Hypothesis Tests
F Critical Value Calculator
An F critical value is the upper-tail cutoff on the F curve set by the numerator degrees of freedom, the denominator degrees of freedom, and alpha.
Worksheet
Press Calculate to see the result.
An F critical value is the upper-tail cutoff on the F curve set by the numerator degrees of freedom, the denominator degrees of freedom, and alpha.
What this calculator answers
An F critical value marks the upper tail of a ratio of two scaled variances, or of two mean squares. ANOVA and a comparison of two variances both end on this curve, but they arrive with different degrees of freedom, and the order of those degrees of freedom matters because the F family is not symmetric. This page takes numerator df, denominator df, and alpha, and it returns the point that leaves probability alpha to the right. Swapping the two df values is a different curve, so the boxes are labeled by role rather than as an anonymous pair.
The headline is the upper critical value. The two degrees of freedom are reported beside it so a table lookup can be checked. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Upper-tail F critical value uses this relationship.
Find F so the probability of a larger F equals alpha, with numerator df1 and denominator df2.
- df1. Numerator degrees of freedom.
- df2. Denominator degrees of freedom.
- alpha. Upper-tail probability.
- F critical. The cutoff with upper area alpha.
Under the null, a ratio of two independent chi-square variables divided by their degrees of freedom follows F. The page inverts that distribution at the upper probability alpha. Because the curve piles up near zero and stretches to the right, the critical value is an upper cutoff only. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The loaded lookup uses numerator df 3, denominator df 20, and alpha 0.05.
Numerator df is 3, denominator df is 20, and alpha is 0.05. The upper-tail F critical value is 3.0984. Reject when the observed F exceeds 3.0984 on (3, 20) degrees of freedom.
The F curve is upper-tailed for this lookup. Probability 0.05 sits to the right of the cutoff. Swapping the degrees of freedom would build a different curve. This answer keeps 3 on top. An F statistic below 1 cannot pass an upper cutoff of 3.0984.
ANOVA uses numerator df equal to groups minus 1. A two-variance ratio uses n minus 1 for each sample. The numbers typed here are 3 and 20. This page returns the cutoff only. The p-value from F page is the tool that starts from an observed F.
| F critical value | 3.0984 |
|---|---|
| Numerator df | 3 |
| Denominator df | 20 |
A second situation
Numerator df 5 and denominator df 10 at alpha 0.05
Numerator df is 5, denominator df is 10, and alpha is 0.05. The upper-tail F critical value is 3.3258. Reject when the observed F exceeds 3.3258 on (5, 10) degrees of freedom.
The F curve is upper-tailed for this lookup. Probability 0.05 sits to the right of the cutoff. Swapping the degrees of freedom would build a different curve. This answer keeps 5 on top. An F statistic below 1 cannot pass an upper cutoff of 3.3258.
ANOVA uses numerator df equal to groups minus 1. A two-variance ratio uses n minus 1 for each sample. The numbers typed here are 5 and 10. This page returns the cutoff only. The p-value from F page is the tool that starts from an observed F.
| F critical value | 3.3258 |
|---|---|
| Numerator df | 5 |
| Denominator df | 10 |
Changing the denominator df moves the cutoff even when the numerator df and alpha stay fixed. A variance test that puts the larger sample on top is using a different df order than one that keeps a fixed group in the numerator. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
Denominator degrees of freedom moved from 20 to 10. Numerator df and alpha stayed put. The headline result went from 3.0984 to 3.7083.
The nudged inputs
Numerator df is 3, denominator df is 10, and alpha is 0.05. The upper-tail F critical value is 3.7083. Reject when the observed F exceeds 3.7083 on (3, 10) degrees of freedom.
The F curve is upper-tailed for this lookup. Probability 0.05 sits to the right of the cutoff. Swapping the degrees of freedom would build a different curve. This answer keeps 3 on top. An F statistic below 1 cannot pass an upper cutoff of 3.7083.
ANOVA uses numerator df equal to groups minus 1. A two-variance ratio uses n minus 1 for each sample. The numbers typed here are 3 and 10. This page returns the cutoff only. The p-value from F page is the tool that starts from an observed F.
| F critical value | 3.7083 |
|---|---|
| Numerator df | 3 |
| Denominator df | 10 |
How to read the result
Reject for an observed F at least as large as this cutoff when the numerator and denominator df match the statistic. An F below 1 is on the lower side of the curve and is not past an upper cutoff. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write F critical with df1, df2, and alpha, in that order. A single number without the df pair cannot be found again in a table. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The numerator and denominator mean squares are independent under the null.
- Each mean square is scaled by the df you typed.
- Alpha is the upper-tail error rate you intend.
- Both df values are positive.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Swapping df1 and df2 and expecting the same cutoff.
- Using a two-sided z habit and halving alpha without deciding that you truly want the upper tail at alpha/2.
- Looking up F with n in place of n minus 1 for a two-variance test.
- Comparing an ANOVA F with the critical value from a different number of groups.
Where a different calculator is the honest choice
The ANOVA calculator computes F from group lists and then the p-value. The F-test calculator compares two sample variances. The p-value from F calculator starts from an F you already calculated. This page is the cutoff alone. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: P-Value from F Calculator, F-Test Calculator, ANOVA Calculator.
Check yourself
What is the upper F critical value at alpha 0.05 with numerator df 3 and denominator df 20?
The F critical value is 3.0984.
Using the result in a sentence
One-way ANOVA and two-variance tests are the standard callers. Regression mean-square tests use the same curve with their own numerator and denominator df. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
The cutoff assumes normality of the underlying measurements, or a solid large-sample excuse. A significant F does not say which group mean differs. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the F Critical Value Calculator commit to?
This page uses Upper-tail F critical value: Find F so the probability of a larger F equals alpha, with numerator df1 and denominator df2.. Under the null, a ratio of two independent chi-square variables divided by their degrees of freedom follows F. The page inverts that distribution at the upper probability alpha. Because the curve piles up near zero and stretches to the right, the critical value is an upper cutoff only. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Numerator df, Denominator df, Alpha, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the f critical value is 3.0984. Reject for an observed F at least as large as this cutoff when the numerator and denominator df match the statistic. An F below 1 is on the lower side of the curve and is not past an upper cutoff. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Swapping df1 and df2 and expecting the same cutoff. The cutoff assumes normality of the underlying measurements, or a solid large-sample excuse. A significant F does not say which group mean differs. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
The ANOVA calculator computes F from group lists and then the p-value. The F-test calculator compares two sample variances. The p-value from F calculator starts from an F you already calculated. This page is the cutoff alone. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The F critical value is 3.0984. Write F critical with df1, df2, and alpha, in that order. A single number without the df pair cannot be found again in a table. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.