Snedecor's F distribution is a sampling distribution derived from the ratio of sample variances from two independent normal distributions. Suppose we take n samples from a normal(μX, σX²) distribution, and m samples from a normal(μY, σY²) distribution. Let SX² and SY² be the respective sample variances. The random variable (SX²/σX²)/(SY²/σY²) has the F(n - 1, m - 1) distribution.
| Parameter | Range | Description |
|---|---|---|
| ν₁ | ν₁ = 1, 2, ... | Numerator degrees of freedom |
| ν₂ | ν₂ = 1, 2, ... | Denominator degrees of freedom |
Probability Density Function
Support
Mean
Variance
| Example | ν₁ | ν₂ |
|---|---|---|
| Two independent random variables Y₁ and Y₂ have chi-squared(3) and chi-squared(6) distributions respectively. Then X = (Y₁/3)/(Y₂/6) has an F(3, 6) distribution. | 3.000 | 6.000 |
| Heights have a normal(63.7, 5.8) distribution for women and normal(69.1, 7.6) distribution for men (in inches). Samples of 10 women and 20 men have variances S₁² and S₂². Then X = (S₁²/5.8)/(S₂²/7.6) has an F(9, 19) distribution. | 9.000 | 19.00 |
| Two random groups of students from the same school take the SAT. Group 1 with 50 students has sample variance S₁², while Group 2 with 80 students has sample variance S₁². Then X = S₁²/S₂² has an F(49, 79) distribution. | 49.00 | 79.00 |
X ~ F(ν₁, ν₂)
E(X) = , Var(X) =
Note that the F distribution is constructed as the ratio of two independent scaled chi-squared random variables. The expected values and variances of such ratio distributions are often undefined. In this case, the expected value is only defined for ν₂ > 2, and the variance is only defined for ν₂ > 4.
The illustration above shows a sample of n red points from a normal(μx, σx2) distribution, and a sample of m purple points from a normal(μy, σy2) distribution. Both samples are chosen independently and at random. The random variable X = (Sx2/σx2)/(Sy2/σy2) has an F(n - 1, m - 1) distribution, where Sx2 and Sy2are the sample variances.
The simulation above shows a sample of n points (marked red) on the x-axis chosen independently and at random from a normal(μx, σx2) distribution, and a sample of m points (marked purple) on the y-axis chosen independently and at random from a normal(μy, σy2) distribution. The light blue circle shows the value of X = (Sx2/σx2)/(Sy2/σy2), where Sx2 and Sy2are the sample variances. The random variable X has an F(n - 1, m - 1) distribution. The histogram accumulates the results of each simulation.