Calculator
Example data table
| df1 | df2 | x | CDF P(F ≤ x) | Right-tail P(F ≥ x) |
|---|---|---|---|---|
| 5 | 10 | 2.5 | 0.897998 | 0.102002 |
| 2 | 20 | 1.8 | 0.808936 | 0.191064 |
| 10 | 8 | 3.2 | 0.943718 | 0.056282 |
Numbers above are computed using the same functions as the calculator.
Formula used
PDF:
f(x) = ((d1/d2)^(d1/2) · x^(d1/2 − 1)) / (B(d1/2, d2/2) · (1 + (d1/d2)x)^((d1+d2)/2)), x > 0
CDF:
P(F ≤ x) = Iz(d1/2, d2/2), where z = (d1·x)/(d1·x + d2)
- B(·) is the beta function.
- Iz(·) is the regularized incomplete beta function.
- Critical values are found by numerically inverting the CDF.
How to use this calculator
- Enter numerator and denominator degrees of freedom (df1, df2).
- Type your F statistic value (x), which must be positive.
- Set alpha (for example 0.05) to compute critical values.
- Choose a tail focus if you want right, left, or two-tail bounds.
- Click Calculate to see PDF, CDF, p-values, and cutoffs.
- Use Download CSV or Download PDF to export the summary.
Article
1) What the F curve represents
The F curve describes the F distribution, used when you compare two variances. It appears in ANOVA, regression model tests, and quality control studies. The curve depends on two degrees of freedom: df1 for the numerator and df2 for the denominator. As df values increase, the curve becomes less skewed and more concentrated near 1.
2) Degrees of freedom and curve shape
Small degrees of freedom create a heavy right tail, which means large F values are more plausible. For example, with df1=2 and df2=5, the distribution is very skewed. With df1=20 and df2=40, it tightens and the tail shrinks. This is why df inputs are critical in any variance-ratio conclusion.
3) PDF vs CDF outputs
The calculator shows both PDF and CDF. The PDF, f(x), is the curve’s height at your chosen F value. The CDF, P(F ≤ x), gives the accumulated probability up to x. Together, they help you see whether your F statistic sits in a common region or deep in the tail.
4) p-values for hypothesis testing
In many tests you care about the right-tail p-value, P(F ≥ x). If p ≤ α, the result is statistically significant under that alpha level. Typical alpha choices are 0.10, 0.05, and 0.01. The left-tail value is useful when the alternative hypothesis focuses on unusually small variance ratios.
5) Critical values and decision boundaries
Critical values answer: “What F cutoff leaves α probability in the tail?” For a right-tail test, the calculator finds the F1−α point so that P(F ≥ cutoff)=α. For two-tail style reporting, it also shows lower and upper bounds using α/2 on each side. These cutoffs are the decision boundaries you compare against your statistic.
6) Interpreting the plot with your marker
The plot draws the PDF curve and places a marker at your input x. If your marker sits far to the right where the curve is near zero, the right-tail probability will be small. If it sits near the peak, the result is more typical. Use the plot as a visual “sanity check” alongside the numeric p-value.
7) Practical data tips for better results
Always confirm how your software defines df1 and df2. In one-way ANOVA with k groups and N total samples, a common setup is df1=k−1 and df2=N−k. Keep units consistent when you compute mean squares, and report both the F statistic and chosen alpha in your write-up.
FAQs
1) What is an F statistic in simple terms?
It is a ratio that compares two variance estimates. If the ratio is much larger than 1 (given df1 and df2), it suggests the numerator variance is unusually large under the null hypothesis.
2) Which p-value should I use: left or right?
Most ANOVA and variance-ratio tests use the right-tail p-value, P(F ≥ x). Use the left-tail only when your alternative hypothesis expects an unusually small ratio.
3) Why does changing degrees of freedom change the result?
Degrees of freedom control the F curve’s shape and tail thickness. Smaller df values produce heavier tails, so the same F statistic can be less “surprising” than it would be under larger df values.
4) What does the critical value mean?
The critical value is a cutoff based on alpha. For a right-tail test, values above the cutoff fall in the rejection region because only α probability remains beyond that point under the null model.
5) What alpha should I choose?
Common choices are 0.05 for general research, 0.01 for stricter evidence, and 0.10 for exploratory work. Pick alpha before seeing results, and match it to your field’s reporting standards.
6) Can this calculator be used for two-tailed tests?
Yes. It reports lower and upper cutoffs using α/2 on each side and a common two-tail p-value approximation. For formal designs, confirm your specific two-sided test definition for the F distribution.