# Expected Value vs. Mean: What’s the Difference?

Two terms that are sometimes used interchangeably in statistics are expected value and mean.

In general, we use the following terms in different situations:

• Expected value is used when we want to calculate the mean of a probability distribution. This represents the average value we expect to occur before collecting any data.
• Mean is typically used when we want to calculate the average value of a given sample. This represents the average value of raw data that we’ve already collected.

The following examples illustrate how to calculate the expected value and the mean in practice.

### Example: Calculating Expected Value

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 calculate the expected value of this probability distribution, we can use the following formula:

Expected Value = Σx * P(x)

where:

• x: Data value
• P(x): Probability of value

For example, we would calculate the expected value for this probability distribution to be:

Expected Value = 0*0.18  +  1*0.34  +  2*0.35  +  3*0.11  +  4*0.02  =  1.45 goals.

This represents the expected number of goals that the team will score in any given game.

### Example: Calculating Mean

We typically calculate the mean after we’ve actually collected raw data.

For example, suppose we record the number of goals that a soccer team scores in 15 different games:

Goals Scored: 1, 1, 0, 2, 2, 1, 0, 3, 1, 1, 1, 2, 4, 3, 1

To calculate the mean number of goals scored per game, we can use the following formula:

Mean = Σxi / n

where:

• xi: Raw data values
• n: Sample size

For example, we would calculate the mean number of goals scored as:

Mean = (1+1+0+2+2+1+0+3+1+1+1+2+4+3+1) / 15 = 1.533 goals.

This represents the mean number of goals scored per game by the team.