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# How to Calculate the Variance 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 variance of a probability distribution, we can use the following formula:

Ïƒ2 = Î£(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 variance as:

The variance is simply the sum of the values in the third column. Thus, we would calculate it as:

Ïƒ2 = .3785 + .0689 + .1059 + .2643 + .1301 = 0.9475

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

### Example 1: Variance 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:

To find the variance of this probability distribution, we need to first calculate the mean number of expected failures:

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 variance as:

The variance is the sum of the values in the third column. Thus, we would calculate it as:

Ïƒ2 = .2305 + .0002 + .1665 + .1224 = 0.5196

### Example 2: Variance 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:

To find the variance of this probability distribution, we need to first calculate the mean number of expected sales:

Î¼ = 10*.24Â  +Â  20*.31Â  +Â  30*0.39Â  +Â  40*0.06Â  =Â  22.7 sales.

We could then calculate the variance as:

The variance is the sum of the values in the third column. Thus, we would calculate it as:

Ïƒ2 = 38.7096 + 2.2599 + 20.7831 + 17.9574 = 79.71

Note that we could also use the Probability Distribution Calculator to automatically calculate the variance of this distribution:

The variance is 79.71. This matches the value that we calculated by hand.