F Test Variance Calculator — Free & Interactive

The F-Test for Variances. Can two instruments be shown to have different measurement variance?

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.

Under H naught, sigma one squared divided by sigma two squared equals 1 and normality, F equals s one squared divided by s two squared follows an F distribution with df one equals n one minus 1 and df two equals n two minus 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. 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. Report both sample variances, F, df one, df two, 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 H naught supports a variance difference under the model. Failing to reject means the observed ratio is not sufficiently extreme at alpha; 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.

Two independent instruments measure reference specimens under controlled normal-error conditions. Instrument A errors are minus 2, 1, 0, 2, minus 1, 1 and Instrument B errors are minus 5, 3, 0, 4, minus 4, 2. The plan uses a two-sided F-test at alpha equals 0.05. Enter Instrument A as Sample 1 and Instrument B as Sample 2. The second sample is visibly more dispersed, so the observed F equals s A squared divided by s B squared will be below one. The calculator reports each sample variance, F, df one equals 5, df two equals 5, the two critical bounds, and the doubled smaller-tail p-value. Compare the result with alpha while also inspecting the variance ratio. With only six measurements per instrument, a large observed precision difference may still have substantial sampling uncertainty.

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.

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