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Walk-through

How to Use This Calculator

4 steps

1

Choose From Data or From Variances

Pick From Data if you have the raw values for each sample and want the calculator to compute each group's variance for you. Pick From Variances if you already know each group's variance and sample size and just want to run the test.

2

Enter your two samples

For From Data, paste each sample's values (comma, space, or newline separated). For From Variances, enter each group's variance (s²) and sample size (n). The calculator updates instantly as you type.

3

Pick one-tailed or two-tailed

Two-tailed tests whether the variances differ in either direction and is the standard, conservative choice. One-tailed only makes sense if you predicted in advance which sample would have the larger variance — it uses the ratio exactly as entered instead of reordering to put the larger variance on top.

4

Read the F-statistic, df, and p-value

The result card shows F, the numerator and denominator degrees of freedom, and the p-value, plus the reject/fail-to-reject call at α = 0.05. If p is less than 0.05, the variances are significantly different; if not, the data don't give enough evidence to say the variances differ.

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Reference

Formula & Methodology

3 formulas

F-statistic (two-tailed)

F = s₁²(larger) / s₂²(smaller) df₁ = n(larger) − 1, df₂ = n(smaller) − 1

For a two-tailed test, the two sample variances are reordered so the larger one sits on top, making F ≥ 1. The p-value is twice the upper-tail probability of that F, capped at 1, since either group could plausibly be the one with the larger true variance.

F-statistic (one-tailed)

F = s₁² / s₂² df₁ = n₁ − 1, df₂ = n₂ − 1

For a directional (one-tailed) test of Ha: σ₁² > σ₂², the ratio is used exactly as entered — no reordering — and the p-value is the upper-tail probability of that F under the F-distribution with df₁ and df₂ degrees of freedom.

Sample variance (From Data mode)

s² = Σ(xᵢ − x̄)² / (n − 1)

When working from raw data, each group's sample variance is computed with the standard (n − 1) divisor before the F-ratio is formed.

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Glossary

Key Terms Explained

7 terms

F-test A statistical hypothesis test that compares the variances (spread) of two samples by forming their ratio and comparing it to the F-distribution, to decide whether the two populations have significantly different variability.

Variance ratio The quotient of two sample variances (s₁²/s₂²) — the core quantity the F-test evaluates. A ratio near 1 suggests similar variability; a ratio far from 1 suggests the underlying populations have different spread.

F-statistic The computed variance ratio used as the test statistic. Under the null hypothesis of equal variances, it follows an F-distribution with (df₁, df₂) degrees of freedom.

Degrees of freedom (df) For each group, df = n − 1. The F-distribution needs two degrees-of-freedom values — one for the numerator (the group whose variance is on top) and one for the denominator.

p-value The probability of observing an F-statistic at least this extreme if the two populations actually had equal variance. A small p-value means the observed spread difference would be unusual under that assumption.

Equality of variances The null hypothesis (H₀) that both samples come from populations with the same true variance. The F-test either rejects this assumption (variances differ) or fails to reject it (no significant evidence of a difference).

Link to ANOVA One-way ANOVA is, in effect, a generalization of the F-test to more than two groups — it also computes an F-statistic (between-groups variance ÷ within-groups variance) to test whether group means differ, using the same F-distribution machinery.

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Scenarios

Real-World Examples

3 worked examples

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Two production machines

From Data — two-tailed

Machine A output 23, 25, 21, 24, 26, 22, 24 Machine B output 20, 24, 19, 22, 25, 18, 21

Both machines produce similar averages, but the F-test checks whether one machine's output is more variable (less consistent) than the other's. With small samples like these, the test often fails to reject equal variance unless the spread difference is large — a useful sanity check before assuming one machine is 'less reliable.'

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Two lab measurement methods

From Variances — one-tailed

Method 1 variance (s₁²) / n₁ 24.5 / 12 Method 2 variance (s₂²) / n₂ 8.1 / 10

If you predicted in advance that Method 1 would be noisier than Method 2, a one-tailed test directly evaluates Ha: σ₁² > σ₂² using F = 24.5/8.1 ≈ 3.02 with df = (11, 9). A small p-value here would support switching to the more consistent Method 2.

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Reading degrees of freedom

From Variances — df walkthrough

Group 1 variance / n₁ s₁² / 11 Group 2 variance / n₂ s₂² / 11

df₁ and df₂ are each just sample size minus one (n − 1). With n₁ = n₂ = 11, both degrees of freedom equal 10. If the larger variance turns out to belong to group 2 in a two-tailed test, the calculator swaps which df is 'numerator' and which is 'denominator' to match — the F-statistic is always reported as larger ÷ smaller in that mode.

