What is the F-distribution?

The F-distribution is a continuous sampling distribution of the ratio of two independent random variables with chi-square distributions, each divided by its degrees of freedom.
ν1 = 1 and ν2 = 1
ν1 = 1 and ν2 = 9
ν1 = 9 and ν2 = 1
ν1 = 9 and ν2 = 9

The F-distribution is right skewed and described by its numerator (ν1) and denominator (ν2) degrees of freedom. The plots below show the effect of different values of degrees of freedom on the shape of the distribution.

Use the F-distribution when a test statistic is the ratio of two variables that each have a chi-square distribution. For example, use the F-distribution in the analysis of variance and in hypothesis testing to determine whether two population variances are equal.

Calculate probabilities for an F-distribution with infinite denominator degrees of freedom

Suppose X follows an F-distribution with 5 numerator degrees of freedom and infinite denominator degrees of freedom, and you want the probability that X is less than or equal to 2. You can find the probability that Y is less than or equal to 2, where Y follows an F-distribution with 5 numerator and 99999 denominator degrees of freedom and Y approximates X.

  1. Open the cumulative distribution function dialog box.
    • Mac: Statistics > Probability Distributions > Cumulative Distribution Function
    • PC: STATISTICS > CDF/PDF > Cumulative Distribution Function
  2. In Form of input, select A single value.
  3. In Value, enter 2.
  4. In Distribution, select F.
  5. In Numerator degrees of freedom, enter 5.
  6. In Denominator degrees of freedom, enter 99999.
  7. Click OK.

The CDF for 2 is 0.924755. This value gives the area under the curve up to 2.

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