Uniform distribution
Model an outcome equally likely across a finite interval.
Uniform distribution is one of 7 distributions formulas in the probability section of this library, and it is used at high school · university level.
Why uniform distribution works
On this interval every slice of equal length is equally likely, so probability is proportional to length. The height has to be 1 over the interval width for the total area to be 1. Symmetry then puts the mean at the midpoint, and integrating the squared distance from that midpoint produces the division by 12.
What each symbol means
$a$ and $b$ are lower and upper endpoints.
Uniform distribution: when it holds
$a<b$ and the density is zero outside $[a,b]$.
When it stops applying
It assumes every part of the range is genuinely equally likely. Waiting time for a bus that runs every 10 minutes is only uniform if you arrive at a random moment and the buses truly keep to schedule; real timetables bunch, which piles probability into some stretches and empties others.
Uniform distribution: a worked example
For $U(0,10)$, $P(2<X<5)=3/10$.
The mistake to avoid
What people do: Reporting the density value as the probability of a single exact value.
Why it goes wrong: For a continuous variable, any single point has probability zero, since a point has no width. The height 1 over the interval width is a rate of probability per unit, not a probability.
Do this instead: Ask for a range instead. Multiply the density by the width of the interval you care about, so 3 units of width on a range of 10 gives 0.3.
Uniform distribution: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a$ and $b$ are lower and upper endpoints.
- Check the conditions before substituting. $a<b$ and the density is zero outside $[a,b]$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Distributions
- Level
- High school · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where uniform distribution comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about uniform distribution
Does strictly less than give a different answer from less than or equal to?
No. The single boundary point contributes zero probability, so the two are identical for any continuous distribution. That is not true for discrete variables.
Where does the 12 in the variance come from?
From integrating the squared distance to the midpoint across the interval. It is not a rounding or a convention; it falls out of the calculus exactly.
How is the discrete uniform different?
It spreads probability over a list of separate values rather than a continuous stretch. For n equally likely whole numbers the variance is n squared minus 1, all over 12, which is about 2.917 for a fair die.
How do I find the chance of landing in a sub-interval?
Divide the length of that sub-interval by the length of the whole range. Between 2 and 5 on a range from 0 to 10 that is 3 over 10, or 0.3.