorem 1 of F Distribution can be used to test whether the variances of two populations are equal, using the Excel functions and tools which follows. In order to deal exclusively with the right tail of the distribution, when taking ratios of sample variances from the theorem we should put the larger variance in the numerator of In order to use this test, the following must hold:

• Both populations are normally distributed
• Both samples are drawn independently from each other.
• Within each sample, the observations are sampled randomly and independently of each other.

Excel Functions: The following Excel function can be used to carry out this test:

FTEST(R1, R2) = two-tailed F-test comparing the variances of the samples in ranges R1 and R2 = the two-tailed probability that the variance of the data in ranges R1 and R2 are not significantly different.

Thus FTEST(R1, R2) = 2 ∙ FDIST(x, df1, df2) where df1 = the number of elements in R1 – 1, df= the number of elements in R2 – 1 and x = var1 / var2 where var1 is the variance of the data in range R1 and var2 = the variance of the data in range R2. FTEST is a two-tail test, while FDIST and FINV are one-tailed.

Also FTEST(R1, R2) = FDIST(x, df1, df2) + FDIST(1/x, df2, df1), i.e. the sum of the right tail starting from x and the left tail starting from 1/x. This is true since FDIST(1/x, df2, df1) = 1 − FDIST(x, df1, df2).

This function ignores all empty and non-numeric cells.

Excel 2010/2013 also provide a new function F.TEST which is equivalent to FTEST (see Built-in Statistical Functions).

In addition Excel provides an F-Test Two-Sample for Variances data analysis tool which automates the process of comparing two variances.

Example 1: A company is comparing methods for producing pipes and wants to choose the method with the least variability. It has taken a sample of the lengths of the pipes using both methods as shown on the left side of Figure 1. Figure 1 – Excel’s two sample F-test to compare variances

We test the following null hypothesis:

H0σ1 – σ2 = 0 (equivalently: σ1 = σ2; i.e. both methods have the same variability)

and use the statistic with 11, 14 degrees of freedom, as described on the right side of Figure 1. Since this is a two-tail test, we note that

p-value = 2 * FDIST(F, df1, df2) = 2 * FDIST(1.85, 11, 14) = 0.279 > 0.05 = α

F-crit = FINV(α/2, df1, df2) = FINV(.025, 11, 14) = 3.09 > 1.85 = F

Either of the above tests shows there is no significant difference in the variance between the two methods with 95% confidence. Note that we needed to double the value for FDIST or halve α since this is a two-tail test.

Alternatively we can use FTEST which is a two-tail test:

FTEST(A4:A18, B4:B18) = .279 > 0.05 = α

We can also use the F-Test Two-Sample for Variances data analysis tool: Figure 2 – Comparing variances using Excel’s data analysis tool

This tool only performs a one-tail test, and so the p-value (0.1393) needs to be doubled to get 0.279, which is the same value we calculated in Figure 1. The critical value for F is calculated based on α = .05.

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