Variance Ratio F Test Calculator

StatisticsLast updated: August 22, 2026

The variance ratio F test asks whether two independent samples come from populations with equal variances. The null hypothesis is σ₁² = σ₂² and the alternative is that the variances differ. The test statistic is the ratio of the two sample variances, and under H₀ it follows an F(n₁−1, n₂−1) distribution; the further the ratio moves away from 1, the stronger the evidence that the variances differ.

Paste two data sets and this calculator returns the F value, the degrees of freedom, the critical F bounds, the p-value, and a confidence interval for the ratio σ₁²/σ₂². It also shows how far the result depends on the normality assumption and runs the Brown–Forsythe (median-centered Levene) test on the same data so that the two can be read side by side. It is used in quality-control work comparing measurement consistency, in laboratory analyses comparing the variability of two methods, and in checking assumptions before a t test.

The sample that goes in the numerator. You can separate the numbers with commas, spaces, or line breaks.
The sample that goes in the denominator. The two group sizes do not have to be equal.
A directional hypothesis should be chosen only when the direction of the difference can be justified in advance.
Sets the false-positive risk. In the social sciences the common choice is 5%.

Variance Ratio F Test Formulas

Hypotheses:          H₀: σ₁² = σ₂²   H₁: σ₁² ≠ σ₂²
Sample variance:     s² = Σ(xᵢ − x̄)² / (n − 1)
Test statistic:      F = s₁² / s₂²
Degrees of freedom:  df₁ = n₁ − 1 ,  df₂ = n₂ − 1
Two-tailed p:        p = 2 · min[ P(F ≤ F₀) , P(F ≥ F₀) ]
Critical region:     F < F₍α/₂₎(df₁,df₂)  or  F > F₍₁₋α/₂₎(df₁,df₂)
Confidence interval: F / F₍₁₋α/₂₎ ≤ σ₁²/σ₂² ≤ F / F₍α/₂₎

Because the F distribution is not symmetric, the two-tailed p-value is found by doubling the one-tail probability, and the lower and upper critical values are linked by a reciprocal relation: F₍α/₂₎(df₁,df₂) = 1 / F₍₁₋α/₂₎(df₂,df₁). The test is sensitive to the normality assumption; its robust alternative is the Brown–Forsythe test.

How to Calculate

  1. Paste the raw data of the two samples into the separate boxes; each needs at least 3 values.
  2. Choose deliberately which sample goes in the denominator: since F = s₁²/s₂², the order is critical for a one-tailed hypothesis.
  3. Set the hypothesis direction and the significance level; the confidence interval is always produced two-sided at the 1 − α level.
  4. Compare the F value with the critical F bounds and check whether the confidence interval for the variance ratio contains 1.
  5. Read the Shapiro–Wilk summaries in the results section: if normality does not hold, the p-value of the F test is not trustworthy.
  6. If normality is in doubt, base your decision on the Brown–Forsythe (Levene) row and state clearly in your report which test you used.

Worked Examples

Variability of two measurement methods

The variance of the first method is 4.9000 and that of the second is 26.9444; F = 0.1819 with df = (9, 9). The two-tailed p = 0.018, so the variances differ at α = 0.05: the 95% confidence interval for σ₁²/σ₂², 0.045 – 0.732, does not contain 1. The second method is about 5.50 times more variable.

F statistic (s₁² / s₂²): 0.1819 · Degrees of freedom — numerator (df₁): 9 · Degrees of freedom — denominator (df₂): 9

A case where the variances come out equal

Although the means are very different (50.625 and 60.875), the variances are close: 7.1250 and 8.4107, giving F = 0.8471 with df = (7, 7) and p = 0.832. Homogeneity of variance is not rejected; the 95% confidence interval, 0.170 – 4.231, contains 1 comfortably. This shows that the pooled-variance t test is appropriate for comparing the means.

F statistic (s₁² / s₂²): 0.8471 · Degrees of freedom — numerator (df₁): 7 · Degrees of freedom — denominator (df₂): 7

One-tailed hypothesis — is the new process more consistent?

The variance of the new process is 0.0770 and that of the old one is 0.9579; F = 0.0804 with df = (11, 9). With a left-tail p < 0.001, the variance of the new process is significantly smaller — the old process is about 12.45 times more variable. The critical value is 0.345, and the observed F falls far below it.

F statistic (s₁² / s₂²): 0.0804 · Degrees of freedom — numerator (df₁): 11 · Degrees of freedom — denominator (df₂): 9

Frequently Asked Questions

What is the difference between the F test and the Levene test, and which should I use?
Both test the equality of variances, but they differ sharply in robustness. The F test assumes the data is normally distributed, and when that assumption fails the true Type I error rate can climb from 5% to around 20%. The Levene test (particularly the median-centered Brown–Forsythe form) works on absolute deviations and is far more robust to departures from normality. If you cannot demonstrate normality with confidence, prefer Levene; the F test is more powerful only when the data really is normal.
Why is the two-tailed p-value taken as twice the one-tail probability?
The F distribution is not symmetric like the normal or t distributions; it is right-skewed. Because it is asymmetric, a straightforward "add both tails" approach does not apply; the common method, used by software including R's var.test, is to double the probability of the tail in which the observed value lies. The method is slightly conservative, but it is treated as standard in practice.
What does it mean if F comes out below 1?
Since F = s₁²/s₂², a value below 1 means the first sample is less variable than the second. Some textbooks always place the larger variance in the numerator so that F ≥ 1; that makes one-tailed tables easier to use but risks doubling the two-tailed p-value by mistake. This calculator computes the ratio in the order you entered and takes the p-value from the correct tail, so you do not need to reorder anything.
If the variances turn out to be different, how should I run my t test?
Use the Welch-corrected t test instead of the classic (Student) t test with pooled variance. Welch treats the two variances separately and adjusts the degrees of freedom with the Satterthwaite formula. For three or more groups, the counterpart is Welch's ANOVA. Many sources now recommend using Welch directly without a preliminary test, because choosing a test based on the outcome of a pre-test distorts the overall error rate.
How do I interpret the confidence interval for the variance ratio?
The interval is for the true ratio σ₁²/σ₂². If it contains 1, the hypothesis of equal variances is not rejected. If the whole interval lies above 1, the first population is more variable; if it lies entirely below 1, the second is. For the ratio of standard deviations, take the square root of the interval; that is usually easier to interpret because it is on the same scale as the measurements.
What are the equivalents in SPSS, Excel, and R?
In R, var.test(x, y) returns exactly the same F, degrees of freedom, p-value, and confidence interval as this calculator. In Excel, F.TEST(array1, array2) returns only the two-tailed p-value; to see the F value you need the 'F-Test Two-Sample for Variances' tool in the Data Analysis add-in. SPSS has no separate variance ratio F test in its menus; the Levene test in the t-test output serves that role.
Can I use this test on more than two groups?
No. The F ratio is defined for two variances only, and comparing the variances of three or more groups pairwise inflates the Type I error rate. For homogeneity of variance across several groups, use Levene/Brown–Forsythe or (if the data is normal) the Bartlett test. These evaluate all groups in a single test.