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Deep Dive

Understanding the F-Test Calculator

3 chapters · 4 min read

The F-test answers a narrower question than a t-test: not whether two samples have different averages, but whether they have different amounts of spread. That matters on its own — inconsistent output, noisy measurements, and unstable processes are all 'variance' problems — and it's also the test many textbooks recommend running before a two-sample t-test, to decide whether to assume equal variances.

What the F-statistic measures

The F-statistic is simply the ratio of two sample variances. For a two-tailed test — the standard, conservative choice — the calculator always puts the larger variance on top, so F is always ≥ 1: a value near 1 means the two samples have similar spread, while a large F means one sample is considerably more variable than the other. Because the test is symmetric (either group could be the noisier one), the two-tailed p-value doubles the upper-tail probability of that ratio.

From data vs. from variances

If you have the raw measurements, use the From Data tab — the calculator computes each group's sample variance (using the standard n − 1 divisor) automatically. If you already have summary statistics — say, from a lab report or a prior analysis — use the From Variances tab and enter each group's variance and sample size directly. Both paths feed the same underlying F-test math.

One-tailed vs. two-tailed, and what the F-test assumes

Two-tailed is the standard default: it tests whether the variances differ in either direction, without assuming which group is noisier ahead of time. A one-tailed test is only valid if you committed, before seeing the data, to a specific direction (e.g., "Method 1 is noisier than Method 2") — it uses the ratio exactly as entered rather than reordering to put the larger variance on top. The classic F-test assumes both samples are drawn from normally distributed populations; it is fairly sensitive to that assumption, so with clearly non-normal data (heavy skew or outliers), treat a borderline p-value with caution and consider a more robust alternative like Levene's test.

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Questions

Frequently Asked Questions

6 questions

What's the formula for the F-test?+

F is the ratio of two sample variances. For a two-tailed test, F = (larger variance) / (smaller variance), so F is always ≥ 1. For a one-tailed, directional test, F = s₁²/s₂² exactly as entered, without reordering.

How are the degrees of freedom calculated?+

Each group contributes df = n − 1, where n is that group's sample size. The F-distribution needs both values: df₁ for whichever group's variance is in the numerator, and df₂ for the group in the denominator.

What does p < 0.05 mean here?+

It means that if the two populations actually had equal variance, you'd see an F-statistic this extreme (or more) only about 5% of the time by chance. Conventionally, that's called statistically significant, and you reject the null hypothesis of equal variances in favor of the two samples having genuinely different spread.

When should I use the F-test?+

Use it whenever you want to compare the variability (not the average) of two samples — comparing measurement consistency between two instruments or methods, checking whether one production process is less predictable than another, or (classically) deciding whether to use the pooled or Welch version of a two-sample t-test.

Should I use one-tailed or two-tailed?+

Two-tailed is the standard, conservative default and is appropriate almost all the time — it tests for a difference in either direction without assuming which group is more variable. Use one-tailed only if you predicted, before collecting the data, that a specific group would have the larger variance; it isn't valid to switch to one-tailed after seeing which group turned out noisier.

What assumptions does the F-test make?+

The classic F-test for equality of variances assumes both samples are drawn from approximately normally distributed populations and that observations within each group are independent. It is notably sensitive to violations of normality — with skewed data or outliers, the reported p-value can be unreliable, and a more robust test (such as Levene's test) is usually preferred in practice.

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F-Test Calculator

From Sample Data

From Summary Variances

How to Use This Calculator

Choose From Data or From Variances

Enter your two samples

Pick one-tailed or two-tailed

Read the F-statistic, df, and p-value

Formula & Methodology

Key Terms Explained

Real-World Examples

Two production machines

Two lab measurement methods

Reading degrees of freedom

Understanding the F-Test Calculator

What the F-statistic measures

From data vs. from variances

One-tailed vs. two-tailed, and what the F-test assumes

Frequently Asked Questions

Get a branded PDF of your results

F-Test Calculator

From Sample Data

From Summary Variances

How to Use This Calculator

Choose From Data or From Variances

Enter your two samples

Pick one-tailed or two-tailed

Read the F-statistic, df, and p-value

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the F-Test Calculator

What the F-statistic measures

From data vs. from variances

One-tailed vs. two-tailed, and what the F-test assumes

Frequently Asked Questions

Get a branded PDF of your results

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