1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.5. Quantitative Techniques

1.3.5.9.

F-Test for Equality of Two Variances

Purpose:
Test if variances from two populations are equal

An F-test (Snedecor and Cochran, 1983) is used to test if the variances of two populations are equal. This test can be a two-tailed test or a one-tailed test. The two-tailed version tests against the alternative that the variances are not equal. The one-tailed version only tests in one direction, that is the variance from the first population is either greater than or less than (but not both) the second population variance. The choice is determined by the problem. For example, if we are testing a new process, we may only be interested in knowing if the new process is less variable than the old process.

Definition

The F hypothesis test is defined as:

H0:	\( \sigma_{1}^{2} \) = \( \sigma_{2}^{2} \)
Ha:	\( \sigma_{1}^{2} < \sigma_{2}^{2} \)     for a lower one-tailed test \( \sigma_{1}^{2} > \sigma_{2}^{2} \)     for an upper one-tailed test \( \sigma_{1}^{2} \ne \sigma_{2}^{2} \)     for a two-tailed test
Test Statistic:	F = \( s^{2}_{1}/s^{2}_{2} \) where \({s^{2}_{1}}\) and \({s^{2}_{2}}\) and are the sample variances. The more this ratio deviates from 1, the stronger the evidence for unequal population variances.
Significance Level:	α
Critical Region:	The hypothesis that the two variances are equal is rejected if \( F > F_{\alpha,N_1 - 1,N_2 - 1} \)    for an upper one-tailed test \( F < F_{1 - \alpha,N_1 - 1,N_2 - 1} \)    for a lower one-tailed test \( F < F_{1 - \alpha/2,N_1 - 1,N_2 - 1} \) or \( F > F_{\alpha/2,N_1 - 1,N_2 - 1} \)    for a two-tailed test where Fα, N1-1, N2-1 is the critical value of the F distribution with N1-1 and N2-1 degrees of freedom and a significance level of α. In the above formulas for the critical regions, the Handbook follows the convention that Fα is the upper critical value from the F distribution and F1-α is the lower critical value from the F distribution. Note that this is the opposite of the designation used by some texts and software programs.

F Test Example

The following F-test was generated for the JAHANMI2.DAT data set. The data set contains 480 ceramic strength measurements for two batches of material. The summary statistics for each batch are shown below.

BATCH 1:
   NUMBER OF OBSERVATIONS      =      240
   MEAN                        =    688.9987
   STANDARD DEVIATION          =    65.54909
  
BATCH 2:
   NUMBER OF OBSERVATIONS      =      240
   MEAN                        =    611.1559
   STANDARD DEVIATION          =    61.85425

We are testing the null hypothesis that the variances for the two batches are equal.

H0:  σ12 = σ22 
Ha:  σ12 ≠ σ22 

Test statistic:  F = 1.123037
Numerator degrees of freedom:  N1 - 1 = 239
Denominator degrees of freedom:  N2 - 1 = 239
Significance level:  α = 0.05
Critical values:  F(1-α/2,N1-1,N2-1) = 0.7756
                  F(α/2,N1-1,N2-1) = 1.2894
Rejection region:  Reject H0 if F < 0.7756 or F > 1.2894

The F test indicates that there is not enough evidence to reject the null hypothesis that the two batch variancess are equal at the 0.05 significance level.

Questions

The F-test can be used to answer the following questions:

1. Do two samples come from populations with equal variancess?
2. Does a new process, treatment, or test reduce the variability of the current process?

Related Techniques

Quantile-Quantile Plot
Bihistogram
Chi-Square Test
Bartlett's Test
Levene Test

Case Study

Ceramic strength data.

Software

The F-test for equality of two variances is available in many general purpose statistical software programs. Both Dataplot code and R code can be used to generate the analyses in this section. These scripts use the AUTO83B.DAT data file.