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F Ratio — Definition, Formula & Examples

The F ratio is a test statistic that compares how much group means differ from each other with how much individual values vary inside the groups. A large F ratio suggests the differences between groups are bigger than random scatter alone would produce.

In one-way ANOVA, the F ratio is F=MSB/MSWF = \text{MSB}/\text{MSW}. Here MSB (mean square between groups) is the between-group sum of squares divided by k−1k-1. MSW (mean square within groups) is the within-group sum of squares divided by N−kN-k. If the null hypothesis of equal population means is true and the conditions hold, FF follows an F distribution with (k−1,N−k)(k-1, N-k) degrees of freedom. In regression, the analogous statistic is F=MSR/MSEF = \text{MSR}/\text{MSE}.

Key Formula

F=MSBMSW=SSB/(k−1)SSW/(N−k)F = \frac{\text{MSB}}{\text{MSW}} = \frac{\text{SSB}/(k-1)}{\text{SSW}/(N-k)}
Where:
  • SSB\text{SSB} = Sum of squares between groups (spread of group means around the overall mean)
  • SSW\text{SSW} = Sum of squares within groups (spread of values around their own group mean)
  • kk = Number of groups
  • NN = Total number of observations across all groups

Worked Example

Problem: You compare three teaching methods with 5 students each, for N=15N = 15 students in total. The ANOVA gives SSB = 80 and SSW = 120. Find the F ratio. Then decide whether it exceeds the 5% critical value, F0.05(2,12)≈3.89F_{0.05}(2, 12) \approx 3.89.
Step 1: Find the degrees of freedom for each source of variation.
dfbetween=k−1=3−1=2,dfwithin=N−k=15−3=12df_{\text{between}} = k - 1 = 3 - 1 = 2, \qquad df_{\text{within}} = N - k = 15 - 3 = 12
Step 2: Turn each sum of squares into a mean square by dividing by its degrees of freedom.
MSB=802=40,MSW=12012=10\text{MSB} = \frac{80}{2} = 40, \qquad \text{MSW} = \frac{120}{12} = 10
Step 3: Divide the mean squares to get F, then compare it with the critical value.
F=4010=4>3.89F = \frac{40}{10} = 4 > 3.89
Answer: F=4F = 4 with (2,12)(2, 12) degrees of freedom. This exceeds the critical value of about 3.89, so you reject the null hypothesis at the 5% level. At least one teaching method has a different mean score.

Why It Matters

Computer regression printouts on the AP Statistics exam often list an F statistic next to the slope's t statistic. In simple linear regression, the two tests are equivalent because F=t2F = t^2 for the slope, and they give the same p-value. Knowing this lets you read the whole printout instead of guessing what the extra numbers mean.

Common Mistakes

Mistake: Dividing the sums of squares directly, for example 80/120≈0.6780/120 \approx 0.67, instead of the mean squares.
Correction: Each sum of squares must first be divided by its own degrees of freedom. Groups and individuals contribute different numbers of independent pieces of information, and dividing by degrees of freedom puts both on the same scale before you compare them.

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