What is a Uniform Distribution?
The uniform distribution is a symmetric probability distribution where all outcomes have an equal likelihood of occurring. All values in the distribution have a constant probability, making them uniformly distributed. This distribution is also known as the rectangular distribution because of its shape in probability distribution plots, as I’ll show you below.
Uniform distributions come in both discrete and continuous varieties of probability distributions.
Discrete: Each discrete value has an equal probability. For example, the chances of obtaining any of the six values on a die are equal.
Continuous: Models symmetric, continuous data where all equal sized ranges have the same probability. For example, values are equally like to fall in the range of 0.1 – 0.2 as they are in 0.4 – 0.5.
Both forms of the uniform distribution have two parameters, a and b. These values represent the smallest and largest values in the distribution.
Related post: Understanding Probability Distributions
Uniform Distribution Examples
In real life, analysts use the uniform distribution to model the following outcomes because they are uniformly distributed:
- Rolling dice and coin tosses.
- The probability of drawing any card from a deck of cards.
- Random sampling because that method depends on population members having equal chances.
- P-values in hypothesis tests follow the uniform distribution when the null hypothesis is true under certain conditions.
- Random number generators use the uniform distribution because no number should be more common than other numbers.
- Radioactive decay of particles over time.
Analysts can use the uniform distribution to approximate new processes when there is insufficient data to estimate the actual distribution of outcomes. In other cases, analysts use this distribution because it’s a close approximation and the formula is simple, as I show below.
Graphing Distributions and Finding Probabilities
Graphing is a great way to see what uniform distributions look like and find probabilities. Below, I’ll graph discrete and continuous forms of the distribution.
For discrete uniform distributions, finding the probability for each outcome is 1/n, where n is the number of outcomes. Rolling dice has six outcomes that are uniformly distributed. Therefore, each one has a likelihood of 1/6 = 0.167. The bar chart below displays the rectangular-shaped distribution.
For continuous probability density functions, you obtain probabilities for ranges of values by finding the area under the curve for that range. Usually, that involves complex calculations. However, in continuous uniform distributions, the formula is simple because you’re finding areas of a rectangle instead of a curve. You just divide the number of units of interest by the total number of units.
In the example below, the distribution ranges from 5 to 10, which covers 5 units. The shaded area is one unit out of five or 1 / 5 = 20% of the total area. Hence, the probability for a value falling between 6 and 7 is 0.2. In fact, all one-unit ranges in this distribution have the same likelihood of 0.2. You can use the same formula to calculate the probabilities for other sized ranges when the outcomes are uniformly distributed.
The mean of the uniform distribution is (a + b) / 2. This mean is the expected value for a uniform distribution. Learn more about Expected Values: Definition, Formula & Finding.