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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:

**Ïƒ ^{2} = Î£(x_{i}-Î¼)^{2} * P(x_{i})**

where:

**x**The i_{i}:^{th}value**Î¼:**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:

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:

Ïƒ^{2} = .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:

Ïƒ^{2} = .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:

Ïƒ^{2} = 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.