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Judge F-test significance fast: enter numerator df, denominator df 1-1000, and an F value for the PDF, CDF, and right-tail p-value plus a p<0.05 verdict.

📘 How to Use

  1. Enter the numerator degrees of freedom d1 as an integer from 1 to 1000
  2. Enter the denominator degrees of freedom d2 as an integer from 1 to 1000
  3. Enter your observed F value as a number greater than or equal to 0
  4. Read the PDF, CDF, right-tail p-value, and significance verdict

F-Distribution Probability Calculator

Integer 1 to 1000 (model or factor degrees of freedom).

Integer 1 to 1000 (residual or error degrees of freedom).

Non-negative real number. Enter the observed F statistic.

※ CDF uses the regularized incomplete beta function via continued fraction expansion.

※ Designed for df between 1 and 1000. Extreme values may lose numerical precision.

PDF f(x)
—
CDF P(X<=x)
—
Right-tail p-value P(X>x)
—

—

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F-Distribution Probability Calculator | Read F-Test Significance Without a Table

Enter the numerator df, the denominator df, and an observed F value to get the F-distribution's probability density (PDF), cumulative probability (CDF), and right-tail p-value at once. The tool also flags whether the result clears the p<0.05, p<0.01, or p<0.001 threshold, so you can judge an ANOVA or regression F-statistic without flipping through a printed F-table.

💡 About this tool

Run an ANOVA or a regression and you get an F-statistic back. On its own that number is only a ratio of variances — bigger means more signal relative to noise, but whether it's actually surprising depends on the two degrees of freedom. The number you really want is the p-value: how unlikely this F value is if the null hypothesis is true.

The old way was to look up a critical value in an F-table at the row for d1 and the column for d2. The problem: those tables only print a few alpha levels (0.05, 0.01) and never the exact p-value. This calculator returns the right-tail p-value P(X>x) to six decimals the moment you type an F value, so you can say "p = 0.088, so it doesn't clear the 0.05 bar" with no ambiguity.

There are always two degrees of freedom. The numerator df (d1) is the between-group or model df — in one-way ANOVA it's the number of factor levels k minus 1; in regression it's the number of predictors. The denominator df (d2) is the within-group or residual df — n minus k in ANOVA, the residual df in regression. Swap the two and the p-value changes completely, so check the hint under each field before you type.

🧐 Frequently Asked Questions

What's the difference between the CDF and the right-tail p-value? The CDF is P(X≤x), the probability of landing at or below your value. The F-test uses the right-tail p-value P(X>x) = 1−CDF, the probability of seeing an F value as extreme or more extreme by chance. This tool shows both.

Which df goes in d1? The between-group (factor or predictor) df goes in d1; the within-group (residual or error) df goes in d2. In one-way ANOVA, d1 = k−1 and d2 = n−k. Reversing them gives a different answer.

Does a smaller p-value mean a stronger result? Within the same d1 and d2, a larger F value produces a smaller right-tail p-value. A smaller p means the result is less likely under the null hypothesis of no difference, so the difference is more likely to be statistically significant.

Can I enter a negative F value? No. An F statistic is a ratio of two variances, so it is always 0 or greater. Negative or non-numeric input leaves the output fields showing "—".

What happens with very large degrees of freedom? As both d1 and d2 grow, the F distribution concentrates near 1. The tool is designed for df between 1 and 1000; at extreme values numerical precision can drop slightly.

📚 Why the F-Distribution Shows Up Everywhere

The "F" honors Ronald Fisher, who built the analysis-of-variance framework in the 1920s. Because the F-test compares two variances rather than two means, it sits in a different family from the t-test and underpins quality control, design of experiments, and the overall significance test in regression.

If you have ever read a results line like "F(2, 27) = 4.21, p < .05" in a paper, the two numbers in parentheses are exactly d1 and d2. Type 2, 27, and 4.21 into this tool and you'll reproduce the paper's p-value — handy for checking your own ANOVA output, replicating a published test, or just building intuition for how df shifts the curve.