7.2.1 Mode and Mean of Grouped Data
(A) Modal Class
(B) Class Midpoint
The class midpoint is the value of data that lies at the centre of a class.
Class midpoint=Lower limit + Upper limit2
Class midpoint=Lower limit + Upper limit2
(C) Calculating the Mean of Grouped Data
The steps to calculate the mean of grouped data are as follows.
Step 1: Calculate the midpoint value of each class.
Step 2: Calculate the value of (frequency × midpoint value) of each class.
Step 3: Calculate the sum of the values of (frequency × midpoint value) of all the classes.
Step 4: Calculate the sum of all the frequencies of all the classes.
Step 5: Calculate the value of the mean using the formula below.
Mean of grouped data, ˉx=Sum of (frequency × midpoint)Sum of frequencies=∑fx∑fWhere Σ is the notation of summation,x is the midpoint of a class and f is its frequency.
Mean of grouped data, ˉx=Sum of (frequency × midpoint)Sum of frequencies=∑fx∑fWhere Σ is the notation of summation,x is the midpoint of a class and f is its frequency.
Example:
The following frequency table shows the number of magazines sold at a bookshop for 30 days in April 2013.
Number of magazines
|
Frequency
|
220 – 229 |
3 |
230 – 239 |
5 |
240 – 249 |
11 |
250 – 259 |
6 |
260 – 269 |
5 |
Based on the data given,
(a) calculate the size of class,
(a) calculate the size of class,
(b) state the modal class,
(c) calculate the mean number of magazine sold per day. Solution:
(a) Size of the class
= upper boundary – lower boundary
= 229.5 – 219.5
= 10
(b) Modal class = 240 – 249 (Highest frequency)
(c)
Number of magazines
|
Frequency (f)
|
Class midpoint (x)
|
220 – 229 |
3 |
224.5 |
230 – 239 |
5 |
234.5 |
240 – 249 |
11 |
244.5 |
250 – 259 |
6 |
254.5 |
260 – 269 |
5 |
264.5 |
Good