A probability distribution tells us the probability that a random variable takes on certain values.

For example, the following probability distribution tells us the probability that a certain soccer team scores a certain number of goals in a given game:

To find the **standard deviation **of a probability distribution, we can use the following formula:

**σ = √Σ(x _{i}-μ)^{2} * P(x_{i})**

where:

**x**The i_{i}:^{th}value**μ:**The mean of the distribution**P(x**The probability of the i_{i}):^{th}value

For example, consider our probability distribution for the soccer team:

The mean number of goals for the soccer team would be calculated as:

μ = 0*0.18 + 1*0.34 + 2*0.35 + 3*0.11 + 4*0.02 = **1.45** goals.

We could then calculate the standard deviation as:

The standard deviation is the square root of the sum of the values in the third column. Thus, we would calculate it as:

Standard deviation = √(.3785 + .0689 + .1059 + .2643 + .1301) = **0.9734**

The variance is simply the standard deviation squared, so:

Variance = .9734^{2} = **0.9475**

The following examples show how to calculate the standard deviation of a probability distribution in a few other scenarios.

**Example 1: Mean Number of Vehicle Failures**

The following probability distribution tells us the probability that a given vehicle experiences a certain number of battery failures during a 10-year span:

**Question: **What is the standard deviation of the number of failures for this vehicle?

**Solution:** The mean number of expected failures is calculated as:

μ = 0*0.24 + 1*0.57 + 2*0.16 + 3*0.03 = **0.98 **failures.

We could then calculate the standard deviation as:

The standard deviation is the square root of the sum of the values in the third column. Thus, we would calculate it as:

Standard deviation = √(.2305 + .0002 + .1665 + .1224) = **0.7208**

**Example 2: Mean Number of Sales**

The following probability distribution tells us the probability that a given salesman will make a certain number of sales in the upcoming month:

**Question: **What is the standard deviation of the number of sales for this salesman in the upcoming month?

**Solution:** The mean number of expected sales is calculated as:

μ = 10*.24 + 20*.31 + 30*0.39 + 40*0.06 = **22.7 **sales.

We could then calculate the standard deviation as:

The standard deviation is the square root of the sum of the values in the third column. Thus, we would calculate it as:

Standard deviation = √(38.7096 + 2.2599 + 20.7831 + 17.9574) = **8.928**

**Additional Resources**

How to Find the Mean of a Probability Distribution

Probability Distribution Calculator