Uniform Distribution Formula and Worked Example

Q: What is a uniform distribution?

A uniform distribution is a probability distribution in which every value in a stated range is equally likely. For a continuous variable on [a, b] the density is a constant 1/(b − a); for a discrete variable each of n outcomes has probability 1/n.

Q: What is the formula for a uniform distribution?

The probability density is f(x) = 1/(b − a) for a ≤ x ≤ b. The mean is (a + b)/2 and the variance is (b − a)²/12.

Q: What are examples of a uniform distribution?

A fair die (discrete, probability 1/6 per face), a random number generator drawing from [0, 1] (continuous), and p-values under a true null hypothesis with a continuous test statistic.

Q: How do you find the probability between two values?

Subtract the lower value from the upper value and divide by b − a. For Uniform(2, 10), P(3 < X < 5) = 2/8 = 0.25.

Q: Is the uniform distribution the same as a normal distribution?

No. The uniform distribution is flat with sharp limits and excess kurtosis of −6/5; the normal distribution is bell-shaped with unbounded tails. Averages of many uniform draws approach a normal shape, as described in the central limit theorem guide.

The uniform distribution formula for a continuous variable on the interval from a to b is f(x) = 1 / (b − a). Every value between a and b is equally likely, so the density is a flat line. This guide gives the pdf, cdf, mean and variance, one fully worked example with the arithmetic shown, the discrete case, and the parameter conventions that differ between R and SciPy.

What Is a Uniform Distribution? Definition

A uniform distribution assigns equal probability to every outcome in a range. The NIST/SEMATECH e-Handbook of Statistical Methods puts it this way: “The uniform distribution defines equal probability over a given range for a continuous distribution.” The range is set by two parameters, a lower limit a and an upper limit b, and the distribution is written X ~ Uniform(a, b) or U(a, b).

Two versions are in everyday use. The continuous uniform distribution covers every real number between a and b. The discrete uniform distribution covers a finite list of outcomes, such as the faces of a fair die. Most of this page is about the continuous case; the discrete case is covered under examples.

Uniform Distribution Formula: PDF, CDF, Mean and Variance

For X ~ Uniform(a, b) with a < b:

Quantity	Formula	Notes
Probability density f(x)	1 / (b − a) for a ≤ x ≤ b; 0 otherwise	NIST; Pishro-Nik
Cumulative distribution F(x)	0 if x < a; (x − a) / (b − a) if a ≤ x ≤ b; 1 if x > b	Derived here by integrating the pdf. NIST states the standard form, F(x) = x on [0, 1].
Mean	(a + b) / 2	NIST; Pishro-Nik
Variance	(b − a)² / 12	NIST; Pishro-Nik
Standard deviation	(b − a) / √12	Square root of the variance
Skewness	0	NIST
Kurtosis	9/5 (excess kurtosis −6/5)	NIST reports 9/5; subtracting 3 gives the excess value

What each symbol means

a is the minimum value the variable can take; b is the maximum.
b − a is the width of the interval. The density 1/(b − a) is exactly the height that makes the total area under the curve equal 1.
x is the value at which you evaluate the density or the cumulative probability.

The mean is the midpoint of the interval. Pishro-Nik derives the variance as E[X²] − (E[X])², where E[X²] = (a² + ab + b²) / 3, which simplifies to (b − a)² / 12. The probability that X falls between two values c and d inside the interval is simply (d − c) / (b − a): the width of the sub-interval divided by the total width.

Worked Example: Uniform(2, 10)

Suppose X ~ Uniform(2, 10), so a = 2 and b = 10.

Width: b − a = 10 − 2 = 8.
Density: f(x) = 1/8 = 0.125 for every x from 2 to 10.
Mean: (2 + 10) / 2 = 6.
Variance: 8² / 12 = 64 / 12 = 5.3333.
Standard deviation: √5.3333 = 2.3094, which rounds to 2.31.
P(3 < X < 5): (5 − 3) / 8 = 2/8 = 0.25.
P(X ≤ 7): F(7) = (7 − 2) / 8 = 5/8 = 0.625.
90th percentile: solve (x − 2) / 8 = 0.90, so x = 2 + 0.90 × 8 = 9.2.

Check: P(X > 8) = (10 − 8) / 8 = 0.25, and P(X ≤ 8) = 6/8 = 0.75, which add to 1.

