# 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:

σ2 = 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 variance as:

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

σ2 = .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:

μ = 0*0.24  +  1*0.57  +  2*0.16  +  3*0.03 =  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:

σ2 = .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:

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

σ2 = 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.

## 3 Replies to “How to Calculate the Variance of a Probability Distribution”

1. Monty says:

No sure how to ask this. The distribution is tabular – that it doesn’t fit any known distribution, and your method looks reasonable. How do you extends this to more than one variable?

2. Monty says:

I think I know. I think this would be like a normal covariance calculation. If I’m given a joint distribtion of let’s say 2 variables, I’d get 2d table of probabilities (P(x1[i], x2[i])) – that’d add up to one. I think I’d compute the marginal mean for variable 1 and variable 2 and make it a vector MU. Then define DX(i) = (x1[i], x2[i]) – MU. Then cov = sum_i P(x1[i], x2[i]) DX DX^T. I think that’d work.

3. prince william says:

the queen died rip