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)
- 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
- 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.
The following tutorials provide more information on probability distributions: