Uniform Distribution Calculator
When Every Outcome in a Range Is Equally Likely
Not every distribution has a peak or a most-likely value. The uniform distribution describes situations where every outcome in a defined range is exactly as likely as every other — a random number generator producing decimals between 0 and 1, or a fair die where each face is equally probable. This calculator covers both the continuous version (any value in a range) and the discrete version (whole numbers only).
The Formulas
Continuous, on [a, b]:
Mean = (a+b) ÷ 2 Variance = (b−a)² ÷ 12 PDF f(x) = 1 ÷ (b−a)
Discrete, on integers [a, b] with n = b−a+1:
Mean = (a+b) ÷ 2 Variance = (n²−1) ÷ 12 PMF = 1 ÷ n
Mean = (a+b) ÷ 2 Variance = (b−a)² ÷ 12 PDF f(x) = 1 ÷ (b−a)
Discrete, on integers [a, b] with n = b−a+1:
Mean = (a+b) ÷ 2 Variance = (n²−1) ÷ 12 PMF = 1 ÷ n
Worked Examples
| Type | Mean | Variance | Std Dev |
|---|---|---|---|
| Continuous [0, 10] | 5.0 | 8.333 | 2.887 |
| Discrete {1, 2, 3, 4, 5, 6} | 3.5 | 2.917 | 1.708 |
For the continuous example, P(3 ≤ X ≤ 7) = (7−3) ÷ (10−0) = 0.40 exactly — probability in a continuous uniform distribution is just the proportion of the interval covered.
Where This Matters
- Random number generation — most pseudo-random number generators produce a continuous uniform distribution as their base output, which is then transformed into other distributions.
- Simulation and Monte Carlo methods — uniform sampling is the starting point for simulating countless other probability distributions.
- Scheduling problems — modeling an arrival or event time as uniformly likely across a known window.
- Fair games of chance — a single die roll or spin of an unweighted wheel is a discrete uniform distribution by design.
How to Use This Calculator
- Choose Continuous Uniform Distribution or Discrete Uniform Distribution from the dropdown.
- Enter the lower bound (a) and upper bound (b) of the distribution.
- Enter the range (x1 to x2) you want the probability for.
- Select Calculate to get the probability, along with the distribution's mean, variance, and standard deviation.
Related Calculations
For non-uniform outcomes like dice sums, use the Dice Roll Calculator. For the continuous waiting-time analog, see the Exponential Distribution Calculator.