The continuous uniform distribution is the simplest distribution there is: a random value is equally likely to land anywhere between a lower bound a and an upper bound b, and never outside. Its density is a flat rectangle of height 1/(b − a), so every probability is just a length divided by the total length.
It models situations with no reason to prefer one value over another: the arrival time of a bus that runs every 15 minutes when you turn up at random, rounding errors, the position of a break along a rod, or the angle of a spun pointer. The standard uniform U(0, 1) is the building block of computer simulation – random numbers from any other distribution are made by feeding U(0, 1) values through an inverse CDF.
This calculator gives P(X ≤ x), P(X > x), the probability between or outside two values, quantiles, random values and the summary measures. Because the density is flat there is no single mode – every value in [a, b] is equally likely – and the distribution is symmetric with skewness 0 and excess kurtosis −1.2.
How to use the Uniform Distribution Calculator
- Lower bound a
- Upper bound b
- What to calculate
- Value x (x₁ for a range)
- Upper value x₂
- Cumulative probability p
- How many random values
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Probabilities are lengths: the overlap of the requested interval with [a, b] divided by b − a. Quantiles are a + p(b − a), which also produces the random sample. Choose what to calculate: a cumulative probability, an upper tail, the probability between or outside two values, a quantile (inverse CDF) or a random sample generated by inverse-transform sampling. The chart shades exactly the area you asked for.
Worked example
For U(0, 10): the density is 1/10 = 0.1, P(X ≤ 3) = 3/10 = 30%, P(3 ≤ X ≤ 7) = 40% and P(X < 3 or X > 7) = 60%. The mean and median are 5, the variance 100/12 = 8.333 and the standard deviation 2.887. The 25th percentile is 0 + 0.25 × 10 = 2.5.
When to use it
Use the uniform distribution when only the range of a quantity is known and no value is more likely than another – random arrival within a timetable gap, rounding errors of ±0.5 units, or a first rough model in a simulation. If values near the middle are more likely, a triangular, beta or normal model fits better.
Formula
f(x) = 1/(b − a) for a ≤ x ≤ b · P(X ≤ x) = (x − a)/(b − a) · mean = median = (a + b)/2 · variance = (b − a)²/12 · quantile = a + p(b − a)
Worked example
Inputs — Lower bound a: 0; Upper bound b: 10; What to calculate: P(x₁ ≤ X ≤ x₂) – probability between two values; Value x (x₁ for a range): 3; Upper value x₂: 7.
Results — Result = P(3 ≤ X ≤ 7) = 0.4 (40%); Probability density f(x) = 0.1; P(X ≤ x) – cumulative (CDF) = 30 %; P(X > x) – survival = 70 %; P(x₁ ≤ X ≤ x₂) = 40 %; P(X < x₁ or X > x₂) = 60 %.
Key terms
- Continuous uniform distribution
- A distribution with constant density on [a, b] and zero elsewhere.
- Standard uniform
- U(0, 1), the basis of random-number generation.
- Inverse-transform sampling
- Turning a U(0, 1) value u into a value of another distribution with its quantile function.
Frequently asked questions
What is the probability of a range in a uniform distribution?
Its length divided by b − a. For U(0, 10), P(3 ≤ X ≤ 7) = 4/10 = 40%.
What are the mean and variance of U(a, b)?
Mean (a + b)/2 and variance (b − a)²/12. For U(0, 10): 5 and 8.333 (SD 2.887).
What is the mode?
There is none – every value between a and b has the same density, so the calculator leaves the mode blank.
What is the standard uniform distribution?
U(0, 1), with mean 0.5 and variance 1/12. Random-number generators produce it.
If a bus comes every 15 minutes, what is the chance of waiting under 5 minutes?
With a wait uniform on 0 to 15 minutes, 5/15 = 33.3%.
What is the difference from the discrete uniform distribution?
The discrete version has a finite list of equally likely values (a fair die: 1 to 6); the continuous one allows any real number in the interval.