# The Average

The average is a measure of the center of a distribution of values. It is computed by summing the values and dividing by the number of values. For example, the average of

0 1 1 2 9

is

(0+1+1+2+9)/5 = 13/5 = 2.6

Here's an example from real data. The Health and Nutrition Examination Survey (HANES) of 1976-80, studied a sample of 20,322 Americans between the ages of 1 and 74. They collected all kinds of physiological data like height, weight, sex, race, blood pressure, etc.

Looking at just the adults (ages 18-74), they found

• The average height of the men was 5 feet 9 inches, and their average weight was 171 pounds.
• The average height of the women was 5 feet 3.5 inches, and their average weight was 146 pounds.

The following diagrams show average height and weight broken down by age:

Figure. Age-specific average heights and weights for men and women 18-74 in the HANES sample. The panel on the left shows height, the panel on the right shows weight.

In the figure, the average height of men appears to decrease after age 20, dropping about 2 inches in 50 years. Same for women. Does this mean people get shorter as they get older? Well, that may be true, but you couldn't prove it by these data. That's because these data are cross-sectional, rather than longitudinal. In a cross-sectional study, you collect all the data at the same time, so the old people in the sample are not the same people as the young people. In contrast, in a longitudinal study, the same subjects are tracked over time, so the old people are the same people as the young people, at a different point in time.

Cross-sectional data are much easier to collect, but it is sometimes hard to conclude things from them. For example, the people aged 18-24 in the figure are completely different from those age 65-74. The first group was born around 1955, the second around 1905. They were exposed to a completely different environment. In 1955, the US was much wealthier, nutrition was much better, medical care was better, and so on. The people born in 1905 lived through the Depression, and two world wars. So the reason why the old people in the sample are so short could be because of these kinds of factors, not because of aging. In fact, longitudinal data suggests that in fact, Americans are getting taller with every generation (this is called a secular trend), so that if we compare 18 year olds today with 18 year olds 50 years, the current ones are quite a bit taller.

(based on Statistics, by Freedman, Pisani, Purves, Adhikari).