F-Test for Equality of Two Variances

The F-Test (Variance Ratio Test) is a hypothesis test used to compare the variances of two independent, normally distributed populations to evaluate whether one population has greater variability than another.

F-Statistic Formula:

F = \frac{s_1^2}{s_2^2} \quad (s_1^2 \ge s_2^2)

Where:

$s_1^2$ is the larger sample variance with $df_1 = n_1 - 1$ .
$s_2^2$ is the smaller sample variance with $df_2 = n_2 - 1$ .

Example of F-Test

Comparing Sample Variances

n_1 = 10, s_1^2 = 25.0 \quad | \quad n_2 = 12, s_2^2 = 10.0

Test at $\alpha = 0.05$

Step 1 & 2: Formulate Hypotheses & Assign Degrees of Freedom

H_0: \sigma_1^2 = \sigma_2^2, \quad H_1: \sigma_1^2 \neq \sigma_2^2

df_1 = 9, \quad df_2 = 11

Step 3 & 4: Compute F Statistic & Compare

F = \frac{25.0}{10.0} = 2.50

F_{\text{calc}} = 2.50 < 2.896

Conclusion

\boxed{\text{Fail to Reject } H_0 \text{ at } \alpha = 0.05}

Therefore, because $F_{\text{calc}} = 2.50 < 2.896$ , we fail to reject the null hypothesis and conclude that there is no statistically significant difference between the two population variances.

Interactive F-Test Calculator

F-Test for Equality of Variances

PreviousChi-Square Test

Unit 5•28 of 28

F-Test for Equality of Two Variances

F-Statistic Formula:

F = \frac{s_1^2}{s_2^2} \quad (s_1^2 \ge s_2^2)

Where:

$s_1^2$ is the larger sample variance with $df_1 = n_1 - 1$ .
$s_2^2$ is the smaller sample variance with $df_2 = n_2 - 1$ .

Example of F-Test

Comparing Sample Variances

n_1 = 10, s_1^2 = 25.0 \quad | \quad n_2 = 12, s_2^2 = 10.0

Test at $\alpha = 0.05$

Step 1 & 2: Formulate Hypotheses & Assign Degrees of Freedom

H_0: \sigma_1^2 = \sigma_2^2, \quad H_1: \sigma_1^2 \neq \sigma_2^2

df_1 = 9, \quad df_2 = 11

Step 3 & 4: Compute F Statistic & Compare

F = \frac{25.0}{10.0} = 2.50

F_{\text{calc}} = 2.50 < 2.896

Conclusion

\boxed{\text{Fail to Reject } H_0 \text{ at } \alpha = 0.05}

Interactive F-Test Calculator

F-Test for Equality of Variances

PreviousChi-Square Test

Unit 5•28 of 28

F-Test for Equality of Two Variances

Example of F-Test

Step 1 & 2: Formulate Hypotheses & Assign Degrees of Freedom

Step 3 & 4: Compute F Statistic & Compare

Conclusion

Interactive F-Test Calculator

F-Test for Equality of Variances

NETZ Engine

F-Test for Equality of Two Variances

Example of F-Test

Step 1 & 2: Formulate Hypotheses & Assign Degrees of Freedom

Step 3 & 4: Compute F Statistic & Compare

Conclusion

Interactive F-Test Calculator

F-Test for Equality of Variances