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Uniform Distribution Probability Calculator

Calculate the probability of a value falling within a range, for a continuous uniform distribution.

Result

P(3 ≤ X ≤ 7)
40%
Mean
5
Variance
8.333

About the Uniform Distribution

The Uniform Distribution Probability Calculator finds the probability that a value falls within a specified range, for a continuous uniform distribution where every outcome between a minimum and maximum is equally likely. It also reports the distribution's mean and variance, useful for characterizing any process modeled as uniformly random within known bounds.

How It Works

Enter the distribution's overall minimum and maximum, then the lower and upper bounds of the specific range you want the probability for. The calculator clamps your range bounds to stay within the distribution's minimum and maximum, then divides the width of that clamped range by the total width of the distribution to get the probability. It also computes the distribution's mean as the midpoint of the range and its variance from the total width.

P(lower <= X <= upper) = (clamped upper - clamped lower) / (b - a); Mean = (a + b)/2; Variance = (b - a)^2 / 12, where a is the minimum and b is the maximum.

Formula & Methodology

To calculate by hand, first make sure your range of interest does not extend past the distribution's own minimum and maximum, since any part outside those bounds contributes no probability and should be clamped back to the boundary. Subtract the clamped lower bound from the clamped upper bound to get the width of interest, then divide by the total width (maximum minus minimum) for the probability. The mean is the midpoint of the minimum and maximum, and the variance is the square of the total width divided by 12.

Examples

Random Number Generator Range

For a distribution from 0 to 10, the probability that a value falls between 3 and 7 is (7-3)/(10-0) = 4/10 = 40%, with a mean of 5 and a variance of 100/12, about 8.33.

Bus Arrival Window

If a bus arrives uniformly at random between 0 and 15 minutes after the hour, the probability it arrives within the first 5 minutes is (5-0)/(15-0) = 33.33%, with a mean arrival time of 7.5 minutes.

Advantages

  • Automatically clamps out-of-range bounds to the distribution's own minimum and maximum, preventing an invalid probability from a mistyped range.
  • Reports the distribution's mean and variance alongside the requested probability, giving a fuller picture of the distribution in one result.
  • Handles any continuous range within defined bounds without requiring a distribution table, since uniform probability is just proportional width.

Common Mistakes

  • Entering a range that extends beyond the distribution's own minimum and maximum without realizing the calculator will clamp it, which can produce a different probability than expected.
  • Confusing a continuous uniform distribution, where any real number in the range is possible, with a discrete uniform distribution over a fixed set of outcomes like dice rolls, which uses different formulas.
  • Forgetting that the variance formula divides the squared total width by 12, not by 4 or another factor, leading to hand-calculation errors.

Edge Cases to Watch For

  • The maximum must be strictly greater than the minimum, or the calculator returns an error rather than dividing by a zero or negative total width.
  • If the entered lower or upper bound falls outside the distribution's minimum and maximum, the calculator clamps it back to the nearest boundary before computing probability.
  • If, after clamping, the upper bound is still less than the lower bound, the calculator returns an error rather than reporting a negative probability.

Common Use Cases

  • Students and instructors working through continuous uniform distribution probability problems in introductory statistics.
  • Analysts modeling processes assumed to be equally likely across a known range, such as arrival times within a fixed window or measurement rounding error.
  • Simulation and modeling work that needs quick probability, mean, and variance figures for a uniformly distributed variable without building a full simulation.
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

What is a uniform distribution?

Every value between the minimum and maximum is equally likely - probability over any sub-range is simply that range's width divided by the total width. It's the simplest continuous distribution, often used to model complete uncertainty within known bounds, like a random number generator's raw output.

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Conclusion

The Uniform Distribution Probability Calculator turns a defined range and a distribution's bounds into a probability, mean, and variance using the proportional-width logic of the continuous uniform distribution. It is a quick way to check probability calculations for any process assumed to be equally likely across a known interval.