Estimating the Mean From a Frequency Distribution

Sometimes we are given a chart showing frequencies of certain groups instead of the actual values. We can still come up with a good estimate of a typical value for the set of data, provided that we make some assumptions. We assume that the values in each class or group are spread evenly throughout the group. If this is the case, then the mean for each class should be approximately equal to the midpoint for each class. Recall that the midpoint is found by adding the lower class boundary to the higher class boundary, then dividing that sum by 2. So for each class, we have a mean and a number of values (this is a frequency distribution after all). We now call on our friend the weighted mean. If we multiply each midpoint by its frequency, and then divide by the total number of values in the frequency distribution, we have an estimate of the mean.

Let's try an example.

Ages of Statistics Students in Spring 1997
Ages
Frequency
17-21
12
22-26
15
27-31
7
32-36
4
37-41
2

Estimate the mean for this set of data.
Class
Midpoint
Frequency
m f
17-21
19
12
228
22-26
24
15
360
27-31
29
7
203
32-36
34
4
136
37-41
39
2
78

The sum of the product of the midpoints and frequencies is 1005. (Just add the values in the last column). Divide this number by 40 (the total of the frequencies), and we estimate the mean to be 25.125. If you look at the frequency distribution, this value will look like a typical for this set of data.