About F Distribution Calculator
This F distribution calculator gives upper tail probabilities and critical values for a numerator degrees of freedom d1 and a denominator degrees of freedom d2. It opens on d1 = 3 and d2 = 12 and finds the p-value of F = 3.49, the 5% critical value. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
An F variable is the ratio of two independent chi-square variables, each divided by its degrees of freedom. ANOVA, regression overall tests and variance comparisons all use it, and the test is right-tailed.
The order of the degrees of freedom matters. In a one-way ANOVA, d1 is the number of groups minus 1 and d2 is the total sample size minus the number of groups.
How to use F Distribution Calculator
- Choose the distribution from the list. Each has its own parameters, such as the mean and standard deviation for the normal.
- Enter the parameters. The hint under each field says what is allowed.
- Choose what to calculate: a lower tail, an upper tail, an interval, a density or point probability, or the inverse.
- Enter x, or a and b for an interval, or the probability for the inverse. The result updates as you type.
- Read the answer, the mean and variance, and the shaded chart. Open the working to see which CDF values were used.
- Copy the result with the copy button.
Features
- Nine distributions: normal, binomial, Poisson, geometric, exponential, uniform, Student t, chi-square and F.
- Lower and upper tails, strict and non-strict, and intervals P(a ≤ X ≤ b).
- Density for continuous and point probability for discrete distributions.
- Inverse (quantile) calculation, including critical values for t, chi-square and F.
- Mean, variance and standard deviation, shown as undefined where they do not exist.
- Bell curve with shaded tail or interval for continuous distributions, and a bar chart for discrete ones.
- Upper tails computed directly, so very small p-values keep their precision.
- Formula of the distribution in KaTeX and a short worked step for every answer.
F density and moments
The density uses the beta function B. The mean exists only when d2 is above 2 and the variance only when d2 is above 4.
f(x) = sqrt((d1·x)^d1 × d2^d2 / (d1·x + d2)^(d1+d2)) / (x × B(d1/2, d2/2))
mean = d2 / (d2 − 2)Worked examples
d1 = 3, d2 = 12: P(F ≥ 3.49) gives 0.050011
About 0.05, so 3.49 is the 5% critical value.
d1 = 2, d2 = 15: 95th percentile gives 3.6823
The critical value for a one-way ANOVA with three groups and 18 observations.
Privacy: how it works
- Everything is calculated by code running in your browser tab. Nothing you enter is sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- For a discrete distribution, P(X < 3) and P(X ≤ 3) are different. For a continuous one they are the same.
- For a two-sided critical value, such as the 95% value of t, put the inverse probability at 0.975.
- A density can be above 1 when the spread is small. It is not a probability.
- If your question is "at least k", use P(X ≥ x), not P(X > x).
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I find the p-value for an F statistic?
Choose P(X ≥ x), enter d1 and d2 and the F value. The upper tail is the p-value.
Which degrees of freedom go first?
The numerator degrees of freedom first, then the denominator. Swapping them gives a different distribution.
Why can the mean be undefined?
The mean exists only when d2 is above 2. With fewer denominator degrees of freedom the tail is too heavy.
More Probability Distribution Calculator pages
- Probability Distribution CalculatorPick a distribution, set its parameters and get PDF, CDF, interval probabilities, quantiles, mean and variance with a chart.
- Normal Distribution CalculatorEnter a mean and standard deviation to get tail probabilities, intervals and percentiles on a bell curve.
- Binomial Distribution CalculatorEnter the number of trials and the chance of success to get P(X = k), P(X ≤ k), P(X ≥ k) and ranges.
- Poisson Distribution CalculatorEnter the average rate λ to get the chance of exactly k events, at most k, at least k or a range.
- Geometric Distribution CalculatorEnter the success probability to get the chance that the first success happens on or by trial k.
- Exponential Distribution CalculatorEnter the rate λ to get the chance a waiting time exceeds, or stays below, a value, and its percentiles.
- Uniform Distribution CalculatorEnter the minimum a and maximum b to get the chance of landing in any sub-interval.
- Student t Distribution CalculatorEnter the degrees of freedom to get t critical values, tail areas and p-values on a shaded t curve.
- Chi-Square Distribution CalculatorEnter the degrees of freedom to get chi-square p-values, critical values and tail areas.