Uniform Distribution Calculator. Enter your lower bound (a) and upper bound (b) to define a continuous uniform distribution, then specify an interval [x₁, x₂] to get the probability P(x₁ ≤ X ≤ x₂). Results include the probability density (PDF), cumulative probability (CDF), mean, variance, and standard deviation — all computed from your inputs. Also try the Permutation Calculator.
The continuous uniform distribution is a probability distribution where all outcomes in a given interval [a, b] are equally likely. It is sometimes called the rectangular distribution because its probability density function forms a perfect rectangle when graphed. Any sub-interval of the same length within [a, b] has the same probability of occurring. See also our Probability Fraction Calculator.
The probability that a uniformly distributed random variable X falls between x₁ and x₂ is P(x₁ ≤ X ≤ x₂) = (x₂ − x₁) / (b − a), where a and b are the lower and upper bounds of the distribution. For example, if a = 0, b = 10, x₁ = 2, x₂ = 7, then P = (7 − 2) / (10 − 0) = 0.5.
The PDF of the continuous uniform distribution is f(x) = 1 / (b − a) for a ≤ x ≤ b, and 0 otherwise. This means the density is constant across the entire interval, which is what makes every outcome equally likely. The total area under the PDF always equals 1.
The mean of a uniform distribution U(a, b) is simply the midpoint of the interval: μ = (a + b) / 2. For example, if a = 0 and b = 10, the expected value is (0 + 10) / 2 = 5.
The median of a uniform distribution equals its mean, which is (a + b) / 2. Because the distribution is perfectly symmetric, the midpoint divides the probability mass exactly in half.
The variance of a uniform distribution is σ² = (b − a)² / 12, and the standard deviation is σ = (b − a) / √12. For example, with a = 0 and b = 10, the variance is 100 / 12 ≈ 8.333 and the standard deviation is approximately 2.887.
No. A normal distribution is bell-shaped, with outcomes near the mean being far more likely than those in the tails. A uniform distribution assigns equal probability to all values in its interval — it has no peak or tails. They are fundamentally different distributions used in different contexts.
Yes — 'rectangular distribution' is simply another name for the uniform distribution. The name comes from the shape of its PDF, which forms a perfect rectangle over the interval [a, b] when plotted on a graph. You might also find our calculate Monte Carlo Simulation useful.