median formula for grouped data The idea of grouped data can be illustrated by considering the following raw data set:

Table 1: Time taken (in seconds) by a group of students

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 grouped to construct the frequency distribution in several ways. One of the methods is to use the intervals as a basis.

The smallest value in the above data is 8, and the largest is 34. The distance from 8 to 34 is divided into smaller sub-intervals (called class intervals). For each class interval, the amount of data belonging to this interval is counted. This number is called the frequency of this class interval. The results are summarized in the frequency table as follows:

Busy time (in seconds) Frequency

####
5 ≤ t <10 1

10 ≤ t <15 4

15 ≤ t <20 6

20 ≤ t <25 4

25 ≤ t <30 2

30 ≤ t <35 3

Table 2: Distribution of the frequency of time (in seconds) from the group of students to

answer a simple math question

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

Frequency

Below the standard 5

Normal 10

Above standard 5

Table 3: Frequency distribution of three types of students

Average grouped data

The respect of {\ display style {\ bar {x}}} {\ bar {x}} of the average population from which data is drawn can be calculated from grouped data as:

{\ display style {\ bar {x}} = {\ frac {\ sum {f \, x}} {\ sum {f}}}.} {\ bar {x}} = {\ frac {\ sum {f \, x}} {\ sum {f}}}.

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

Frequency of classes Frequency (f) Center point (x) f x

5 and above, below 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

So the average of the grouped data is

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