The F-test for variances is used to decide whether two populations appear to have the same spread. It compares two sample variances by forming a ratio, so it is useful when variability itself is the question. This matters in science, engineering, manufacturing, and statistics because two processes can have the same average but very different consistency.
The test helps answer whether an observed difference in sample variance is large enough to be unlikely from random sampling alone.
The test statistic is F = s1^2 / s2^2, where s1^2 and s2^2 are sample variances from two independent samples. Under the null hypothesis that the population variances are equal, this ratio follows an F-distribution with degrees of freedom df1 = n1 - 1 and df2 = n2 - 1. Because variances are always nonnegative, F-values are always positive and the F-distribution is right-skewed.
A worked example usually places the larger sample variance in the numerator, computes F, then compares it with a critical value or p-value.
Key Facts
- Test statistic: F = s1^2 / s2^2.
- Null hypothesis for equal variances: H0: sigma1^2 = sigma2^2.
- Degrees of freedom: df1 = n1 - 1 and df2 = n2 - 1.
- If the larger sample variance is placed in the numerator, then F >= 1.
- The F-test assumes independent random samples from populations that are approximately normal.
- For a two-tailed test at significance level alpha, reject H0 if F is too large or too small compared with F critical values.
Vocabulary
- Variance
- Variance measures the average squared distance of data values from their mean.
- F-statistic
- The F-statistic is the ratio of two sample variances used to test whether population variances are equal.
- F-distribution
- The F-distribution is a right-skewed probability distribution used for ratios of variances.
- Degrees of freedom
- Degrees of freedom describe how many independent pieces of information are available for estimating a statistic.
- Significance level
- The significance level alpha is the chosen probability of rejecting a true null hypothesis.
Common Mistakes to Avoid
- Using standard deviations directly in F = s1^2 / s2^2 is wrong because the F-test compares variances, not standard deviations. Square each sample standard deviation before forming the ratio.
- Forgetting the degrees of freedom order is wrong because df1 must match the numerator variance and df2 must match the denominator variance. Swapping them changes the critical value and p-value.
- Applying the F-test to strongly nonnormal data is risky because the test is very sensitive to departures from normality. For skewed data or outliers, consider a more robust method such as Levene's test.
- Treating a large F-value as proof of different means is wrong because the F-test for variances tests spread, not center. A separate test is needed to compare means.
Practice Questions
- 1 Sample A has n1 = 12 and s1^2 = 45. Sample B has n2 = 10 and s2^2 = 18. Compute F using the larger variance in the numerator and state df1 and df2.
- 2 Two machines produce parts with sample standard deviations 3.2 mm and 2.0 mm from samples of sizes 16 and 11. Compute the F-statistic using the larger variance in the numerator and give the corresponding degrees of freedom.
- 3 A data set has several extreme outliers and a strongly right-skewed distribution. Explain why a standard F-test for equal variances may be unreliable and what assumption is being threatened.