Calculator Inputs
Example Data Table
| Observed F | df1 | df2 | α | CDF | Right Tail | |
|---|---|---|---|---|---|---|
| 0.75 | 4 | 12 | 0.05 | 0.587203 | 0.423283 | 0.576717 |
| 1.50 | 5 | 15 | 0.05 | 0.298137 | 0.751850 | 0.248150 |
| 2.25 | 8 | 20 | 0.10 | 0.104377 | 0.932377 | 0.067623 |
| 3.00 | 10 | 18 | 0.01 | 0.029999 | 0.979466 | 0.020534 |
Formula Used
The F distribution compares two scaled variance estimates. Its density and cumulative probability depend on the observed F value and two degrees of freedom parameters.
f(x) = [(d1/d2)^(d1/2) × x^(d1/2 - 1)] / [B(d1/2, d2/2) × (1 + (d1/d2)x)^((d1 + d2)/2)]
F(x) = I(d1x)/(d1x+d2)(d1/2, d2/2)
P(F ≥ x) = 1 - F(x)
p ≈ 2 × min[F(x), 1 - F(x)]
Find x such that F(x) = p, where p may be α, 1-α, α/2, 1-α/2, or any chosen target probability.
How to Use This Calculator
- Enter the observed F statistic from your test or variance ratio.
- Provide numerator and denominator degrees of freedom.
- Set the significance level for critical values and decisions.
- Enter a target cumulative probability if you also need a quantile.
- Optionally define a graph maximum x range.
- Choose how many decimals you want displayed.
- Press the calculate button to see probabilities, critical values, summary measures, and the graph.
- Use CSV or PDF export buttons to save the output.
Frequently Asked Questions
1. What does this calculator return?
It returns the F density, cumulative probability, right-tail probability, approximate two-tailed p value, critical values, selected quantile, and summary measures such as mean, variance, and mode when defined.
2. When is the F distribution used?
It is used in ANOVA, variance ratio testing, regression model comparison, and other procedures where a statistic follows an F distribution under the null hypothesis.
3. Why are there two degrees of freedom values?
The F distribution is built from a ratio of two scaled chi-square variables. Each component contributes its own degrees of freedom, so both values affect the curve shape.
4. What is the difference between PDF and CDF?
The PDF shows density at a specific F value. The CDF shows the probability that the random variable is less than or equal to that value.
5. How should I read the right-tail probability?
It is the probability of observing an F statistic at least as large as your input, assuming the reference F distribution is correct.
6. Why can mean, variance, or mode be undefined?
These moments only exist under certain denominator or numerator degrees of freedom conditions. If those conditions are not met, the value is mathematically undefined.
7. What does the target cumulative probability do?
It finds the quantile whose cumulative probability matches your chosen level. This is useful for thresholds, percentiles, and custom reference points.
8. Is the two-tailed p value always appropriate?
Not always. Many F tests are naturally upper-tailed. The doubled smaller-tail approach is mainly helpful when interpreting variance ratios in a two-sided way.