Interactive module guide

F-Test for Equal Variances Calculator

The F-test for equal variances compares two independent population variances by placing the first sample variance over the second.

This calculator supports two-sided and directional alternatives and keeps Sample 1 in the numerator.

Under the null ratio of one and independent normal populations, the statistic follows an F distribution.

The test is highly sensitive to non-normality, so it should not be used as an automatic preliminary switch for choosing a pooled t-test.

Free · No sign-up · Calculations stay in your browser

Is this the right module for my question?

Use it when

Use the test when the scientific question directly concerns variability in two independent normal populations, such as precision under two controlled measurement systems. The outcome must be quantitative and each sample must contain at least two distinct values.
Choose two-sided when either larger or smaller variance matters. A right-sided alternative asks whether Var(Sample 1)>Var(Sample 2), while a left-sided alternative asks the reverse. Sample order and tail must be pre-specified.
The classical F reference is exact under independent normal observations. It can be useful in designed laboratory settings where normality is credible and variance itself is the target.

Choose another method when

Avoid this test with skewed, heavy-tailed, contaminated, or outlier-prone data. Even moderate departures from normality can substantially alter its Type I error because sample variance squares deviations.
Do not use a non-significant F-test to conclude variances are equal or as the sole reason to pool variances in a subsequent mean test. Welch’s t-test is generally robust without this pretest workflow.

Interactive tool

The calculator loads as you approach this section so the guide remains fast on mobile connections.

How to read the result

Report both sample variances, F, df₁, df₂, test direction, and p. F greater than one means Sample 1 has the larger observed variance; F below one means Sample 2 does, given the fixed numerator order.

Rejecting H₀ supports a variance difference under the model. Failing to reject means the observed ratio is not sufficiently extreme at α; it does not establish equal precision or interchangeability.

Practical interpretation should use the variance ratio or standard-deviation ratio and domain tolerances. A p-value can be small for a minor difference in a large sample or inconclusive for a consequential ratio in a small sample.

Inputs and parameter meaning

Enter independent raw observations for each sample in the same measurement unit. The calculator computes unbiased sample variances with n−1 denominators and forms F=s₁²/s₂² without automatically placing the larger variance on top.

Set alpha and Test Type to the planned variance-ratio hypothesis. Swapping samples replaces F by 1/F and exchanges numerator and denominator degrees of freedom; a two-sided p-value should remain substantively consistent, while directional claims reverse.

At least two distinct values are required in each sample. Zero sample variance makes the ratio undefined or degenerate and cannot support the standard reference distribution.

Formula, hypotheses, and assumptions

F = \frac{s_1^2}{s_2^2}

Conditions to review

Both populations are normally distributed
Samples are independent
Data is continuous

Calculator parameters

Significance Level (α): default 0.05.
Test Type: default Two-tailed.

What the method is doing

Under H₀:σ₁²/σ₂²=1 and normality, F=(s₁²/s₂²) follows an F distribution with df₁=n₁−1 and df₂=n₂−1. The distribution is asymmetric and depends on both degrees of freedom.

For a two-sided test, evidence can lie in either tail, so the calculator doubles the smaller tail probability and uses lower and upper critical values. Directional tests use the corresponding single tail.

The p-value is a conditional tail probability under equal variances and normal populations. It is not the probability the variances are equal and does not quantify an acceptable equivalence margin.

Worked example: comparing measurement precision

Two independent instruments measure reference specimens under controlled normal-error conditions. Instrument A errors are −2, 1, 0, 2, −1, 1 and Instrument B errors are −5, 3, 0, 4, −4, 2. The plan uses a two-sided F-test at α=0.05.

1Enter Instrument A as Sample 1 and Instrument B as Sample 2. The second sample is visibly more dispersed, so the observed F=sA²/sB² will be below one.
2The calculator reports each sample variance, F, df₁=5, df₂=5, the two critical bounds, and the doubled smaller-tail p-value.
3Compare the result with α while also inspecting the variance ratio. With only six measurements per instrument, a large observed precision difference may still have substantial sampling uncertainty.

Interpretation

The test evaluates whether the population error variances differ under normal independent measurement errors. A non-significant result would not certify equal precision, and a significant result would not identify calibration bias because variance and mean error are different properties. Replication across conditions and uncertainty intervals are needed for qualification decisions.

Common mistakes

Always placing the larger sample variance in the numerator without adjusting the hypothesis and p-value changes the defined statistic. This calculator preserves input order.
Outliers can dominate squared deviations and create apparent variance differences. Investigate data provenance rather than deleting points solely to change significance.
Variance is expressed in squared units; standard deviation is in the original unit. Do not compare or report them interchangeably.

Limits and independent validation

The calculator does not provide robust Levene or Brown–Forsythe alternatives, confidence intervals for the variance ratio, normality diagnostics, or equivalence testing.

The exact F model is fragile outside normal populations and does not address dependent samples, repeated measurements, or nested sources of variation.

Before using the result

Review independence, collection order, normal quantile behavior, and influential observations. State why variance, rather than a mean or total error measure, is the target.
Recompute s₁², s₂², F, both degrees of freedom, and the selected F tails independently. For non-normal data, use a robust method instead of treating this result as definitive.
Repeat the descriptive comparison with standard deviations and a robust spread summary such as median absolute deviation in an independent package. These do not replace the planned F-test, but disagreement can expose tail sensitivity or an influential observation that makes the exact normal-theory variance ratio a poor summary of practical precision.

two sample t-test →Compare means with Welch’s option rather than automatically pretesting variances.one-way ANOVA →Study how an F ratio compares between- and within-group mean squares.methodology →Review scope, reporting, and verification limits.

← Chi-Square Test of Independence One Sample Z-Test →

See every option in the Statistical Hypothesis Test Calculator or review the DistriScope methodology. Educational information; last reviewed 2026-08-06. Verify consequential calculations independently.