Probability and Statistics > Critical Values & Hypothesis Testing > F Test to Compare Two Variances

### What is an F Test to Compare Two Variances?

A **Statistical F Test** uses an F Statistic to compare two variances, s_{1} and s_{2}, by dividing them. The result is always a positive number (because variances are always positive). The equation for comparing two variances with the f-test is:

F = s^{2}_{1} / s^{2}_{2}

## F Test to compare two variances by hand: Steps

**Warning**: F tests can get really tedious to calculate by hand, especially if you have to calculate the variances. You’re much better off using technology (like Excel — see below).

**These are the general steps to follow. Scroll down for a specific example (watch the video underneath the steps).**

**Step 1**: *If you are given standard deviations, go to Step 2. If you are given variances to compare, go to Step 3.*

**Step 2:** *Square both standard deviations to get the variances.* For example, if σ_{1} = 9.6 and σ_{2} = 10.9, then the variances (s_{1} and s_{2}) would be 9.6^{2} = **92.16** and 10.9^{2} = **118.81**.

**Step 3:***Take the largest variance, and divide it by the smallest variance to get the f-value.* For example, if your two variances were s_{1} = 2.5 and s_{2} = 9.4, divide 9.4 / 2.5 = **3.76**.

Why? Placing the largest variance on top will force the F-test into a right tailed test, which is much easier to calculate than a left-tailed test.

**Step 4:**Find your degrees of freedom. Degrees of freedom is your sample size minus 1. As you have two samples (variance 1 and variance 2), you’ll have two degrees of freedom: one for the numerator and one for the denominator.

**Step 5:***Look the f-value you calculated in Step 3 in the f-table. *Note that there are several f-tables, so you’ll need to locate the right table for your alpha level. Unsure how to read an f-table? Read What is an f-table?.

**Step 4:**Compare your calculated value (Step 3) with the table f-value in Step 5. If the f-table value is smaller than the calculated value, you can reject the null hypothesis.

*That’s it!*

## F Test to Compare Two Variances in Excel

If you have Excel on your computer, you can use it to run an F Test to Compare Two Variances. If you have to do the calculations by hand, you can use Excel to check your work. Check out this short video:

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I am having a REALLY hard time finding the tables for the f test. I have incomplete tables for a=0.05 and a=0.01, but its obviously not enough. Where can I find full tables, including ones for a=0.10 or a=0.001?

Those tables should all be on the disk that came with the textbook.

is there any other place to find them?

This blog is really helpful, but for once the mathzone explained how to do the chapter 8 problems in the second homework.

I’m still not understanding how to tell when it’s a one-tailed or two-tailed test for the F distribution. I have tried to compare the questions to see if there is a difference in wording, but there doesn’t seem to be one.

Stephanie N, did you get the complete tables? I have them from mathzone – appendix. Daniel.

Phil,

Have you tried this?

http://www.statisticshowto.com/articles/how-to-decide-if-a-hypothesis-test-is-a-one-tailed-test-or-a-two-tailed-test/

Stephanie

Even after looking at that site, I’m still confused about left tail, right tail, one tail, two tail. I feel like Dr. Seuss

I had a hard time too, but once I found it I labeled it so that way I remember that it was used for this chapter, but I’m still confused about the tails, how do you know when you need to consider it a left tail and a right tail and let’s not go to when its two tail. I know when its two tails can the section that you right your answer will would have a negative and positive sign in it.

I really do not understand this, now that I have the value, what do I do with it?

Nevermind, I think I may have finally figured it out. :)

A simple way to distinguish one-tailed and two-tailed tests.

Ha tells you if the test is one-tailed or two-tailed.

Let PP1=population parameter 1; PP2=population parameter 2.

a) If PP1>PP2, this is right tailed. It becomes left-tailed if you rewrite it as PP2<PP1.

b) If PP1PP1.

c) If PP1≠PP2, this is two tailed. This is because it means that either PP1>PP2 or PP1<PP2, which are two alternatives, one in each tail.

In the tests of variance, think of PP1=variance 1, PP2=variance2.

Hope this helps.

How is that test (comparing two variances) done with two time series?

Hi, Janne,

Could you post your question on our forum? One of our mods will be glad to answer your question.

Regards,

Stephanie