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Wize University Statistics Textbook > Test of Equality of Two Variances

Levene’s Test for the Equality of Two Variances

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Levene’s Test for the 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
We use this test the equality of variances for two or more groups.

  • The Levene’s Test is preferred over the F-test because it may not be appropriate to assume that the two population populations are normal.
  • The F-test is not robust because it uses means (appropriate for normal populations). The Levene’s Test is robust because it uses medians (appropriate for non-normal populations).

Hypotheses:

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

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


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Steps to Conduct Levene's Test
  1. Find the sample medians X~i\tilde{X}_i for each group.
  2. Subtract the associated median for the data for each sample.
  3. Take the absolute values from all the differences. Then solve for the means of the absolute values for each sample.
  4. Perform a two-tailed pooled t-test to compare the means.
  5. Find the p-value based on the t-test statistic and df = n1 + n2 − 2.
  6. Draw your conclusion.


Exam Tip
It is impossible to conduct a Levene's Test with raw data without access to software. If you are given software output, try to find the p-value to draw your conclusion.

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Example: Levene’s Test for the Equality of Two Variances

We wish to compare variances of women and men. We cannot assume that the populations are normal. Conduct a Levene's with a 10% significance level.
  • The median for women is 105.5.
  • The median for men is 69.5.
  • There are 34 women and 26 men in the respective samples.


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Using the pooled/equal variances t-test method:

xˉ1 =37.35, xˉ2=46.19x̄_1\ =37.35,\ x̄_2=46.19
s1=24.31 , s2=28.55s_1=24.31\ ,\ s_2=28.55
n1=34 , n2=26n_1=34\ ,\ n_2=26

t=x‾1−x‾2sp1n1+1n2\boxed{t=\frac{\overline{x}_1-\overline{x}_2}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}}

sp=(n1−1)s12+(n2−1)s22n1+n2−2s_p=\sqrt{\frac{\left(n_1-1\right)s_1^2+\left(n_2-1\right)s_2^2}{n_1+n_2-2}}

sp=(34−1)(24.31)2 +(26−1)(28.55)234+26−2=26.2218s_p=\sqrt{\frac{\left(34-1\right)\left(24.31\right)^{2\ }+\left(26-1\right)\left(28.55\right)^2}{34+26-2}}=26.2218

Plug sps_p into the tt formula:
t=37.35−46.19(26.2218)134+126=−1.29t=\frac{37.35-46.19}{\left(26.2218\right)\sqrt{\frac{1}{34}+\frac{1}{26}}}=-1.29

Degrees of freedom:
df=n1+n2−2df=n_1+n_2-2

df=n1+n2−2 =34+26−2=58df=n_1+n_2-2\ =34+26-2=58


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Find the range of p-values. (Use the closest available dfdf provided in the table.)


Pooled variance (t=-1.29, df=58 →\rightarrow50) *More conservative to round down the df.*
  • From t-table, 0.2<0.2< p-value <0.3<0.3
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Draw your conclusion. Based on the Levene's T-test, is there evidence that the variances are different?

p-value >0.10 →>0.10\ \rightarrow Fail to reject HoH_o.
  • At the 10% level of significance, we do not have enough evidence to reject HoH_o.
  • There is no evidence that the variances are not equal.


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Minitab for 2-sample pooled t-test:



  • p-value = 0.201
  • The p-value is large>0.1→>0.1\rightarrow Fail to reject HoH_o
  • There is no evidence that the variances are not equal.
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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Minitab for Levene's Test for Two Variances:

  • p-value = 0.201 (same p-value as the 2-sample t-test!)
  • The p-value is large>0.1→>0.1\rightarrow Fail to reject HoH_o
  • There is no evidence that the variances are not equal.

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: Levene’s Test for the Equality of Two Variances

We want to test if the variances in Group 1 and Group 2 are equal or not. We cannot assume that both population distributions are normal.

Raw data:

(a) State the hypotheses.






(b) Refer to the software output below. At the 10% significance level, is the evidence that the variances differ? (Hint: find the p-value.)



Note: The above Minitab output is for a Hypothesis Test for Two Variances without the assumption that the populations are normal.


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.