# t distribution

When sigma is unknown, replace it by the sample SD s and use the t distribution instead of the normal. It has heavier tails, shaped by degrees of freedom n minus 1. The interval is mean plus or minus t* x s / square root of n.

Statistics (university) · Part 05: Sampling and estimation · Card 7 of 8 · Last updated 2026-10-19

## Analogy

The t distribution is a cautious normal curve, wider to admit that s is only an estimate.

## Example

**Small sample**

n = 16, mean 20, s = 4
SE = 4 / 4 = 1
df = 15, t* = 2.131 (95%)
Interval: 20 plus or minus 2.131
= 17.87 to 22.13

## In practice

Standard for small samples, where the extra uncertainty in s really matters.

## Practice

### Flashcards

- **When do you use t instead of z?** When the population SD is unknown and you use the sample SD s.
- **Degrees of freedom for a one-sample t interval?** n minus 1.

## Questions people ask

### Why is the t distribution wider than the normal?

The sample SD varies from sample to sample, which adds uncertainty beyond the sample mean. The t distribution has heavier tails to reflect that. With few degrees of freedom the effect is large, and the critical value is noticeably above 1.96.

### When does the t distribution become the same as the normal?

As degrees of freedom grow, the estimate s becomes very reliable, and the t curve approaches the normal curve. By about 30 degrees of freedom they are close, and with infinite degrees of freedom they are identical, so t* becomes 1.96.

## Related concepts

- [Point estimate and bias](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-5/statistics-university-part-5-concept-5)
- [Confidence interval for a mean](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-5/statistics-university-part-5-concept-6)
- [Confidence interval for a proportion](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-5/statistics-university-part-5-concept-8)

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