Calculate probabilities, percentiles, and statistics for continuous and discrete uniform distributions. Includes PDF/CDF visualization, quantile table, order statistics, and sampling distribution.
The uniform distribution calculator handles both continuous and discrete uniform distributions. In the continuous case, every value in an interval is equally likely in the sense of density; in the discrete case, every integer in the range has the same probability.
Enter bounds and query values to compute interval probabilities, review the PDF or PMF, and inspect the quantile and order-statistic tables. For continuous distributions, the page also shows how the sampling distribution of the mean tightens as sample size grows.
The result is a compact way to work with the simplest "flat" probability model and to see how it behaves in both theory and simulation.
Uniform models are a good baseline because the mathematics is simple and the assumptions are explicit. They are useful whenever you know the range but do not have a reason to prefer one value over another.
This page gives you the core calculations in one place so you can move from the interval assumption to probabilities, quantiles, and sample-size effects without rebuilding the formulas manually.
Continuous: f(x) = 1/(b−a) for a ≤ x ≤ b. Mean = (a+b)/2. Variance = (b−a)²/12. Discrete: P(X=k) = 1/n where n = b−a+1. Variance = (n²−1)/12.
Result: P(3 ≤ X ≤ 7) = 40%, mean = 5, σ = 2.887
For Uniform(0, 10), the PDF height is 1/10 = 0.1 everywhere. P(3 ≤ X ≤ 7) = (7−3)/(10−0) = 40%. The mean is the midpoint 5, and variance is 100/12 ≈ 8.33.
In Bayesian statistics, when you have no prior information about a parameter except its range, the uniform distribution is the maximum entropy prior — it encodes "I know nothing about which values are more likely." This makes it the default choice for uninformative priors in Bayesian analysis.
Every probability distribution can be generated from uniform random numbers. If U ~ Uniform(0,1), then F⁻¹(U) follows the distribution F. This inverse transform method is the basis of all Monte Carlo simulation. For example, −ln(U)/λ generates exponential random variables.
The sum of n independent Uniform(0,1) random variables follows the Irwin-Hall distribution. For n=2, the "triangular" distribution. For n=12, it closely approximates the standard normal — this was historically used to generate normal random numbers before more efficient algorithms existed.
Last updated:
Use it when the model really implies equal likelihood across the interval or set: fair dice, random number generation, simple timing assumptions, and other cases where you do not want to weight one value more than another.
Continuous uniform applies to real numbers in interval [a,b] — any value is possible. Discrete uniform applies to integers from a to b, each with equal probability. A die roll is discrete; a random time is continuous.
The denominator 12 comes from integration. The variance of U(0,1) is the integral of (x − 0.5)² from 0 to 1, which equals 1/12. Scaling by (b−a) gives (b−a)²/12. The factor of 12 is fundamental, not arbitrary.
If you take n samples from a distribution and sort them, the k-th smallest is the k-th order statistic X₍ₖ₎. For uniform distributions, E[X₍ₖ₎] divides the interval into n+1 equal parts, placing the k-th value at position k/(n+1).
The mean of n uniform samples has mean (a+b)/2 (same as individual) but standard error σ/√n. By the CLT, the sample mean is approximately normal for large n, even though individual values are uniform.
Yes! U(−5, 5) is perfectly valid. The mean would be 0, variance 100/12 ≈ 8.33. Negative bounds are common when modeling measurement errors centered around zero.