Definition

The median divides a frequency distribution into two halves. The median of set of values arranged either in ascending order or descending order of their magnitude is referred as the middle value. Median is denoted by clip_image002 ( X- childa).

Explanation

Where the number of values in a data is odd, the median shall be the middle value. And where the number of values is even, the median shall be the mean of two middle values. Once the distribution is divided into two halve by way of median then the number of values greater than the median is equal to the values smaller than the median.

For Example

i. The median of values; 4, 5, 6, 8, 10, 11 and 12 is 8.

ii. The median of values; 4, 6, 7, 9, 11 and 13 is clip_image004

From the above examples we have learned that when the number of values is odd, the median is the middle values and when the number of values is even, the median is the mean of the two middle values present in the data. In both the cases the median is the value of clip_image006th item from either ends of the data which is being arranged in ascending or descending order.

In example (i) n=7, and the clip_image008th item or the 4th item from either sides is 8. And in example (ii), n=6 and the median which is clip_image010th or 3.5th i.e. half way between the 3rd and 4th from either end, is clip_image012.

In an array of values the position, the position of median could be determined by the following formula;

clip_image014

Median for Grouped Data

In case of frequency distribution the median is the value of clip_image016th item from either end. Therefore, if we have 100 items in a frequency distribution, the median will be the value of the 50th item. In order to find the median from a frequency distribution, we need to form a separate column for cumulative frequency. The median will lie in the class which corresponds to the cumulative frequency in which clip_image018 lies.

The formula for median in case of frequency distribution is as follows;

clip_image020

Where,

l = lower class boundary of the median class, which is corresponding to the cumulative frequency in which clip_image018[1] lies.

h = class interval size of the median class.

f = frequency of the median class.

n = number of values, or the total frequency.

C.F = cumulative frequency of the class preceding the median class.

For Example

The median for the following Table 14 will be;

Table 14

Classes

Frequency (f)

Class Boundaries

Cumulative Frequency (C.F)

0 – 9

2

-0.5 – 9.5

2

10 – 19

3

9.5 – 19.5

5

20 – 29

11

19.5 – 29.5

16

30 – 39

24

29.5 – 39.5

40

40 – 49

32

39.5 – 49.5

72

50 – 59

40

49.5 – 59.5

112

clip_image022

Here,

n =112

i = 10

f = 32

l = 39.5

First we find, clip_image024

So,

clip_image026

clip_image028

clip_image030

clip_image032

clip_image034

clip_image036

Another example for calculating median by the Table 15 is provided as follows;

Table 15

Class Marks (X)

Frequency (f)

Class Boundaries

Cumulative Frequency (C.F)

18.5

7

13.5 – 23.5

7

28.5

12

23.5 – 33.5

19

38.5

23

33.5 – 43.5

42

48.5

35

43.5 – 53.5

77

58.5

25

53.5 – 63.5

102

68.5

8

63.5 – 73.5

110

clip_image022[1]

Here,

n =110

h = 10

f = 35

l = 43.5

First we find, clip_image038

So,

clip_image040

clip_image042

clip_image044

clip_image046

clip_image048

clip_image050

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