1.5625
19
24
1.5625
1.25
9
25
16
1.5625
19
24
1.5625
1.25
9
25
16
The F-Test Calculator compares the variances of two independent samples to determine whether they are significantly different. Enter each sample's variance and size, and the calculator computes the F-statistic and degrees of freedom for hypothesis testing.
The F-test for equality of variances uses the ratio: $$F = \frac{s_1^2}{s_2^2}$$ where the larger variance is conventionally placed in the numerator. This ensures $$F \geq 1$$, simplifying comparison with upper-tail critical values of the F-distribution.
The null hypothesis states that the population variances are equal: $$H_0: \sigma_1^2 = \sigma_2^2$$. Under this hypothesis, the F-statistic follows an F-distribution with $$df_1 = n_1 - 1$$ (numerator) and $$df_2 = n_2 - 1$$ (denominator) degrees of freedom, where $$n_1$$ corresponds to the sample with the larger variance.
The F-distribution is a right-skewed probability distribution that arises as the ratio of two chi-squared distributions divided by their degrees of freedom. Its shape depends on both $$df_1$$ and $$df_2$$. As both degrees of freedom increase, the distribution becomes more symmetric and concentrated around 1.
For a two-tailed test at significance level α, reject the null hypothesis if the computed F exceeds the critical value $$F_{\alpha/2, df_1, df_2}$$. Alternatively, compute the p-value as $$P(F > f_{observed})$$ and compare with α/2. Common significance levels are 0.05, 0.01, and 0.10.
The F-test for variances is sensitive to departures from normality. Even moderate non-normality can substantially affect its Type I error rate. For non-normal data, Levene's test or the Brown-Forsythe test provide more robust alternatives. The F-test remains important as a prerequisite for choosing between pooled and Welch's t-tests, and is fundamental to ANOVA, regression analysis, and many other statistical procedures.
Places the larger variance in the numerator. F = larger variance / smaller variance. Degrees of freedom: df₁ = n(larger) - 1 for the numerator, df₂ = n(smaller) - 1 for the denominator. Compare F to critical F-distribution values for significance testing.
F close to 1 suggests equal variances. F much greater than 1 suggests unequal variances. Compare against F-critical for your chosen α: e.g., F(19, 24) at α = 0.05 two-tailed requires F > 2.33 approximately. If F exceeds the critical value, reject the null hypothesis of equal variances.
Inputs
Results
F = 1.5625 with df (19, 24). Likely not significant at α = 0.05.
Inputs
Results
F = 5.0 with df (14, 14). F_critical ≈ 2.98 at α = 0.05 two-tailed. Reject H₀.
The F-test compares two population variances to determine if they are significantly different. It is commonly used as a preliminary test before performing a t-test, to decide between pooled and Welch versions.
Placing the larger variance in the numerator ensures F ≥ 1, allowing us to use only the upper tail of the F-distribution for critical values. This is a convention that simplifies the testing procedure.
The F-test assumes both populations are normally distributed and samples are independent. It is highly sensitive to non-normality, making Levene's test a better alternative when normality is questionable.
ANOVA uses an F-test to compare between-group variance to within-group variance. The variance equality F-test compares two specific sample variances, while the ANOVA F-test compares variance components.
A two-tailed test checks if variances differ in either direction (σ₁² ≠ σ₂²). A one-tailed test checks a specific direction (σ₁² > σ₂²). Two-tailed is more common for equality testing; use α/2 for each tail.
df₁ = n₁ - 1 for the numerator sample and df₂ = n₂ - 1 for the denominator sample. Together they determine the shape of the F-distribution used to find critical values and p-values.
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