Page 106 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 6: Measures of Central Tendency

2. Given that there are n observations, the median is given by: Notes
n 1
(a) The size of th observations, when n is odd.
2
n n 1
(b) The mean of the sizes of th and th observations, when n is even.
2 2

Example: Find median of the following observations:
20, 15, 25, 28, 18, 16, 30.
Solution:
Writing the observations in ascending order, we get 15, 16, 18, 20, 25, 28, 30.

7 1
Since n = 7, i.e., odd, the median is the size of th i.e., 4th observation.
2

Hence, median, denoted by Md = 20.
Note: The same value of Md will be obtained by arranging the observations in descending order
of magnitude.

Example: Find median of the data : 245, 230, 265, 236, 220, 250.
Solution:

Arranging these observations in ascending order of magnitude, we get
220, 230, 236, 245, 250, 265. Here n = 6, i.e., even.

6 6
Median will be arithmetic mean of the size of th, i.e., 3rd and 1 th , i.e., 4th observations.
2 2
236 245
Hence M 240.5
d 2
When ungrouped frequency distribution is given

In this case, the data are already arranged in the order of magnitude. Here, cumulative frequency
is computed and the median is determined in a manner similar to that of individual observations.

Example: Locate median of the following frequency distribution:
Variable (X) : 10 11 12 13 14 15 16
Frequency (f ) : 8 15 25 20 12 10 5
Solution:
X : 10 11 12 13 14 15 16
f : 8 15 25 20 12 10 5
c. f . : 8 23 48 68 80 90 95
th
95 1
Here N = 95, which is odd. Thus, median is size of i.e., 48th observation. From the table
2
48th observation is 12, M = 12.
d

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