F-Test in Statistics: Formula, Examples & When to Use It

Your ANOVA output just spit out an F-static. You’re looking at it. It shows 4.87, but you haven’t been told what this value means and if you should be conducting this test to begin with.

This tutorial will take you through how the formula for the F-test works, how you conduct it in Excel, R, and Python, how to interpret the outcome, and when you should choose another test.

What Is an F-Test?

An F-test is a kind of statistical test designed to determine whether the difference between the variances of two or more populations is greater than the variance resulting from random variation.

While variance is a measure of dispersion, two samples can be completely alike in terms of mean yet widely different with respect to dispersion. F-tests examine such differences between variances of two samples. Not their means. Their variances.

In any case, F-ratios are always computed as the ratio of two variances. In other words, you take the larger variance and divide it by the smaller variance. When the quotient is near 1, the variances are virtually the same. As the F-ratio rises from there, it indicates stronger and stronger evidence of an underlying cause behind that difference in variances.

Ronald Aylmer Fisher proposed the ‘variance ratio’ or F-distribution in the mid-1920s. It was George W. Snedecor who came up with the name ‘F-distribution’ and dedicated it to Fisher. The origin of this distribution is explained in Fisher’s book Statistical Methods for Research Workers (first published in 1925), where references can be made to his papers printed in the Journal of Agricultural Science, volume 14, 1924.

The F-Test Formula

There are three versions of the F-test formula. Using the wrong one is a common source of confusion.

F-Statistic for Comparing Two Variances

Where s₁² is the variance from the first sample, and s₂² is the variance from the second. Put the larger variance in the numerator to always make sure that the output is ≥ 1, which makes sure that you are working on the right-hand side. 

In the case of big sample sizes where you know the value of the population variance, use population variance (σ²) in place of sample variance(s²).

F-Statistic in ANOVA

F = Explained variance (between groups) / Unexplained variance (within groups)

You’re asking whether the differences between your group averages are bigger than the natural noise within each group. If yes, something beyond chance is separating those groups.

F-Statistic in Regression

This version gets skipped in most guides, which is frustrating because it’s what practitioners actually see in Python and Excel output.

The SSE represents the squared error of each individual model; ‘m’ refers to the number of constraints, ‘n’ indicates the size of the sample, while ‘k’ refers to the number of parameters in the total model. The significance value that comes after the F statistic in your regression analysis will indicate if the entire model has explained any significant variation at all.

Assumptions of the F-Test

A secret hidden by many textbooks is that the F-test does have assumptions, and violating them does make a difference.

The populations must be approximately normally distributed. F tests are sensitive to deviations from normality, even more so than t-tests. As noted in a paper by G.E.P. Box from 1953, modest deviation from normality may significantly affect the test’s type I error rate. When your data is skewed or leptokurtic, you should not be performing this test.

Samples must be independent. The two groups can’t be paired or related.

Data must be on an interval or ratio scale. Ordinal data means a different test.

If your distribution fails normality, conduct a Levene test or the Brown-Forsythe test. Both are much more robust to non-normal distributions. The Bartlett test is technically a choice but is in fact even less robust to non-normality than the F-test. Choose the Levene test. It’s your mainstay.

Homoscedasticity means that variances among groups must be equal. Important to note: you are conducting an F-test to validate an assumption which other tests (such as a two-sample t-test) rely on. This circular reasoning problem is often overlooked in introductory statistics classes.

How to Perform an F-Test — Step by Step

Let’s use a real scenario. A manufacturing plant runs two production lines making steel bolts. Line A has a standard deviation of 10.47 microns in bolt diameter across 41 measurements. Line B shows 8.12 microns across 21 measurements after a process change. The question: did the change reduce variability, or is the difference just noise?

Step 1: State your hypotheses

H₀: σ₁² = σ₂² (variances are equal, no real difference) H₁: σ₁² ≠ σ₂² (variances differ, the change had an effect)

Two-tailed test, because we’re checking for any difference.

Step 2: Compute the variances

s₁² = (10.47)² = 109.63 s₂² = (8.12)² = 65.99

Step 3: Calculate the F-statistic

F = 109.63 / 65.99 = 1.66

The larger variance always goes in the numerator.

Step 4: Find your degrees of freedom

df₁ = 41 – 1 = 40 (numerator) df₂ = 21 – 1 = 20 (denominator)

Understanding Degrees of Freedom

Degrees of freedom gets defined badly in most textbooks. Here’s the practical version: it’s the number of values free to vary once you’ve calculated the mean. With 10 data points and a known average, only 9 values can be anything; the 10th is locked in. So df = n – 1.

Step 5: Find the critical F-value

How to Use the F-Table

Two-tailed test means divide your alpha in half. At α = 0.05, use α/2 = 0.025. Look up the F-table for α = 0.025 (not 0.05, which is the one-tailed table). Find the column for df₁ = 40, the row for df₂ = 20.

If your exact degrees of freedom aren’t in the table, use the next larger critical value. It keeps you conservative and reduces false rejections.

