Grouped data

Grouped data is a statistical term used in data analysis. A raw dataset can be organized by constructing a table showing the frequency distribution of the variable (whose values are given in the raw dataset). Such a frequency table is often referred to as grouped data.

Example

The idea of grouped data can be illustrated by considering the following raw dataset:

Table 1: Time taken (in seconds) by a group of students to
answer a simple math question

20	25	24	33	13
26	8	19	31	11
16	21	17	11	34
14	15	21	18	17

The above data can be organised into a frequency distribution (or a grouped data) in several ways. One method is to use intervals as a basis.

The smallest value in the above data is 8 and the largest is 34. The interval from 8 to 34 is broken up into smaller subintervals (called class intervals). For each class interval, the amount of data items falling in this interval is counted. This number is called the frequency of that class interval. The results are tabulated as a frequency table as follows:

Table 2: Frequency distribution of the time taken (in seconds) by the group of students to
answer a simple math question

Time taken (in seconds)	Frequency
5 and above, below 10	1
10 and above, below 15	4
15 and above, below 20	6
20 and above, below 25	4
25 and above, below 30	2
30 and above, below 35	3

Another method of grouping the data is to use some qualitative characteristics instead of numerical intervals. For example, suppose in the above example, there are three types of students: 1) Smart, if the response time is 5 to 14 seconds, 2) normal if it is between 15 and 24 seconds, and 3) below normal if it is 25 seconds or more, then the grouped data looks like:

Table 3: Frequency distribution of the three types of students

	Frequency
Smart	5
Normal	10
Below normal	5

Mean of grouped data

An estimate, $\bar{x}$, of the mean of the population from which the data are drawn can be calculated from the grouped data as:

$\bar{x}=\frac{\sum{f\,x}}{\sum{f}} .$

In this formula, x refers to the midpoint of the class intervals, and f is the class frequency. Note that the result of this will be different from the sample mean of the ungrouped data. The mean for the grouped data in the above example, can be calculated as follows:

Class Intervals	Frequency ( f )	Midpoint ( x )	f x
5 and above, below 10	1	7.5	7.5
10 and above, below 15	4	12.5	50
15 and above, below 20	6	17.5	105
20 and above, below 25	4	22.5	90
25 and above, below 30	2	27.5	55
30 and above, below 35	3	32.5	97.5
TOTAL	20		405

Thus, the mean of the grouped data is

$\bar{x}=\frac{\sum{f\,x}}{\sum{f}} = \frac{405}{20} = 20.25$

See also

• Data binning
• Level of measurement
• Frequency distribution
• Discretization of continuous features

References

• Newbold, P., W. Carlson and B. Thorne (2009) Statistics for Business and Economics, Seventh edition, Pearson Education. ISBN 978-0-13-507248-6.