Interpretation: How to Read the Results

Result	What it tells you
f(x) = 0.125	This is a density, not a probability. Any single exact value has probability 0; only intervals carry probability.
Mean = 6, median = 6	The distribution is symmetric (skewness 0), so the mean and median both sit at the midpoint.
SD = 2.31	The standard deviation depends only on the width, 8 divided by √12. Doubling the width doubles the SD.
P(3 < X < 5) = 0.25	Two units out of an eight-unit range: probability is proportional to length.
Excess kurtosis −6/5	Lighter tails than the normal distribution; there is a hard cut-off at both ends.

Uniform Distribution Examples: Continuous and Discrete

Continuous: random numbers on [0, 1]

The standard uniform distribution, Uniform(0, 1), is the base case that random number generators aim to reproduce. Monte Carlo methods turn these draws into other distributions; see Monte Carlo simulation. When a null hypothesis is true and the test statistic is continuous, the resulting p-values are themselves uniform on [0, 1], a fact that underlies p-curve analysis and is why a flat histogram of p-values is the expected pattern under no effect. See also the dictionary entry for the p-value.

Discrete: a fair six-sided die

Each face 1 to 6 has probability 1/6. The mean is (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21/6 = 3.5. The mean of the squares is (1 + 4 + 9 + 16 + 25 + 36) / 6 = 91/6 = 15.1667, so the variance is 15.1667 − 3.5² = 15.1667 − 12.25 = 2.9167. This is not (b − a)² / 12, which would give 25/12 = 2.0833; the continuous variance formula must not be applied to discrete outcomes.

Assumptions and Common Mistakes

Finite, known limits. Both a and b must be finite, and a must be less than b. In R, dunif returns NaN when min equals max.
Equal likelihood is an assumption, not a default. Use it when you have a reason to believe every value in the range is equally plausible. Data that cluster around a centre are better described by other models; see normality of distribution.
Density is not probability. f(x) = 0.125 does not mean a 12.5% chance of X = 5. Probability comes from area: width times height.
Endpoints do not matter. For a continuous variable P(X = a) = 0, so whether the interval is written open or closed leaves every probability unchanged.
Parameter conventions differ between software. R takes the lower and upper limits (min, max). SciPy takes a start and a width: loc = a and scale = b − a. Passing the upper limit as scale shifts the whole distribution.
Continuous and discrete variance formulas differ. See the die example above.

To test whether a sample is consistent with a uniform distribution, for example when checking a random number generator, a one-sample goodness-of-fit test against Uniform(0, 1) is the usual approach; see the Kolmogorov-Smirnov test.

How to Report a Uniform Distribution

State the distribution, its parameters, and the summary statistics you computed, and report them to a consistent number of decimals. For the worked example:

Waiting times were modeled as uniformly distributed between 2 and 10 minutes, U(2, 10), mean = 6.00, SD = 2.31. The probability of a wait between 3 and 5 minutes was 0.25.

Follow the symbol, italics and decimal conventions of the style guide your journal or institution requires. If you used a uniform distribution as a prior or as a sampling assumption, say so and give the limits, so the reader can reproduce the calculation.

How to Calculate It in Excel, R or Python

Excel

Put a in cell B1 and b in cell B2, then use ordinary cell formulas:

Density:   =1/(B2-B1)
Mean:      =(B1+B2)/2
Variance:  =(B2-B1)^2/12
SD:        =SQRT((B2-B1)^2/12)
P(c<X<d): =(d-c)/(B2-B1)
Random draw: =B1+RAND()*(B2-B1)

With a = 2 and b = 10 these return 0.125, 6, 5.3333, 2.3094, 0.25 for c = 3 and d = 5, and a random value between 2 and 10.

R

dunif(5, min = 2, max = 10)                      # 0.125
punif(5, 2, 10) - punif(3, 2, 10)                # 0.25
qunif(0.9, 2, 10)                                # 9.2
(10 - 2)^2 / 12                                  # 5.3333
runif(5, min = 2, max = 10)                      # five random draws

The R manual documents dunif, punif, qunif and runif, with defaults min = 0 and max = 1.

Python (SciPy)

from scipy.stats import uniform
u = uniform(loc=2, scale=8)        # Uniform on [2, 10]
u.pdf(5)                           # 0.125
u.cdf(5) - u.cdf(3)                # 0.25
u.mean(), u.var(), u.std()         # 6.0, 5.3333, 2.3094
u.ppf(0.9)                         # 9.2
u.rvs(size=5)                      # five random draws

The SciPy documentation states that, using loc and scale, one obtains the uniform distribution on [loc, loc + scale].

Frequently Asked Questions

What is a uniform distribution?