Uniform Distribution

A uniform distribution is a continuous probability distribution where all events have equal likelihood of occurring. It is represented by a rectangular function with lower cut-off 'a' and upper cut-off 'b'. The probability density function creates a rectangular shape on a graph, where the total area under the curve always equals one.

In a uniform distribution, probability equals the area under the curve segment for that event. Calculate this by multiplying the width of the segment by its height. For example, if voltage ranges from 122 to 126 volts with height 0.25, the probability of voltage below 123 volts is 1 volt × 0.25, or 0.25.

The mean of a uniform distribution is calculated by adding the lower cut-off 'a' and upper cut-off 'b', then dividing by two: μ = (a + b) / 2. For example, if voltage ranges from 122 to 126 volts, the mean is (122 + 126) / 2, which equals 124 volts.

Standard deviation in a uniform distribution equals the range divided by the square root of twelve: σ = (b - a) / √12. For a 4-volt range, the standard deviation is 4 / √12, or approximately 1.2 volts. This measures how spread out the values are within the distribution.

The area under the probability density curve represents the total probability of all possible outcomes. Since one of the outcomes must occur with certainty, the total probability equals one. In a uniform distribution, this rectangular area is calculated as width (range) multiplied by height (probability density).

When data follows a uniform distribution between values 'a' and 'b', every outcome within that range has an equal probability of occurring. For instance, baby smiling times uniformly distributed between 0 and 23 seconds means any duration in that range is equally likely, with mean 11.5 seconds and standard deviation 6.64 seconds.

A histogram constructed from sample data creates an empirical distribution that closely matches the theoretical uniform distribution when the data truly follows uniform distribution rules. This visual comparison helps verify whether observed data aligns with the expected rectangular shape and equal probability assumptions of the uniform model.