How to Find the Standard Deviation of a Probability Distribution


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(xi-μ)2 * P(xi)

where:

  • xi: The ith value
  • μ: The mean of the distribution
  • P(xi): The probability of the ith 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:

Standard deviation of probability distribution example

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 = .97342 = 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:

Example of finding the mean of a probability distribution

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:

Example of calculating the standard deviation and variance of a probability distribution

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

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