F Critical Value Table

A common mistake is treating "n ≥ 30 means use Z" as a rule for choosing critical values. The real choice between the Z distribution and the t distribution depends on whether the population standard deviation is known or estimated from the sample, not on sample size alone. The F critical value operates under its own logic: it is always positive, used for comparing multiple group variances in ANOVA and for regression F-tests, and it requires a pair of degrees of freedom: numerator df1 and denominator df2. The F critical value is the threshold in the F-distribution that defines the rejection region for a null hypothesis in these tests.

F tables at alpha 0.10, 0.05, and 0.01 give the critical values for the F-distribution at these standard significance levels. The NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.7, provides exact tables for these alpha values. A right-tail critical value is used for hypothesis tests in ANOVA. For example, with df1=5 and df2=10 at alpha=0.05, the F critical value is 3.33. With df1=10 and df2=5 at the same alpha, it is 4.74, showing how swapping the degrees of freedom changes the threshold.

The F table lists critical values for the F-distribution, organised by numerator degrees of freedom (df1) in columns and denominator degrees of freedom (df2) in rows. Textbooks like OpenIntro Statistics (4th ed.) and Moore, McCabe & Craig's Introduction to the Practice of Statistics print these tables for alpha 0.10, 0.05, and 0.01, typically to three or four decimal places. To look up the correct value, find the row for df2 and the column for df1, then read the cell at their intersection.

The F distribution table is used when you need an F cut-off for ANOVA, regression or comparing two variances. The F-distribution is asymmetric, so you must specify the right tail for hypothesis tests. A left-tail value for the F-distribution would be near zero and is not the threshold you need. For a two-tailed F test comparing two variances, the lower critical value is 1 divided by the F critical value with the degrees of freedom swapped: 1 / F(alpha, df2, df1).

In ANOVA, the numerator degrees of freedom (df1) equal the number of groups minus one. The denominator degrees of freedom (df2) equal the total sample size minus the number of groups. For a one-way ANOVA with k groups and n total observations, df1 = k - 1 and df2 = n - k. In regression, df1 is the number of predictors in the model, and df2 is the sample size minus the number of predictors minus one. For a two-variance test, df1 is the degrees of freedom for the numerator variance, and df2 is for the denominator variance.

The numerator denominator degrees of freedom pair defines the shape of the F-distribution. The NIST table gives critical values for df1 from 1 to 10 and for df2 from 1 to 1000 for common alpha levels. Software like Excel and Google Sheets uses the function F.INV.RT(alpha, df1, df2) to return the exact right-tail critical value for any alpha and any degrees of freedom pair. This function accepts non-integer degrees of freedom in most implementations.

A two-tailed F test for comparing two variances uses both an upper and a lower critical value. The upper critical value is F(alpha/2, df1, df2) from a standard F table or software function. The lower critical value is 1 / F(alpha/2, df2, df1). This reciprocal relationship is necessary because the F-distribution is not symmetric. For a test at alpha=0.05, the upper critical value uses the right-tail threshold at 0.025, and the lower critical value uses the same threshold with the degrees of freedom swapped.

Suppose you run a one-way ANOVA with three groups, each with 10 observations. The numerator df1 equals 3 minus 1, which is 2. The denominator df2 equals 30 minus 3, which is 27. For alpha=0.05, the F critical value from a table or software is approximately 3.35. If the F-statistic from your data exceeds 3.35, you reject the null hypothesis that all group means are equal. If the F-statistic falls below 3.35, you fail to reject the null. This threshold remains the same whether you report a p-value or make a decision by hand.

For a one-way ANOVA: df1 = k - 1, where k is the number of groups. df2 = N - k, where N is the total number of observations. For a regression F-test: df1 = number of predictors in the model. df2 = n - number of predictors - 1. For a two-variance F-test: df1 = n1 - 1, where n1 is the sample size for the variance in the numerator. df2 = n2 - 1, where n2 is the sample size for the variance in the denominator.