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Wize University Statistics Textbook > Test of Equality of Two Variances

F-Test for Equality of Two Variances

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F-Test for Equality of Two Variances

Purpose: To test if two population variances are equal.

Assumptions:
  • If two populations are normal →\rightarrow F-test
  • If two populations are not normal →\rightarrow Levene’s Test

Watch Out!
When we ask "Are the two populations normal?", it means the samples are both drawn from normal population distributions. Central Limit Theorem does not apply because having a large sample only makes the sampling distribution normal but it doesn't change the fact that the population distributions are normal or not.


Hypotheses:

Ho:σ12=σ22H_o:\sigma_1^2=\sigma_2^2 “The two populations variances are equal.”

Ha:σ12≠σ22H_a:\sigma_1^2\ne\sigma_2^2 “The two populations variances are not equal.”



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Test Statistic:
F=sL2sS2\boxed{F=\frac{s_L^2}{s_S^2}}
Important: The larger sample variance sL2s_L^2 should be put in the numerator, and the smaller sample variance ss2s_s^2 is put in the denominator. This forces the F-test to always be a right-sided test, which makes things a lot easier.

If the variances are equal sL2=sS2s_L^2=s_S^2, then F=1F=1.

Wize Tip
If you are given sample standard deviations instead (i.e. sLs_L and sss_s), then be sure to square them both when solving for the F-stat:

F=(sL)2(sS)2\boxed{F=\frac{(s_L)^2}{(s_S)^2}}



The F-test has two degrees of freedom:
  • numerator  df=nL−1numerator\ \ df=n_L-1
  • denominator df=ns−1denominator\ df=n_s-1
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P-value method:
  • If you are able to solve for the p-value using software, ensure that the p-value is doubled for two-tailed tests.
  • In order to reject HoH_o, the p-value needs to be smaller than the significance level α\alpha.
Critical value method:
  • Without software, use to F-table to find the critical value.
  • For two-tailed tests, divide α\alpha by 2 to find the critical value Fα2,df1,df2F_{\frac{\alpha}{2},df_1,df_2}.
  • In order to reject HoH_o, the test statistic needs to be bigger than the critical value.
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Example

Ho:σ12=σ22H_o:\sigma_1^2=\sigma_2^2 “The two populations variances are equal.”
Ha:σ12≠σ22H_a:\sigma_1^2\ne\sigma_2^2 “The two populations variances are not equal.”

α=0.10\alpha=0.10
  • numerator  df=9numerator\ \ df=9
  • denomin⁡ator df=20deno\min ator\ df=20
Find the critical value.
This is a two-tailed test, so α2=0.102=0.05\frac{\alpha}{2}=\frac{0.10}{2}=0.05
We will therefore use the 5% F-table.
CV: F0.05,9,20=2.393F_{0.05,9,20}=2.393
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Example: F-Test for Equality of Two Variances

We want to test if the variances in Group 1 and Group 2 are equal or not. Assume both population distributions are normal. The sample standard deviation in Group 1 is s1=4.764s_1=4.764 and the sample standard deviation in Group 2 is s2=4.116s_2=4.116.

Raw data:

(i) State the hypotheses.

Ho:σ12=σ22H_o:\sigma_1^2=\sigma_2^2 “The two populations variances are equal.”

Ha:σ12≠σ22H_a:\sigma_1^2\ne\sigma_2^2 “The two populations variances are not equal.”


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(ii) α=0.10\alpha=0.10. Find the critical value.
This is a two-tailed test, so α2=0.102=0.05\frac{\alpha}{2}=\frac{0.10}{2}=0.05

We will therefore use the 5% F-table.

You need the degrees of freedoms first.

n1=9n_1=9
n2=9n_2=9

numerator  df=nL−1=9−1=8numerator\ \ df=n_L-1=9-1=8
denomin⁡ator df=ns−1=9−1=8deno\min ator\ df=n_s-1=9-1=8
Critical value F0.05,8,8=3.438F_{0.05,8,8}=3.438

In order to reject HoH_o, the test statistic needs to be bigger than the critical value (3.438).

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(iii) Find the test statistic.
F=sL2sS2=(4.764)2(4.116)2=22.6916.94=1.34F=\frac{s_L^2}{s_S^2}=\frac{\left(4.764\right)^2}{\left(4.116\right)^2}=\frac{22.69}{16.94}=1.34

(iv) Refer to the software output below. Is the evidence that the variances differ?

P-value method:

P-value (0.689) > alpha (0.10) →\rightarrow Fail to reject HoH_o. There is no evidence that the variances differ.

Critical value method:

Test statistic (1.34) < critical value (3.438) →\rightarrow Fail to reject HoH_o. There is no evidence that the variances differ.


Minitab Output:


Note: Since this is a two-tailed test, Minitab had doubled the p-value.

Portions of information contained in this publication/book are printed with permission of Minitab, LLC. All such material remains the exclusive property and copyright of Minitab, LLC. All rights reserved.


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Practice: F-Test for Equality of Two Variances

A random sample of 26 women and 34 men were selected. The time spend at the mall (in minutes) are recorded.

Descriptive statistics:


(a) Describe the two distributions.

(b) Looking at the boxplots, what can you say about the variation of time spend at the mall?

(c) Conduct a test for the equality of the two population variances (α=0.05\alpha=0.05). State the hypotheses and draw your conclusion.

Useful information from Minitab:

Sample 1: Men
Sample 2: Women


Note: Since this is a two-tailed test, Minitab had doubled the p-value.

(d) Which t-test should we do: pooled or non-pooled?


Portions of information contained in this publication/book are printed with permission of Minitab, LLC. All such material remains the exclusive property and copyright of Minitab, LLC. All rights reserved.