# 1.3.5.9. F-Test for Equality of Two Variances (2023)

## 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.DATdata set. The data set contains 480 ceramic strength measurementsfor two batches of material. The summary statistics for each batchare 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 forthe two batches are equal.
H0: σ12 = σ22 Ha: σ12 ≠ σ22 Test statistic: F = 1.123037Numerator degrees of freedom: N1 - 1 = 239Denominator degrees of freedom: N2 - 1 = 239Significance level: α = 0.05Critical values: F(1-α/2,N1-1,N2-1) = 0.7756 F(α/2,N1-1,N2-1) = 1.2894Rejection 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.

## FAQs

### What are F-tests for equality of two variances? ›

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.

How do you interpret F-tests to compare two variances? ›

F Test to Compare Two Variances

If the variances are equal, the ratio of the variances will equal 1. For example, if you had two data sets with a sample 1 (variance of 10) and a sample 2 (variance of 10), the ratio would be 10/10 = 1. You always test that the population variances are equal when running an F Test.

What does F value mean in equality of variances? ›

An F-value is the ratio of two variances, or technically, two mean squares. Mean squares are simply variances that account for the degrees of freedom (DF) used to estimate the variance. F-values are the test statistic for F-tests.

What is F in Levene's test for equality of variances? ›

A Levene's Test for Equality of of Variances: This section has the test results for Levene's Test. From left to right: F is the test statistic of Levene's test. Sig. is the p-value corresponding to this test statistic.

What does the F-test value mean? ›

The F value is a value on the F distribution. Various statistical tests generate an F value. The value can be used to determine whether the test is statistically significant. The F value is used in analysis of variance (ANOVA). It is calculated by dividing two mean squares.

What does a significant F-test in an analysis of variance indicate? ›

The F-test of overall significance indicates whether your linear regression model provides a better fit to the data than a model that contains no independent variables.

How do I know if my F-test is significant? ›

You perform the F test by looking for the appropriate p-value in the computer analysis and interpreting the resulting significance level, as we did in Chapter 10. If the p-value is more than 0.05, then the result is not significant. If the p-value is less than 0.05, then the result is significant.

How can you tell if an F-test is significant? ›

If the F statistic is higher than the critical value (the value of F that corresponds with your alpha value, usually 0.05), then the difference among groups is deemed statistically significant.

What is a good F value in ANOVA? ›

The F ratio is the ratio of two mean square values. If the null hypothesis is true, you expect F to have a value close to 1.0 most of the time.

Is the Levene's F-test significant or insignificant? ›

However, Levene's test is statistically significant because its p < 0.05: we reject its null hypothesis of equal population variances.

### How do I report Levene's F-test? ›

To report the results of the Levene's test in APA format, it might look something like this: There was homogeneity of variances, as assessed by the Levene's test for equality of variances, for competence, p > . 05. However, homogeneity of variances was violated for likeability, p < .

What is F Levene's test? ›

Levene's test ( Levene 1960) is used to test if k samples have equal variances. Equal variances across samples is called homogeneity of variance. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples. The Levene test can be used to verify that assumption.

What is a good F-test score? ›

An F statistic of at least 3.95 is needed to reject the null hypothesis at an alpha level of 0.1. At this level, you stand a 1% chance of being wrong (Archdeacon, 1994, p.168). For more details on how to do this, see: F Test. F Values will range from 0 to an arbitrarily large number.

What is a high F-test? ›

The high F-value graph shows a case where the variability of group means is large relative to the within group variability. In order to reject the null hypothesis that the group means are equal, we need a high F-value.

Is a high or low F-test better? ›

The higher the F value, the better the model.

When the F-test value is close to 1 the null hypothesis should be rejected? ›

If the null hypothesis is false, then we will reject the null hypothesis that the ratio was equal to 1 and our assumption that they were equal.

Can F value be less than 1? ›

If the F-score is less than one, or not much greater than one, the variance between the samples is no greater than the variance within the samples and the samples probably come from populations with the same mean.

What does it mean if the obtained F value is close to 1 in ANOVA? ›

The F-distribution is used to quantify this likelihood for differing sample sizes and the confidence or significance we would like the answer to hold. A value of F=1 means that no matter what significance level we use for the test, we will conclude that the two variances are equal.

