The mean as a balance point

Five quiz scores of 6, 8, 9, 7 and 10 add to 40, so their mean is 40 ÷ 5 = 8.

Five friends pool their pocket money

Five friends get different amounts of pocket money this week. The numbers are made up: $6, $8, $9, $7 and $10. They put it all in one jar, $40 in total, and split it evenly. Each takes $8 home. That share is the mean. Only one of the five had exactly $8 to begin with, and nobody minds.

When people say "the average", they nearly always mean this arithmetic mean. Add up the values, then divide by how many there are. Quiz scores of 6, 8, 9, 7 and 10 work the same way as the pocket money: the total is 40, there are 5 scores, and the mean is 40 ÷ 5 = 8. Statistics books write the mean of a sample as x̄ ("x bar") and the mean of a whole population as μ, the Greek letter mu. The recipe is the same for both: x̄ = (sum of the values) ÷ n.

Deviations from 8 add up to zero

Now put the scores on a ruler. Each one is a coin of equal weight sitting at its own number. Where does the ruler balance? At the mean. You can check this with deviations: the distance of each value from the mean, with a sign. For the quiz scores they are −2, 0, +1, −1 and +2. Their total is exactly 0. That happens in every data set, because the values below the mean pull with the same total force as the values above it. Far-off values pull harder, so one extreme value can drag the mean a long way.

Four whole numbers, and a mean of 2.5

The mean does not have to be one of the values. The mean of 1, 2, 3 and 4 is 2.5. The mean number of children per family can be 1.8 even though no family has 1.8 children. Shift every value by the same amount and the mean shifts with it; double every value and the mean doubles. Change a single value by d and the mean changes by d ÷ n. In a set of 5 values, raising one score by 5 points raises the mean by 1.

Because the mean is tied to the total, you can run it backwards. Four tests average 80? Then they total 320. A mean of 82 over five tests needs a total of 410, so the fifth test must be 410 − 320 = 90. Sometimes the parts of a grade count differently, and then you need a weighted mean: multiply each value by its weight, add, and divide by the total weight. A course grade that is 60% homework at 80 and 40% exam at 90 comes to 0.6 × 80 + 0.4 × 90 = 84.

Data often come as a frequency table instead of a list. Multiply each value by how many times it occurs, add the products, and divide by the total count. OpenStax's symmetric set has one 4, one 5, three 6s, six 7s, three 8s, one 9 and one 10. The products add to 4 + 5 + 18 + 42 + 24 + 9 + 10 = 112. The count is 16. So the mean is 112 ÷ 16 = 7, which is a weighted mean again, with the frequencies as the weights.