Uniform Distribution Calculator

Free Online Uniform Distribution Calculator

This uniform distribution calculator finds the density (or probability mass), mean, variance, standard deviation, and range probability for both continuous and discrete uniform distributions. Enter the distribution's bounds and the range you're interested in, and the calculator instantly returns every value, complete with a proportional diagram and a full step-by-step solution.

Whether you're a student solving probability and statistics problems, a teacher preparing worked examples, or an analyst modeling a scenario where every outcome is equally likely, this uniform distribution probability calculator handles both the continuous case (any real number in a range) and the discrete case (whole numbers only) in one tool.

What Is a Uniform Distribution?

A uniform distribution describes a situation where every possible outcome within a given range is equally likely — there's no bias toward any particular value. There are two versions: the continuous uniform distribution, where X can take any real number between a and b (like a random arrival time within an hour), and the discrete uniform distribution, where X can only take whole-number values (like the roll of a fair die, or picking a random integer from a list).

Uniform Distribution Formulas

For a continuous uniform distribution U(a, b):

• Density: f(x) = 1 / (b − a), the same constant value across the whole range
• Mean: μ = (a + b) / 2
• Variance: σ² = (b − a)² / 12
• P(c ≤ X ≤ d) = (d − c) / (b − a)

Discrete Uniform Distribution Formulas

For a discrete uniform distribution over the whole numbers {a, a+1, ..., b}, with n = b − a + 1 possible outcomes:

• Probability mass: P(X = k) = 1 / n for every outcome k
• Mean: μ = (a + b) / 2
• Variance: σ² = (n² − 1) / 12
• P(c ≤ X ≤ d) = (number of whole numbers in [c, d]) / n

How to Use This Uniform Distribution Calculator

Using the calculator takes just a few steps. First, choose continuous or discrete from the dropdown. Second, enter the distribution's lower bound (a) and upper bound (b). Third, enter the range (c to d) you want the probability for. The calculator instantly displays the density (or probability of a single outcome), the mean, variance, standard deviation, and the probability P(c ≤ X ≤ d), along with a proportional diagram.

The diagram is drawn so the highlighted fraction is the literal answer: for the continuous case, the shaded width divided by the total width equals your probability; for the discrete case, the number of highlighted bars divided by the total number of bars equals your probability. There's no separate scale to interpret — what you see is what you get.

Worked Example

Continuous example: A bus arrives at a stop at a uniformly random time between 0 and 10 minutes past the hour (a = 0, b = 10). What's the probability it arrives between minute 3 and minute 7? Using P(c ≤ X ≤ d) = (d − c) / (b − a) = (7 − 3) / (10 − 0) = 4/10 = 0.4, or 40%.

Discrete example: A fair 6-sided die is rolled (a = 1, b = 6, n = 6 outcomes). What's the probability of rolling a 3, 4, or 5? Using P(c ≤ X ≤ d) = count / n = 3 / 6 = 0.5, or 50%, since three of the six equally likely outcomes (3, 4, and 5) fall in that range.

Where Uniform Distributions Are Used

Uniform distributions are one of the simplest but most widely used probability models, appearing in many practical situations:

• Students and teachers use it as an introductory example throughout probability and statistics coursework.
• Random number generators in computer science are built on the continuous uniform distribution over [0, 1].
• Simulation and Monte Carlo methods use uniform distributions to generate random inputs before transforming them into other distributions.
• Quality control and manufacturing sometimes model tolerances or arrival times as uniformly distributed when no value is more likely than another.
• Games of chance — like rolling a fair die or drawing a numbered ball — are classic examples of the discrete uniform distribution.