# Uniform distribution

A uniform distribution on the range a to b makes every stretch of equal width equally likely. Its mean is (a + b) / 2 and its variance is (b - a) squared / 12.

Statistics (university) · Part 04: Continuous distributions and the normal curve · Card 2 of 8 · Last updated 2026-10-19

## Analogy

A spinner with no favourite colour. Every angle is as likely as any other.

## Example

**Waiting for a bus**

Wait uniform from 0 to 20 min
Mean = (0 + 20) / 2 = 10
Variance = 400 / 12 = 33.3
P(wait < 5) = 5 / 20 = 0.25

## In practice

Models a random waiting time when you do not know the timetable, or a random number generator.

## Practice

### Flashcards

- **Mean of a uniform distribution from a to b?** (a + b) / 2.

## Questions people ask

### How do I find probabilities for a uniform distribution?

Find the width of the interval you want and divide by the total width b minus a. For a wait uniform between 0 and 20 minutes, the chance of waiting between 5 and 10 minutes is 5 divided by 20, which is 0.25.

### Is the mean of a uniform distribution always the midpoint?

Yes. The density is flat, so the balance point sits exactly halfway between a and b. The median is the same point. Unlike the normal curve, the uniform has hard edges: no value outside a to b can happen, so the model suits a bounded range only.

## Related concepts

- [Probability density function](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-1)
- [Exponential distribution](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-3)

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