Track 5 · Probability Distributions · Lesson 8 of 10
The t, chi-square and F distributions
Explain where the t, chi-square and F distributions come from, choose their degrees of freedom, and calculate tail probabilities with them.
A quality engineer measures 10 bottles to check a filling machine. She does not know the machine's true standard deviation, so she estimates it from the same 10 bottles — and that extra uncertainty changes which distribution her calculations must use. The t, chi-square and F distributions are the exact answers to questions like hers.
The normal, binomial and Poisson distributions model data. The three distributions in this lesson model statistics calculated from data — they are sampling distributions. Each one appears when we take a sample from a normal population and replace an unknown population quantity with its estimate from the sample.
All three are indexed by degrees of freedom (df): roughly, the number of independent pieces of information left for estimating variability.
Included with a Formula Quest membership
It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:
- Explain that t, chi-square and F are sampling distributions of statistics from normal samples
- Choose the degrees of freedom: n − 1 for one sample, (n₁ − 1, n₂ − 1) in order for F
- Calculate upper-tail probabilities for t, chi-square and F statistics
- Explain why t has heavier tails than the normal and approaches it as df grows