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# 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:

Ïƒ = âˆšÎ£(xi-Î¼)2 * P(xi)

where:

• xi: The ith value
• Î¼: 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:

Î¼ = 0*0.18Â  +Â  1*0.34Â  +Â  2*0.35Â  +Â  3*0.11Â  +Â  4*0.02Â  =Â Â 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 = âˆš(.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: Standard Deviation 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:

Î¼ = 0*0.24Â  +Â  1*0.57Â  +Â  2*0.16Â  +Â  3*0.03 =Â  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 = âˆš(.2305 + .0002 + .1665 + .1224) = 0.7208

### Example 2: Standard Deviation 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:

Î¼ = 10*.24Â  +Â  20*.31Â  +Â  30*0.39Â  +Â  40*0.06Â  =Â  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 = âˆš(38.7096 + 2.2599 + 20.7831 + 17.9574) = 8.928