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 = Σx_{i} / n

where:

**x**: Raw data values_{i}**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.

**Additional Resources**

The following tutorials provide more information on probability distributions:

What is a Probability Distribution Table?

How to Find the Mean of a Probability Distribution

How to Find the Standard Deviation of a Probability Distribution

Probability Distribution Calculator