Calculator
Example Data Table
| Test type | Inputs | Main calculation | Expected F |
|---|---|---|---|
| Variance ratio | s1² = 18.4, df1 = 11, s2² = 7.9, df2 = 9 | 18.4 / 7.9 | 2.3291 |
| ANOVA | SSB = 54.6, dfB = 2, SSW = 124.8, dfW = 21 | (54.6 / 2) / (124.8 / 21) | 4.5949 |
| Nested regression | SSE reduced = 420, SSE full = 360, q = 3, df full = 44 | ((420 - 360) / 3) / (360 / 44) | 2.4444 |
Formula Used
Variance ratio: F = s12 / s22.
ANOVA: F = MSB / MSW, where MSB = SSB / dfB and MSW = SSW / dfW.
Nested regression: F = ((SSE reduced - SSE full) / q) / (SSE full / df full).
Probability: p values are estimated from the F distribution with numerator and denominator degrees of freedom.
How to Use This Calculator
- Select the F test type that matches your problem.
- Enter positive variance, sum of squares, or error values.
- Enter the correct numerator and denominator degrees of freedom.
- Set alpha, such as 0.05, for critical values.
- Press the calculate button.
- Read the result above the form.
- Use the CSV or PDF download for records.
F Test Statistic Guide
What the F Test Measures
An F test compares two estimates of variance. It asks whether the ratio is large enough to suggest a real difference. This calculator supports three common settings. You can compare two sample variances. You can test an ANOVA mean square ratio. You can also test nested regression models. Each setting uses the same core idea. A numerator variance estimate is divided by a denominator variance estimate.
Degrees of Freedom
The F value is always positive. Values near one show similar variance estimates. Large values often show stronger evidence against the null claim. The p value depends on two degrees of freedom. The numerator degrees of freedom describe the top estimate. The denominator degrees of freedom describe the bottom estimate. Good inputs matter because wrong degrees of freedom change the decision.
Supported Test Methods
For a two variance test, the calculator can place the larger variance on top. That method gives an F value of at least one. It is useful for quick classroom work. For ANOVA, enter sums of squares and their degrees of freedom. The tool divides each sum by its degrees of freedom. Then it forms the between to within ratio. For nested regression, enter the reduced and full model error sums. The tool measures whether the extra predictors reduce error enough.
P Values and Critical Values
The calculator also reports right tail, left tail, and two tail probabilities. A right tail result is common for ANOVA and regression. A two tail result is often used for equality of variances. Critical values are estimated with numerical search. They help compare your result with an alpha level. The default alpha is 0.05, but you can change it.
Practical Review Notes
Use this page to check hand calculations. Start by choosing the test type. Enter positive sums, variances, and degrees of freedom. Keep units consistent. Press calculate to view steps above the form. Download the CSV for spreadsheets. Download the PDF for a quick report. The calculator supports study, audit, and review tasks. It does not replace statistical judgment. Always confirm assumptions such as independence, normality, and model fit before drawing conclusions. When data are skewed, consider robust methods. When sample sizes are tiny, interpret p values carefully. Save both inputs and notes so another reader can reproduce the calculation without extra guessing.
FAQs
What is an F test statistic?
It is a ratio of two variance estimates. It is used in variance comparisons, ANOVA, and nested model tests.
Can the F statistic be negative?
No. Variances and mean squares are nonnegative. A valid F statistic is always zero or positive.
Which tail should I use?
Use the right tail for ANOVA and nested regression. Use two tails when testing equality of two variances.
What are numerator degrees of freedom?
They describe the variance estimate placed on top of the ratio. They strongly affect the F distribution.
What are denominator degrees of freedom?
They describe the variance estimate placed on the bottom of the ratio. They also affect p values and critical values.
Why place the larger variance on top?
Many classroom variance tests do this to create an F value at least one. It simplifies two tail comparison work.
Can this calculator test regression models?
Yes. Use the nested regression option. Enter reduced SSE, full SSE, added predictors, and full residual degrees of freedom.
Does a large F always prove significance?
No. The decision depends on degrees of freedom, alpha, tail choice, assumptions, and the matching p value.