The uniform distribution is a continuous probability distribution of events with an equal probability of occurrence. This distribution is rectangular.

Two essential properties of this distribution are

1. The area under the rectangular shape equals 1.
2. There is a correspondence between the probability of an event and the area under the curve.

Further, the mean and standard deviation of the uniform distribution can be calculated when the lower and upper cut-offs, denoted as a and b, respectively, are given. For a random variable x, in a uniform distribution, given a and b, the probability density function is f(x) is calculated as

Consider data of 55 smiling times, in seconds, of an eight-week-old baby:

10.4, 19.6, 18.8, 13.9, 17.8, 16.8, 21.6, 17.9, 12.5, 11.1, 4.9, 12.8, 14.8, 22.8, 20.0, 15.9, 16.3, 13.4, 17.1, 14.5, 19.0, 22.8, 1.3, 0.7, 8.9, 11.9, 10.9, 7.3, 5.9, 3.7, 17.9, 19.2, 9.8, 5.8, 6.9, 2.6, 5.8, 21.7, 11.8, 3.4, 2.1, 4.5, 6.3, 10.7, 8.9, 9.4, 9.4, 7.6, 10.0, 3.3, 6.7, 7.8, 11.6, 13.8 and, 18.6. Assume that the smiling times follow a uniform distribution between zero and 23 seconds, inclusive. Note that zero and 23 are the lower and upper cut-offs for the uniform distribution of smiling times.

Since the smiling times' distribution is a uniform distribution, it can be said that any smiling time from zero to and including 23 seconds has an equal likelihood of occurrence. A histogram that can be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution.

For this example, the random variable, x = length, in seconds, of an eight-week-old baby's smile. The notation for the uniform distribution is x ~ U(a, b) where a = the lowest value (lower cut-off) of x and b = the highest value (upper cut-off) of x. For this example, a = 0 and b = 23.

The mean, μ, is calculated using the following equation:

The mean for this distribution is 11.50 seconds. The smile of an eight-week-old baby lasts for an average time of 11.50 seconds.

The standard deviation, σ, is calculated using the formula:

The standard deviation for this example is 6.64 seconds.

This text is adapted from Openstax, Introductory Statistics, Section 5.2 The Uniform Distribution