Inference about means and variances requires more than the normal distribution.
If the population standard deviation were known, normal theory could be used directly for a mean. In practice, the standard deviation is usually estimated from the same sample.
The t distribution accounts for that extra uncertainty, while chi-square and F distributions arise when working with variances and variance ratios.
Key question
Why do inference about means, variances, and variance ratios require different reference distributions?
Where do the three distributions come from?
t distribution
Inference about means
when standard deviation is estimated from the sample
Chi-square distribution
Inference about variance and counts
sum of squared standard-normal values
F distribution
Ratio of variances
ANOVA and overall regression tests
| Distribution | Range | Typical use |
|---|---|---|
| t | negative to positive | mean differences |
| χ² | 0 and above | variance and categorical counts |
| F | 0 and above | variance ratios and ANOVA |
Why does the t distribution have heavier tails?
Replacing the population standard deviation with a sample estimate adds uncertainty to the denominator. The t distribution therefore assigns more probability to extreme values.
As degrees of freedom increase, the standard deviation estimate becomes more stable and the t distribution approaches the standard normal distribution.
Degrees of freedom
For a sample with one estimated mean, the degrees of freedom are typically n−1.
What do chi-square and F distributions compare?
A chi-square variable is built from squared standard-normal values, so it cannot be negative and is usually right-skewed.
An F variable is the ratio of two independent variance estimates. It is also nonnegative and right-skewed.
The ANOVA F statistic
ANOVA divides between-group variation by within-group variation.
Which tests use each distribution?
| Analysis | Reference distribution |
|---|---|
| One-mean or two-mean comparison | t distribution |
| Variance test or chi-square count test | chi-square distribution |
| ANOVA or overall regression test | F distribution |
Connection to the p-value
The p-value is the probability, under the reference distribution, of a statistic at least as extreme as the observed one.
Change the degrees of freedom
Switch distributions and adjust degrees of freedom to compare tail thickness and skew.
Key lesson
Degrees of freedom reflect the amount of independent information remaining in the sample.