# Uniform Distribution: Definition, How It Works, and Examples

![Uniform Distribution: Probability distributions with equally likely outcomes.](https://www.investopedia.com/thmb/V5KuQyN9rehNkaLAJjm1zs7znMI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/uniform-distribution.asp-final-18a25d70966246ed8eec2ca04602b5d0.png)

*— Theresa Chiechi / Investopedia*

> **Definition**
>
> A uniform distribution is a probability distribution where each outcome in a range has an equal chance of occurring.

## What Is a Uniform Distribution?

In [statistics](https://www.investopedia.com/terms/s/statistics.asp), probability distributions can help you decide the probability of a future event—that is, the likelihood of something happening. A uniform distribution is a type of probability distribution where the probability remains constant across the entire range of possible values (i.e., all outcomes are equally likely).

A deck of cards is expected to have a uniform distribution because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution expectation because the odds of getting either heads or tails are the same.

A uniform distribution can be visualized as a straight horizontal line. For a coin flip, heads or tails each has a probability of occurring 50% of the time (p = 0.50), so it would be plotted on a chart with a line from the y-axis at 0.50.

### Key Takeaways

*   In a uniform distribution, all possible outcomes are equally probable.
*   In a discrete uniform distribution, outcomes are discrete and have the same likelihood.
*   In a continuous uniform distribution, outcomes are continuous and infinite.
*   In a normal distribution, data around the mean (average) occur more frequently than occurrences farther from it.
*   Uniform distributions can be plotted on charts.

## Understanding Uniform Distributions

There are two types of uniform distributions: [discrete](https://www.investopedia.com/terms/d/discrete-distribution.asp) and continuous.

### Discrete Uniform Distributions

The possible results of rolling a die provide an example of a discrete uniform distribution. It is possible to roll a 1, 2, 3, 4, 5, or 6, but it is not possible to roll, for example, a 2.3, 4.7, or 5.5. Therefore, the roll of a die generates a discrete distribution with the probability of 1/6 for each outcome. There are only six possible values to return and nothing in between. The possibilities are finite.

> **Fast Fact**
>
> The term “discrete” in statistics refers to variables with singular, countable, and limited possible values.

### Continuous Uniform Distributions

Continuous uniform distributions have infinite distribution possibilities. An idealized random number generator would be considered a continuous uniform distribution. With this type of distribution, every point in the continuous range between 0.0 and 1.0 has an equal opportunity of appearing, yet there is an infinite number of points between 0.0 and 1.0.

There are several other important continuous distributions, such as the [normal distribution](https://www.investopedia.com/terms/n/normaldistribution.asp), [chi-square](https://www.investopedia.com/terms/c/chi-square-statistic.asp), and Student’s [t-distribution](https://www.investopedia.com/terms/t/tdistribution.asp).

### Distribution Analysis

There are also several data-generating or data-analyzing functions associated with distributions that help explain the variables and their [variance](https://www.investopedia.com/terms/v/variance.asp) within a dataset. These functions include [probability density function](https://www.investopedia.com/terms/p/pdf.asp), cumulative density, and moment generating functions.

## Visualizing Uniform Distributions

A distribution is a simple way to visualize a dataset. It can be shown either as a graph or a list, and reveals which values of a random variable have lower or higher chances of happening.

In a uniform distribution, each value in the set of possible values has the same possibility of happening. When displayed as a [bar](https://www.investopedia.com/terms/b/bar-graph.asp) or [line graph](https://www.investopedia.com/terms/l/line-graph.asp), this distribution has the same height for each potential outcome. In this way, it can look like a [rectangle](https://www.investopedia.com/terms/r/rectangle.asp) and therefore is sometimes described as the rectangular distribution.

If you think about the possibility of drawing a particular suit from a deck of playing cards, there is a random yet equal chance of pulling a heart as there is for pulling a spade—that is, ¼ or 25%. (There are four suits.)

