Grouped frequency tables

In a nutshell

Sometimes data may be split into different classes to make it more manageable. This means that exact values are not given in the frequency table, instead only a range of values that each one could take.

Different kinds of grouped frequency table

Discrete or continuous grouped frequency tables

For discrete grouped frequency tables there are 'gaps' between each of the classes - for example ages might be split into the classes 13-15 and 16-18.

For continuous data there should be no 'gap' and inequalities should be used instead.

Example 1

In this example Mike has been asked to record the heights of his classmates.

*Height ( $cm$ )*	Frequency
$140\lt h \le 145$	$2$
$145\lt h \le150$	$3$
$150\lt h \le 155$	$13$
$155\lt h \le 160$	$21$
$160\lt h \le 165$	$5$

Finding averages from a grouped frequency table

Similarly to non-grouped frequency tables, finding averages from a grouped frequency table means finding out which classes contain values. However in this case, you will be asked to find the modal class and class containing the median, instead of an exact value.

Definition

Mid-interval value

The value in the middle of a class.

PROCEDURE

Mid-interval value	Take the end values of each interval and add them together. Divide by two to get the mid-interval value.
Mean	Create a new column: 'mid-interval value $\times$ frequency'. Divide the column total by total frequency to get the mean
Class containing the median	Work out the position of the median by using the formula: $\frac{(n+1)}{2}$ (where n is the total frequency). Find the class containing the value in this position.
Modal class	Find the class with the highest frequency.
Range	Find the smallest value and highest value in the classes. Work out the difference between these two values.

Note: The mean and range are only estimates as we do not know the exact values within each class.

Example 2

From the table above, work out estimates for the mean (to one decimal place) and range. Find the modal class and the class containing the median.

Mean:

First create the new columns required.

Mid-interval value	Mid-interval value $\times$ frequency
$\frac{140+145}{2}=142.5$	$142.5\times2=285$
$\frac{145+150}{2}=147.5$	$147.5\times3=442.5$
$\frac{150+155}{2}=152.5$	$152.5\times13=1982.5$
$\frac{155+160}{2}=157.5$	$157.5\times21=3307.5$
$\frac{160+165}{2}=162.5$	$162.5\times5=812.5$

Use the formula.

$\frac{\text{sum of (mid-interval value × frequency)} }{\text{total frequency}}=\frac{285+442.5+1982.5+3307.5+812.5}{44}=\frac{6830}{44} = {155.227...}$

Round to one decimal place.

$\underline{155.2}$

Range:

Take the smallest class value: $140$ away from the highest class value: $165$ .

$165-140=\underline{25}$

Modal class:

The class with the highest frequency is $\underline{155 \lt h \le 160}$

Class containing the median:

Find the position of the median using the formula.

$\frac{n+1}{2}=\frac{44+1}{2}=22.5$

To find the class containing the $22.5th$ value, add up the numbers in the frequency column until you reach $22.5$ .

$2+3=5\\5+13=18\\18+21= 39$

Therefore the class containing the $22.5$ th value is: $\underline{155 \lt h \le 160}$ .