What Is the Mean of Grouped Data?

Rule

MeanofGroupedData

Multiply the midpoint ${x}_{m}$ of each interval with its frequency ${f}_{n}$, then add all these products together and divide the sum by the total number $N$ of observations:

 $\overline{x}=\frac{{x}_{{m}_{1}}\cdot {f}_{1}+{x}_{{m}_{2}}\cdot {f}_{2}+\cdots +{x}_{{m}_{N}}\cdot {f}_{N}}{N}$

Rule

MidpointofanInterval

To find the midpoint ${x}_{m}$ of each interval, just add together the lowest and greatest value in the interval and divide the sum by 2:

Example 1

A group of skaters are distributed into different weight classes. Here’s a table showing the classes and the frequency of each class.

 Weight Class Frequency $\phantom{\rule{-0.17em}{0ex}}\left[\text{}55\text{}\phantom{\rule{0.17em}{0ex}}\text{kg},\text{}60\text{}\phantom{\rule{0.17em}{0ex}}\text{kg}\right)$ 7 $\phantom{\rule{-0.17em}{0ex}}\left[\text{}60\text{}\phantom{\rule{0.17em}{0ex}}\text{kg},\text{}65\text{}\phantom{\rule{0.17em}{0ex}}\text{kg}\right)$ 8 $\phantom{\rule{-0.17em}{0ex}}\left[\text{}65\text{}\phantom{\rule{0.17em}{0ex}}\text{kg},\text{}70\text{}\phantom{\rule{0.17em}{0ex}}\text{kg}\right)$ 12 $\phantom{\rule{-0.17em}{0ex}}\left[\text{}70\text{}\phantom{\rule{0.17em}{0ex}}\text{kg},\text{}75\text{}\phantom{\rule{0.17em}{0ex}}\text{kg}\right)$ 9 $\phantom{\rule{-0.17em}{0ex}}\left[\text{}75\text{}\phantom{\rule{0.17em}{0ex}}\text{kg},\text{}80\text{}\phantom{\rule{0.17em}{0ex}}\text{kg}\right)$ 6

Find the mean weight of this group.

The first thing to do is to find the midpoint of each interval. You can use the formula above to do this.

Now you multiply the midpoint ${x}_{m}$ of each interval with the frequency $f$ of that interval. The results are shown in this table:

 Weight (kg) ${x}_{m}$ $f$ ${x}_{m}\cdot f$ $\phantom{\rule{-0.17em}{0ex}}\left[55,60\right)$ $57.5$ 7 $402.5$ $\phantom{\rule{-0.17em}{0ex}}\left[60,65\right)$ $62.5$ 8 500 $\phantom{\rule{-0.17em}{0ex}}\left[65,70\right)$ $67.5$ 12 810 $\phantom{\rule{-0.17em}{0ex}}\left[70,75\right)$ $72.5$ 9 $652.5$ $\phantom{\rule{-0.17em}{0ex}}\left[75,80\right)$ $77.5$ 6 465

Finally, you find the mean weight of the group of skaters by adding together all the values for ${x}_{m}\cdot f$ and dividing that sum by the total number of observations (the sum of all the frequencies). In this case, that’s the number of skaters in the group.

You’ve found the mean weight of the skaters to be $67$ kg.