Probability Distribution Calculator

Use our probability distribution calculator to find the mean, standard deviation, and variance of a distribution by entering each value x and its probability P(x) below. Plus, see the steps to solve below.

Value Probability

Results:

Mean (μ):
 
Standard Deviation (σ):
 
Variance (σ²):
 

How to Find the Mean

To find the mean for a distribution, use the following formula:

μ = ∑x · p(x)

The mean for a distribution is equal to the sum of each value times the probability of the value occurring.

To find the mean, multiply each value times each probability, then add them all together.

How to Find the Variance

To find the variance for a distribution, use the following formula:

σ² = ∑x² · p(x) - μ²

The variance for a distribution is equal to the sum of each value squared times the probability of the value occurring, minus the mean squared.

How to Find the Standard Deviation

The standard deviation is equal to the square root of the variance:

So, to find the standard deviation, take the square root of the variance.

Learn how we calculated this below


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What is a Probability Distribution?

In statistics, a probability distribution is a function that describes possible values and gives the probabilities of occurrence of each for an experiment. Put another way, the likelihood of getting a particular value varies based on the underlying distribution.

Graphic showing discrete and continuous probability distributions.

The most common type of probability distribution is a normal distribution, which is a perfect bell curve when graphed. This is an example of a continuous probability distribution.

The probability of a value in a continuous distribution is defined by its probability density function (PDF). Chi-squared distributions, T distributions, F distributions, and Weibull distributions are other examples of continuous probability distributions.

A discrete probability distribution is a distribution of discrete, or countable, outcomes. In a discrete probability distribution, the probability of a value is defined by its probability mass function (PMF).

A binomial probability distribution is one example of a discrete distribution. In a binomial distribution, the outcomes are binary, or yes/no outcomes, and the number of successes in n trials of the experiment are reflected in the distribution. An example of this is flipping a coin and showing a distribution for the number of times you flipped heads.

A Poisson probability distribution is another example of a discrete distribution. In a Poisson distribution, the outcomes are independent of each other, but are countable numbers of events. For instance, you might show a distribution showing the number of cars that drive past your home every hour.

You can use our probability or our Bayes’ theorem calculators to find the likelihood of an event.

How to Find the Mean of a Probability Distribution

One characteristic of a probability distribution is the mean, or average. You can find the mean for a probability distribution using the following formula:

μ = ∑x · p(x)

Thus, the population mean for a probability distribution is equal to the sum of each value x times the probability of the value occurring p(x).

Put more simply, you can find the mean by multiplying each value times its probability of success, then add them all together.

For example, let’s find the mean for the following probability distribution:

x 1 2 3 4 5
p(x) 0.2 0.25 0.4 0.12 0.03

Start by substituting each value and probability into the mean equation.

μ = (1 · 0.2) + (2 · 0.25) + (3 · 0.4) + (4 · 0.12) + (5 · 0.03)
μ = 0.2 + 0.5 + 1.2 + 0.48 + 0.15
μ = 2.53

So, the mean μ for this probability distribution is equal to 2.53.

How to Find the Variance of a Probability Distribution

Another characteristic of a probability distribution is the variance, which is a measure of the variability within the data. Variance is represented as σ² (pronounced sigma-squared).

You can find the variance using the following formula:

σ² = ∑x² · p(x) – μ²

The variance for a probability distribution is equal to the sum of each value x squared times the probability of the value occurring p(x), minus the mean μ squared.

For example, let’s find the variance for the following probability distribution:

x 5 10 15 20
p(x) 0.1 0.35 0.25 0.3

Start by solving the first portion of the formula, which is the sum of each value squared multiplied by the probability of it occurring.

∑x² · p(x) = (5² · 0.1) + (10² · 0.35) + (15² · 0.25) + (20² · 0.3)
∑x² · p(x) = (25 · 0.1) + (100 · 0.35) + (225 · 0.25) + (400 · 0.3)
∑x² · p(x) = 2.5 + 35 + 56.25 + 120
∑x² · p(x) = 213.75

Then, subtract the mean μ squared from this result. Using the formula above, the mean for this distribution is 13.75.

σ² = 213.75 – 13.75²
σ² = 213.75 – 189.0625
σ² = 24.6875

The variance for this distribution is equal to 24.6875.

How to Find the Standard Deviation of a Probability Distribution

Another major characteristic of a probability distribution is the standard deviation. The standard deviation is represented by the Greek letter sigma σ², and it’s equal to the square root of the variance.

So, to find the standard deviation, find the variance using the steps above, then take the square root.

σ = √σ²

The standard deviation is equal to the square root of the variance.