t-distribution
t-distribution is a symmetric, bell-shaped probability distribution with heavier tails than the standard normal. It arises when estimating a population mean from sample data with unknown variance.
Also known asStudent's t
See it move
The chart shows a probability-density curve of the t-distribution plotted against the t-statistic on the horizontal axis. The t-distribution has heavier tails than the standard normal, reflecting greater uncertainty when the population variance is unknown; as degrees of freedom increase from 5 to 30 and beyond, the curve's tails narrow and it converges to the standard-normal shape. This property means critical values from the t-table are larger than their normal-distribution equivalents, especially for small samples.
The formula
Variables
- Sample mean
- Population mean (hypothesised value under H₀)
- Sample standard deviation
- Sample size (observations)
One-sample t-statistic; follows a t-distribution with df = n − 1 when the population is approximately normal and σ is unknown.
Variables
- Degrees of freedom — controls the shape and tail weight of the t-distribution
- Sample size (observations)
As df increases, the t-distribution's tails thin and it converges to the standard normal N(0,1).
Check yourself
When estimating a population mean from a small sample with unknown population variance, why is the t-distribution used instead of the standard normal distribution?