Uniform Distribution Calculator

The continuous uniform distribution assigns equal likelihood to every value in an interval [a, b]. It is the distribution of a point placed at random with no preference for any region, and it is the starting point of probability theory because its probabilities are lengths.

It is described by exactly two parameters, and even those are redundant in practice: once the width b − a and the position are known, everything else follows. There is no shape parameter, so there is nothing to fit and nothing to tune.

The density is the constant 1/(b − a) across the interval and zero outside it. Because the area under a density must be one, a rectangle of height 1/(b − a) and width b − a has area exactly 1, whatever the bounds are.

The cumulative distribution is the fraction of the interval already passed: (x − a)/(b − a), clamped to 0 below a and 1 above b. Integrating the constant density gives that straight line, which is why the probability scale and the value scale are the same thing here.

The quantile formula inverts it directly: q(p) = a + p(b − a). Move p of the way across the interval and you are at the pth percentile, which is why the median needs no work at all — it is the midpoint.

The moments come from averaging over the interval rather than from any clever integral. The mean is the midpoint, the variance is the width squared over twelve, and the standard deviation is the width over √12 ≈ 3.4641. The distribution is symmetric, so the skewness is exactly zero, and its excess kurtosis is −6/5.

A probability such as 30% means that in the long run 30 out of every 100 independent draws would fall below that value. It says nothing about any single draw.

The density is a rate, not a probability. The constant 0.1 per minute on [0, 10] means probability accumulates at 10% per minute, and a single minute with no width contributes nothing.

The standard deviation is a fixed share of the width: 0.2887 of it. Doubling the interval doubles the spread, so relative variability never changes and only the absolute scale does.

The sample mode is one realisation, so its mean will differ from (a + b)/2. That gap shrinks roughly as 1/√n and is not an error in either number.

• The width b − a. It is the only thing that matters: it sets the density, the standard deviation, the variance and every quantile position. Widths of 20 minutes and 20 kilometres behave identically.
• The position of the interval. Shifting both bounds moves every value but leaves every probability exactly where it was, which is worth checking when a result looks surprising.
• Where you measure. A threshold close to the lower bound gives a small probability, one at the midpoint gives 50%, and one past the upper bound gives 100%, all because of where it falls on that flat line.
• Sample size in sample mode. The spread of the generated values around the true mean narrows as more values are drawn, and empty stretches of the interval fill in.

• Any situation where a value is chosen from a range with no preference, such as a position along a line or a point inside a shape.
• Baseline and worst-case reasoning: if nothing is known about the distribution except that it lies in [a, b], the uniform is the least committal assumption available.
• Simulation inputs that must stay inside known limits, such as arrival times, jitter or randomised scheduling.
• Teaching and checking distributions: the uniform is the clearest illustration of a flat density, a linear CDF and zero skewness.
• Quick estimates where an interval is genuinely all you know, such as a bounded physical quantity with no other information.

• Uniformity is a strong claim. Real quantities are rarely exactly flat over a range; a value that merely looks spread out may be rounded, truncated or clipped.
• Both bounds must be genuine limits. Nothing outside [a, b] is ever produced, so a real process with heavier tails is misrepresented by this model.
• There is no shape parameter, so the model cannot capture any preference within the range. If the data clusters anywhere, it is not uniform.
• Generated samples are independent draws. Real processes are often correlated in time, and consecutive draws from this generator are not.
• Points have zero probability. If your data is rounded to whole units, the continuous model understates the chance of small buckets, which is why the sample mode reports how evenly the sub-intervals were covered.