Uniform Distribution Calculator
Every value between a and b equally likely: interval probabilities by pure geometry, the density 1/(b − a), mean and variance from the endpoints — with x outside [a, b] answered exactly 0 or 1, named.
Uniform Distribution Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Flat within [a, b], zero outside — no bell, no tail
- One interval only; the discrete dice-face uniform is a different object
In short: A bus comes sometime in the next 10 minutes, all times equally likely (a = 0, b = 10): the chance it arrives within 2.5 minutes is (2.5 − 0)/10 = 0.25, the density is 1/10 = 0.1 everywhere, the mean is 5 and the variance (b − a)²/12 = 8.333333. Outside the support there is no smoothing to do: x below a gets exactly 0, x above b exactly 1, and the page says which wall it hit rather than returning a clipped number in disguise. And b ≤ a is refused — an interval needs two ends in order.
Formula
P(X ≤ x) = (x − a)/(b − a) · f(x) = 1/(b − a) · Var = (b − a)²/12
Every probability is a ratio of lengths on one line segment — the only distribution whose answers you can draw to scale and check with a ruler.
Worked Example
- Give the two endpoints a < b and the point x.
- Read P(X ≤ x) as a length ratio; the density card shows the same thing as 1/(b − a).
- The moments come from the endpoints alone: mean (a+b)/2, variance (b − a)²/12.
- If x lies outside [a, b], the answer is exactly 0 or exactly 1 — the card names the wall.
a = 0, b = 10, x = 2.5: P = 0.25, density 0.1, mean 5, variance 8.333333. x = −3 → exactly 0; x = 14 → exactly 1, both named. b ≤ a refused.
Strengths & Limits Of This Model
Where this engine is strong
- Every answer is checkable geometry — length ratios on a line
- Outside-support values answered exactly and named, never clipped silently
Where it stops
- No inverse quantile mode (percentile → x is trivial algebra: x = a + p(b − a))
- No discrete uniform (dice, lottery balls) mode
Practical Use Cases
Scheduling
equally-likely arrival windows
Simulation
the engine behind every random draw
Teaching
probability as literal geometry
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Uniform Distribution Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why is the uniform the “equally likely” distribution?
Because its density is one flat height over the whole interval — no point or region is favoured. That makes every probability a ratio of lengths: the chance of landing in a sub-interval is just its length over the whole. You can check this page with a ruler.
What does variance (b − a)²/12 measure here?
Spread scales with the SQUARE of the window width — double the interval and the variance quadruples (8.333333 for width 10, 33.333333 for width 20). The 12 is the flat shape’s signature: all mass spread evenly, none bunched at a centre.
Why exactly 0 or exactly 1 outside the support?
Because the model SAYS so: no density lives below a or above b, so there is no tail to approximate. P(X ≤ −3) = 0 and P(X ≤ 14) = 1 are exact statements of the model, and the card names which wall was hit instead of handing you a rounded 0.9999999 as if it were computed.
Is the bus example realistic?
Only if the schedule really is memoryless-flat in the window — a bus on a 10-minute headway that you arrive to at a random moment is roughly uniform; a bus with rush-hour bunching is not. The uniform is the model of maximum ignorance WITHIN a known range, and the range must be honest.
How is this different from the exponential wait?
Different ignorance. Uniform: you know the event lands in a bounded window, equally likely anywhere in it. Exponential: you know only a rate, and the wait can stretch arbitrarily. Bounded-flat vs unbounded-decaying — the two pages cross-link because choosing between them is the modelling decision.
Where does the uniform earn its keep in computing?
Every random-number generator produces uniforms first: any other distribution is a transformation of this one. It is the flour of simulation — nothing is baked without it, and this page is the recipe card for its three properties.
What happens at a = x or b = x exactly?
The boundary belongs to the interval: P(X ≤ a) = 0 and P(X ≤ b) = 1, and the page answers them through the same (x − a)/(b − a) arithmetic — 0 and 1 fall out naturally, no special case needed.
Why refuse b ≤ a?
An interval with its ends crossed has no inside: the density 1/(b − a) would go negative or infinite, and every probability is a length ratio that needs b > a to mean anything. The refusal names the order problem — swap the endpoints and the page computes.