What are grouped data?
Grouped data are data formed by aggregating individual observations of a variable into groups,
so that a frequency distribution of these groups serves as a convenient means of summarizing or analyzing the data.
What is frequency?
No of times of occurance of an event/observation in a particular interval is called frequency.
What is Mean?
The mean is the average of a data set.
What is median?
The median is the middle of the set of numbers.
What is mode ?
The mode is the most common number in a data set.
Example:
Consider the following raw data showing the marks obtained by students in maths exam.
So, the grouped data is written as
Marks ($t$)
Frequency ($f$)
60 ≤ t < 70
4
70 ≤ t < 80
5
80 ≤ t < 90
3
90 ≤ t ≤ 100
2
How to find Mean of Grouped Data:
There are three methods for find mean of grouped data.
(i) The Direct Method
(ii) The Assumed Mean Method
(iii) The Step Deviation Method.
(i)Method 1: The Direct Method
mean $x̄$ = ${f_1x_1+ f_2x_2+...+f_nx_n}/{f_1+f_2+..+f_n}$
i.e $x̄$ = ${∑f_ix_i}/{∑f_i}$
where $x_1$ , $x_2$ ,. . ., $x_n$ are observations with respective frequencies $f_1$ , $f_2$ , . . ., $f_n$
Example: Time taken (in seconds) by the group of students to answer a simple
math question is grouped as given below
Time taken (in seconds)
Frequency ($f_i$)
5 ≤ t < 10
1
10 ≤ t < 15
4
15 ≤ t < 20
6
20 ≤ t < 25
4
25 ≤ t < 30
2
30 ≤ t < 35
3
Find the mean of the grouped data.
Solution
Frequency distribution of the time taken (in seconds) by the group of students to answer a simple
math question is grouped as given below.
Time taken (in seconds)
Frequency ($f_i$)
Midpoint($x_i$)
$f_ix_i$
5 ≤ t < 10
1
7.5
7.5
10 ≤ t < 15
4
12.5
50
15 ≤ t < 20
6
17.5
105
20 ≤ t < 25
4
22.5
90
25 ≤ t < 30
2
27.5
55
30 ≤ t < 35
3
32.5
97.5
Total
20
405
Thus the mean of grouped data => $x̄$ = ${∑f_ix_i}/{∑f_i}$
$x̄$ = $405/20$ = $20.25$