For df(40, 20) at α = 0.025: F_critical = 2.287

Step 6: Compare and decide

1.66 < 2.287. Fail to reject H₀. The data doesn’t give enough evidence to say the process change reduced variability. The plant manager needs more data.

F-Test in Python

Two approaches depending on what you’re doing.

For comparing two variances directly:

pythonimport numpy as npfrom scipy.stats import f
var1 = np.var(sample1, ddof=1)var2 = np.var(sample2, ddof=1)
f_stat = var1 / var2  # larger variance in numerator
df1 = len(sample1) - 1df2 = len(sample2) - 1
p_value = 2 * min(f.cdf(f_stat, df1, df2), 1 - f.cdf(f_stat, df1, df2))
print(f"F-statistic: {f_stat:.4f}")print(f"p-value: {p_value:.4f}")

If p_value > 0.05, fail to reject H₀. The variances aren’t statistically different.

For reading the regression F-statistic — the version nobody shows you:

python

import statsmodels.formula.api as smf

model = smf.ols('y ~ x1 + x2 + x3', data=df).fit()

print(model.summary())

In the output, look for F-statistic and Prob (F-statistic). A low p-value means at least one predictor explains real variance. A high p-value means your model isn’t better than predicting the mean every time.

F-Test in Excel (Microsoft 365)

StatisticsHowTo’s walkthrough references Excel 2013. The 365 interface looks different, so here’s the current version.

Step 1: Go to the Data tab. Then look for the Data Analysis command on the far-right side. If it is not there, go to File → Options → Add-ins → Manage Excel Add-ins → check “Analysis ToolPak.

Step 2: Select F-Test Two-Sample for Variances and click OK.

Step 3: In the dialog, set Variable 1 Range to the group with the higher variance. Set Variable 2 Range to the other group. Set Alpha to 0.05. Choose an output location.

Step 4: Click OK and read the output. Find F and P(F<=f) two-tail. If the p-value is below 0.05, reject H₀.

Critical note: Variable 1 must always be the group with the larger variance. Excel won’t sort them for you. If you put the smaller variance first, you get a wrong F-value. Always verify before running.

But there’s no warning in the dialog itself — it just silently calculates the wrong thing.

F-Test in R

The simplest of the three. R’s built-in var.test() handles everything.

r

result <- var.test(line_a, line_b)

print(result)

R automatically places the larger variance in the numerator. The output gives you the F-statistic, degrees of freedom, and p-value. For a one-tailed test, add alternative = “greater” or alternative = “less”.

F-Test vs T-Test

I think this is where most guides get it exactly wrong. They present the two tests as if they’re interchangeable options you choose by preference. They’re not; they test completely different things.

	F-test	T-test
Tests	Whether variances are equal	Whether means are equal
Distribution	F-distribution	Student’s t-distribution
Use when	Comparing variability between groups	Comparing averages between groups
Typical context	Checking ANOVA assumption; regression model fit	Before a two-sample t-test; treatment effect studies

Decision logic in plain terms:

• Comparing two averages? Use a t-test.
• Testing whether two groups have equal variance? Use the F-test.
• Testing whether your regression model explains anything? Use the regression F-test.
• Testing whether multiple group means differ at once? One-way ANOVA uses the F-test internally.

When NOT to Use the F-Test

Most articles on this topic get this part exactly wrong by skipping it entirely.

Your distribution is not normally distributed. Most often this will be the basis for switching from one test to another. In case a Shapiro-Wilks test or a Q-Q graph indicate that the distribution significantly deviates from normality, employ Levene’s test. It operates based on absolute deviations from group medians rather than squared deviations, and as such, is much more robust to outliers and skewed distributions.

You need to compare variances across more than two groups. When you have three or more groups, the regular two-sample F-test won’t do. There is the Brown-Forsythe test, which works for several groups, and does not assume normally distributed data.

Your outcome variable is ordinal. When working with Likert scale questions, rankings, and other ordinal types of data, F-tests are completely out of the question.

You actually want to compare means. This may sound self-explanatory, but sometimes the hypothesis statement can be confusing and unclear.

Real-World Examples

A/B Testing in Marketing

Two variations of a landing page have very similar conversion rates; the control group has 4.1%, and the variation has 4.3%. However, before making any determination about which one performs better, make sure that the dispersion in the conversion rates between sessions is the same for both groups. In case the variation has erratic conversion rates (high dispersion), the difference in averages is likely to be coincidental.

Manufacturing Quality Control

Two assembly lines producing the same product. Your focus here is on variation rather than averages. While an assembly line that has an average that is perfect but has high variability will have more defects than one whose average is off but with low variability. That is what the F-test picks up.

Clinical Research

In a drug experiment, there are two populations of patients. When testing the results with a two-sample t-test, the first thing that the scientist needs to do is to see if the variance is the same in both populations, and this affects the calculation of the t-test. This requirement is termed homoscedasticity and is tested using the F-test.

Frequently Asked Questions