# Uniform Distribution Calculator

Free uniform distribution calculator. Compute mean, variance, PDF, and CDF for the continuous uniform distribution on any interval [a, b].

## What this calculates

Calculate statistics and probabilities for the continuous uniform distribution. Every value in the interval [a, b] is equally likely, making this the simplest continuous probability distribution.

## Inputs

- **Minimum (a)** — Lower bound of the uniform distribution.
- **Maximum (b)** — Upper bound of the uniform distribution.
- **Value (x)** — A value within [a, b] for probability calculations.

## Outputs

- **Mean** — Expected value (a + b) / 2.
- **Variance** — Variance (b - a)^2 / 12.
- **Standard Deviation** — Standard deviation (b - a) / sqrt(12).
- **P(X <= x)** — Probability that X is less than or equal to x.
- **P(X >= x)** — Probability that X is greater than or equal to x.
- **PDF f(x)** — Probability density function at x.

## Details

The continuous uniform distribution on [a, b] assigns equal probability density to every point in the interval. It is the "flat" distribution, and its PDF is a horizontal line at height 1/(b-a).

Key formulas: Mean = (a + b) / 2, Variance = (b - a)^2 / 12, PDF f(x) = 1/(b-a) for a <= x <= b, CDF F(x) = (x - a) / (b - a). The CDF is a straight line from 0 to 1, reflecting the equal probability across the interval.

Applications: Uniform distributions model scenarios where all outcomes in a range are equally likely: random number generators, rounding errors, arrival times when nothing is known about the pattern, and as a non-informative prior in Bayesian statistics. It also serves as the basis for the probability integral transform, which converts any continuous distribution to uniform.

## Frequently Asked Questions

**Q: What is the difference between discrete and continuous uniform distributions?**

A: A discrete uniform distribution has a finite set of equally likely outcomes (like a fair die, where each face has probability 1/6). A continuous uniform distribution covers an interval where every point is equally likely, with probability density 1/(b-a).

**Q: Why is the variance (b-a)^2 / 12?**

A: This comes from integrating (x - mean)^2 times the PDF over [a, b]. The integral of (x - (a+b)/2)^2 / (b-a) from a to b simplifies to (b-a)^2 / 12. The factor of 12 is characteristic of the uniform distribution.

**Q: When should I use a uniform distribution?**

A: Use it when you have no reason to believe any value in a range is more likely than another. Common examples include random number generation, modeling uncertainty when only the range is known, and as a baseline for comparing other distributions.

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Source: https://vastcalc.com/calculators/statistics/uniform-distribution
Category: Statistics
Last updated: 2026-04-21