Averages from grouped frequency tables

In a nutshell

Grouped frequency tables split the data into classes instead of the exact value of the data. This enables them to be used for continuous data types, as different data items can now be grouped up into classes so that these classes have a frequency greater than $1$ . A disadvantage of grouped frequency tables is that instead of being able to calculate exact values for the mean, median and mode, you can now only make estimations of these values.

Grouped frequency tables

These work in the same way as ungrouped frequency tables, except the data values are grouped into classes and then presented in the table. This means it is impossible to recover the exact data values from a grouped frequency table as you only know the frequency of certain data in a specific class. and not the data values themselves.

Example 1

Below is a grouped frequency table representing the height of certain people in a room.

HEIGHT ( $\bold{h\text{ } /cm}$ )	FREQUENCY
$140 < h \le 150$	$3$
$150 < h \le 160$	$4$
$160 < h \le 165$	$2$

This table shows that $3$ people have a height between $140$ and $150\thinspace\text{cm}$ , $4$ people have a height between $150$ and $160\thinspace\text{cm}$ , and $2$ people have a height between $160$ and $165\thinspace\text{cm}$ .

Averages

Mode

Unlike ungrouped frequency tables, it is impossible to find the mode from a grouped frequency table. Instead of finding the mode, you refer to the modal class which is the class with the highest frequency.

Median

Similarly to the mode, it is impossible to find the median from a grouped frequency table, and instead you can find the class containing the median. This is done identically to ungrouped frequency tables, where the median class is the class containing the median position of data.

Mean

This can be estimated using the same process as for ungrouped frequency tables, except to find the sum of the data of a class you need to estimate this by multiplying the midpoint of the class by its frequency.

Example 2

Find the modal class, the class containing the median, and the mean height from the following grouped frequency table.

HEIGHT ( $\bold{h\text{ } /cm}$ )	FREQUENCY
$140 < h \le 150$	$3$
$150 < h \le 160$	$4$
$160 < h \le 165$	$2$

Modal class: The class $\underline{150 < h \le 160}$ has the greatest frequency and so is the modal class.

Class containing the median: There are $3+4+2 = 9$ items of data, hence the median lies in the $\dfrac{9+1}{2} = 5th$ position. This item lies in the class $\underline{150 < h \le 160}$ .

Mean: Find the midpoints of each class, and the total score of each class. This can be done by extending the table as follows:

HEIGHT ( $\bold{h\text{ } /cm}$ )	FREQUENCY	MIDPOINT	MIDPOINT $\bold{\times}$ FREQUENCY
$140 < h \le 150$	$3$	$145$	$435$
$150 < h \le 160$	$4$	$155$	$620$
$160 < h \le 165$	$2$	$162.5$	$325$

Hence, the total sum of scores is $435+620+325 = 1380$ , and so the mean is:

$\dfrac{1380}{3+4+2}=\underline{153.3} \thinspace (1\thinspace d.p)$ .