How this instrument works
The continuous uniform distribution describes a quantity that can take any value between a lower bound a and an upper bound b, with every value in that range being equally likely. Picture a bus that arrives at a perfectly random time somewhere between 2 and 8 minutes after you start waiting, with no minute more likely than another — that waiting time follows a continuous uniform distribution on [2, 8]. It's the continuous counterpart to a fair die or coin: flat, unbiased probability across the whole allowed range.
The density function, f(x) = 1/(b − a), is a constant — it doesn't depend on x at all, as long as x falls between a and b, which is exactly what 'uniform' means. The cumulative function, F(x) = (x − a)/(b − a), grows in a perfectly straight line from 0 at x = a to 1 at x = b, since probability accumulates evenly across the interval with no value more likely than another.
The mean, E[X] = (a + b)/2, sits at the exact midpoint of the interval, which follows from the distribution's perfect symmetry — as many values lie below the midpoint as above it, equally weighted. The variance, Var(X) = (b − a)² / 12, grows with the square of the interval's width: doubling the width of the interval quadruples the variance, since a wider range of equally likely outcomes naturally spreads the distribution's mass further from its mean.
- Enter the smallest possible value into Lower bound (a) — the left edge of the range.
- Enter the largest possible value into Upper bound (b) — the right edge of the range; it must be greater than a.
- Enter the specific value you want to evaluate into x (value to evaluate, a ≤ x ≤ b) — anywhere within the [a, b] range.
- Read Probability density f(x) — a constant value across the whole range, equal to 1/(b − a).
- Read Cumulative probability F(x) = P(X ≤ x), Mean, E[X], and Variance, Var(X) — the full statistical picture for this range.
Worked example — a value of 3 on the range [0, 10]
Enter a = 0 into Lower bound, b = 10 into Upper bound, and x = 3 into x — a value uniformly distributed anywhere between 0 and 10, evaluated at x = 3. Probability density f(x) reads 0.1, computed as 1/(10 − 0) — constant across the whole range, so the density at x = 3 is identical to the density at x = 7 or x = 0.01.
Cumulative probability F(x) reads 0.3, computed as (3 − 0)/10 — a 30% chance the value lands at or below 3, which matches intuition since 3 is 30% of the way from 0 to 10. Mean, E[X], reads 5.0, the exact midpoint of the range, and Variance, Var(X), reads 8.33333333333, computed as 10² / 12.
Questions
Why is the probability density constant instead of peaked somewhere?
Because 'uniform' specifically means every value in the range is equally likely — there is no most-probable value to peak around. Contrast this with a normal distribution, which is densest at its mean and tapers off symmetrically; the uniform distribution has no such preference, which is exactly what makes it the natural model for 'purely random within known bounds' situations like rounding error or a fair random-number generator.
What's the difference between the continuous and discrete uniform distribution?
The continuous uniform distribution (this calculator) allows any real number within [a, b], like a randomly timed arrival that could be 3.14159 minutes. A discrete uniform distribution instead allows only a finite set of equally likely values, like the outcome of a fair six-sided die (1 through 6, no fractions in between) — this site's dice-probability calculators cover that discrete case, with its own separate mean and variance formulas.
Why does the variance use 12 in the denominator?
The 12 comes directly from integrating (x − mean)² × density over the interval [a, b] and simplifying — it's a fixed constant that falls out of the calculus for any uniform distribution, not something you need to derive by hand each time. Practically, it means the variance scales with the square of the range's width: a range twice as wide has four times the variance, since (2w)²/12 = 4 × (w²/12).
What is the standard uniform distribution?
It's the special case where a = 0 and b = 1, often written U(0,1). It's the foundation for a huge amount of computational statistics, since most random-number generators produce values uniformly distributed on [0,1] first, and other distributions (exponential, normal, and many more) are then generated by transforming those standard uniform values mathematically.
What happens if x falls outside the range [a, b]?
The distribution assigns zero density there — a value outside [a, b] simply cannot occur by definition, since the whole premise of the uniform distribution is that outcomes are confined to that range. This instrument requires x to fall between a and b inclusive and will flag an out-of-range entry rather than return a misleading density or cumulative value.
Can a and b be negative, or a itself be negative while b is positive?
Yes — the formulas work identically for any a and b as long as b is strictly greater than a. A uniform distribution on [−5, 5], for instance, has mean 0 and the same width-based variance formula, (b − a)²/12 = 10²/12, as any other range of the same width.