The roll of a die yields one of six numbers: 1, 2, 3, 4, 5, or 6. Because there are only 6 possible outcomes, the probability of you landing on any one of them is 16.67% (1/6). When plotted on a graph, the distribution is represented as a horizontal line, with each possible outcome captured on the x-axis, at the fixed point of probability along the y-axis.

![Uniform Distribution](https://www.investopedia.com/thmb/NsfKOB6WHeOneuhEVcwIzrUz-jo=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/UsingCommonStockProbabilityDistributionMethods3_2-103616406ed64cd5b347eb939fc05853.png)

*Uniform distribution of one six-sided die.*

## Example of a Uniform Distribution

There are 52 cards in a traditional deck of cards. Also in that deck are four suits: hearts, diamonds, clubs, and spades. Each suit contains an ace, two, three, four, five, six, seven, eight, nine, 10, jack, queen, and a king. The deck contains two jokers as well. However, we’ll ignore the jokers and face cards for this example, and focus only on number cards replicated in each suit. As a result, we are left with 40 cards, a set of discrete data.

Suppose you want to know the probability of pulling a two of hearts from the modified deck. The probability of pulling a two of hearts is 1/40 or 2.5%. Each card is unique. Therefore, the likelihood that you will pull any one of the cards in the deck is the same.

Now, let’s consider the likelihood of pulling a heart from the deck. The probability is significantly higher. Why? We are now only concerned with the suits in the deck. Since there are four suits and each suit has the same number of cards, pulling a heart yields a probability of ¼ or 25%.

## Uniform Distribution vs. Normal Distribution

Some of the most common probability distributions are:

*   Discrete uniform
*   Binomial
*   Continuous uniform
*   Normal
*   Exponential

### Normal

One of the most widely used distributions is the [normal distribution](https://www.investopedia.com/terms/n/normaldistribution.asp), which is often depicted as a [bell curve](https://www.investopedia.com/terms/b/bell-curve.asp). Normal distributions show how continuous data is distributed and depict most of the data as concentrated on the [mean](https://www.investopedia.com/terms/m/mean.asp) (average).

In a normal distribution, the area under the curve equals one. 68.27% of all data falls within one standard deviation (how dispersed the numbers are) from the mean. 95.45% of all data falls within two standard deviations from the mean, and approximately 99.73% of all data falls within three standard deviations from the mean. As the data points move away from the mean, the frequency of data occurring decreases.

### Uniform

Discrete uniform distribution shows that variables in a range have the same probability of occurring. There are no variations in probable outcomes, and the data is discrete, rather than continuous. Its shape resembles a rectangle, rather than the normal distribution’s bell. Like a normal distribution, however, the area under the graph is equal to one.

## Explain Like I’m 5 Years Old

With a uniform distribution, you’re just as likely to get one outcome as another. Take a die with six sides, for example. You’re just as likely to roll a 1 as you are to roll a 2 (or 3, 4, 5, or 6).  

## What Is the Formula for a Uniform Distribution?

The formula for a discrete uniform distribution is

$\begin{aligned}&P_x = \frac{ 1 }{ n } \\&\textbf{where:} \\&P_x = \text{Probability of a discrete value} \\&n = \text{Number of values in the range} \\\end{aligned}$

## Is a Uniform Distribution Normal?

No, a uniform distribution is not normal. Normal, which refers to the way data is distributed about the mean (average), shows that the probability of a variable occurring around the mean is higher than for a variable occurring far away from the mean. The occurrence probability is not uniform with normal data, whereas it is constant with a uniform distribution.

## What Is the Expectation of a Uniform Distribution?

The expectation with a uniform distribution is that all possible outcomes have the same probability. The probability for one variable is the same as that for another.

## The Bottom Line

A uniform distribution is a type of probability distribution that is used in statistics. A uniform, or rectangular, distribution is one where all possible outcomes are finite and equally likely to occur. For example, the roll of a single die presents just six possible results, and each result has a 1/6 chance of taking place.

> **Article Sources**
>
> 1.  National Institute of Standards and Technology, Information Technology Laboratory. “[What Do We Mean By ‘Normal’ Data?](https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc51.htm)”