What does a negative F value mean in ANOVA? ›

The value of FIS ranges between -1 and +1. Negative FIS values indicate heterozygote excess (outbreeding) and positive values indicate heterozygote deficiency (inbreeding) compared with HWE expectations.

Is the F-test the same as Levene's test? ›

2 tests are commonly used to check for homogeneity of variance: Fisher's F test and Levene's test. Fisher's F test, which is introduced here, is restricted to comparison of two variances/groups while Levene's test can assess more than two variances/groups.

### How do you know if Levene's test is not significant? ›

If the Levene's Test is significant (the value under "Sig." is less than . 05), the two variances are significantly different. If it is not significant (Sig. is greater than . 05), the two variances are not significantly different; that is, the two variances are approximately equal.

What is the t-value at 5 level of significance? ›

A significance level of (for example) 0.05 indicates that in order to reject the null hypothesis, the t-value must be in the portion of the t-distribution that contains only 5% of the probability mass.

What does it mean if Levene's test is significant? ›

Levene's test is often used before a comparison of means. When Levene's test is significant, modified procedures are used that do not assume equality of variance. Levene's test may also test a meaningful question in its own right if a researcher is interested in knowing whether population group variances are different.

What is the Levene's test p-value? ›

The p-value reported for Levene's Test for Equality of Variance in the table above is p = 0.000, which is well below the 0.05 threshold. So, we can say that “equal variance is not assumed” for this sample and go on to check the significance level reported in the t test for Equality of Means section.

What if Levene's test is significant independent t-test? ›

If Levene's test is significant, this means that the two groups did not show homogeneity of variance on the dependent or outcome variable. In our example, Levene's test is nonsignificant so we can move on to the statistics for the tests under the condition of equal variances assumed.

Is Levene's statistic F? ›

The Levene's test uses an F-test to test the null hypothesis that the variance is equal across groups. A p value less than . 05 indicates a violation of the assumption. If a violation occurs, it is likely that conducting the non-parametric equivalent of the analysis is more appropriate.

What is F in homogeneity of variance? ›

In statistics, an F-test of equality of variances is a test for the null hypothesis that two normal populations have the same variance.

What does an F-test tell us about the relationship of our population variances? ›

The F-test is designed to test if two population variances are equal. It does this by comparing the ratio of two variances. So, if the variances are equal, the ratio of the variances will be 1. If the null hypothesis is true, then the F test-statistic given above can be simplified (dramatically).

What are F tests and t tests for? ›

The F-test compares the population variances while the t-test compares the population means. 1.3 Select all samples using random or stratified random procedures.

What is the F-test for 2 way ANOVA? ›

The F test is a groupwise comparison test, which means it compares the variance in each group mean to the overall variance in the dependent variable.

### What is F-test formula? ›

The f test statistic formula is given below: F statistic for large samples: F = σ21σ22 σ 1 2 σ 2 2 , where σ21 σ 1 2 is the variance of the first population and σ22 σ 2 2 is the variance of the second population.

What is considered a high F value? ›

A large F-value means the between-group variation is larger than your within-group variation. This can be interpreted to mean there is a statistically significant difference in your group means.

Is a high F value good? ›

If you get a large f value (one that is bigger than the F critical value found in a table), it means something is significant, while a small p value means all your results are significant. The F statistic just compares the joint effect of all the variables together.

Can F statistic be less than 1? ›

If the F-score is less than one, or not much greater than one, the variance between the samples is no greater than the variance within the samples and the samples probably come from populations with the same mean.

What is the relationship between T value and F value? ›

It is often pointed out that when ANOVA is applied to just two groups, and when therefore one can calculate both a t-statistic and an F-statistic from the same data, it happens that the two are related by the simple formula: t2 = F.

What is the F ratio in ANOVA? ›

The F ratio is the ratio of two mean square values. If the null hypothesis is true, you expect F to have a value close to 1.0 most of the time. A large F ratio means that the variation among group means is more than you'd expect to see by chance.

Which is better t-test or F-test? ›

The F-test can be applied on the large sampled population. The T-test is used to compare the means of two different sets. It says whether the mean of one group is significantly different from the other group. T-test can be either paired and normal.

How to interpret F-test results? ›

You perform the F test by looking for the appropriate p-value in the computer analysis and interpreting the resulting significance level, as we did in Chapter 10. If the p-value is more than 0.05, then the result is not significant. If the p-value is less than 0.05, then the result is significant.

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