Elementary Statstics

6.3.1 Median

Median of a data is the value which divides the data it into two equal parts. It is the value which exceeds and is exceeded by the same number of observations, i.e. it is the value such that the number of observations above it is equal to the number of observations below it. The median is thus a positional average.

In case of ungrouped data, if the number of observations is odd then median is the middle most value arranged in the ascending or descending order of magnitude. In case of even number of observations, there are two middle terms and median is obtained by taking the arithmetic mean of the middle most terms. For example, the median of the values 25, 20, 15, 35, 18 i.e. 15, 18, 20, 25, 35,25,20 and the median of 8, 20, 50, 25, 15, 30 i.e. of 8,15, 20, 25, 30, 50 is ½ (20+25) = 22.5.

Remark: In case of even number of observations, in fact any value lying between the two middle values can be taken as median but conventionally we take it to be the mean of the middle terms.

In case of discrete frequency distribution median is obtained by considering the cumulative frequencies. The steps for calculating median are given below:

i) Find N/2, where N= ∑ f_i

i) See the cumulative frequency (c.f) is either equal to just greater than N/2.

ii) The corresponding value of x is median

Example: Obtain the median for the following frequency distribution:

x : 1 2 3 4 5 6 7 8 9

f : 8 10 11 16 20 25 15 9 6

Solution

x	f	c.f.
1	8	8
2	10	18
3	11	29
4	16	45
5	20	65
6	25	90
7	15	105
8	9	114
9	6	120
	120

Here N = 120 and have N/2 = 60

Cumulative frequency (c.f.) just greater than N/2, is 65 and the value of x corresponding to 65 is 5. Therefore median is 5.

In the case of continuous frequency distribution, the class corresponding to the c.f. just greater than N/2 is called the median class and the value of median is obtained by the following formula:

l is the lower limit of the median class

f is the frequency of the median class

h is the magnitude of the median class

c is the c.f. of the class preceding the median class

and N= ∑ f