F-test for Comparing Variances Lab

STA-215 · Semester III · Statistics II

intermediate30 minAvailable

F-test for Variance Ratio

Compare two population variances: F = s₁² / s₂².

s₁² (larger variance)s₂² (smaller variance)n₁n₂

F ratio

1.563

25 / 16 with df = (10, 5)

Fail to reject H₀ — 1.56 ≤ 3.33: variances are equal.

Critical F(0.05, 10, 5) = 3.33. Used before a two-sample t-test to decide whether to pool variances.

Lab brief

Objective

Compare the variances of two independent samples with the F-test and use the ratio to judge whether equal-variance assumptions (like the pooled t-test) are safe.

Theory & concepts

The F-test compares two variances: F = s1^2 / s2^2 with s1^2 the larger variance so that F >= 1. Under H0 (sigma1^2 = sigma2^2) the ratio follows an F distribution with df1 = n1 - 1 and df2 = n2 - 1.

Rendering diagram…

The F-test for the ratio of two variances

The F-test is sensitive to non-normality, so treat it as a rough guide. The pooled two-sample t-test assumes equal variances; when the ratio is large, Welch's test is safer.

F ratio	Verdict	Suggestion
close to 1	Variances look equal	Pooled t-test is reasonable
moderately > 1	Some evidence of inequality	Consider Welch's t-test
far above critical F	Variances differ significantly	Use Welch's t-test

Prerequisites

Lesson: Two-sample t-test for independent samples
Lesson: Point estimation

Instructions

1Enter two independent samples and read their variances and degrees of freedom.
2Compute the ratio of the larger variance to the smaller and read the F statistic.
3Look up the critical value for df1, df2 at alpha = 0.05 and compare.
4Adjust one sample to make the variances more alike and watch the ratio move toward 1.
5Decide whether the pooled t-test or Welch's t-test is appropriate for these data.

Tasks

Variance ratio
Compute F = s1^2/s2^2 and state the conclusion for alpha = 0.05.
Assumption check
Judge whether the equal-variance assumption for a pooled t-test is safe.
Degrees of freedom
Report df1 and df2 and explain how they arise.

Self-check

The F statistic matches the ratio of the two variances.
The student correctly uses the one-sided critical value of the F distribution.
The choice between pooled and Welch t-tests is justified from the F result.

Environment

editable paired sample editorlive F statistic and dfF critical value referencepooled vs Welch indicator