Quartiles, Deciles and Percentiles for Ungrouped Data

When data is arranged in ascending or descending order, it can be divided into various parts by different values such as quartiles, deciles and percentiles. These values are collectively called quantiles and are the extension of median formula which divides data into two equal parts. Since the basic purpose of these partition values is to divide data into different parts therefore a relationship exists between them. This relationship is given below and is elaborated with the help of simple problem.

Problem: Following are the number of defective items produced in a month by a machine for the last 24 months. 45, 30, 36, 26, 16, 21 33, 40, 32, 14, 10, 29, 23, 39, 17, 11, 18, 34, 19, 24, 21, 35, 42, 37

 
Prove that Second Quartile= Fifth Decile = 50th Percentile = Median

Solution:

Arrange data in ascending order:
10, 11, 14, 16, 17, 18, 19, 21, 21, 23, 24, 26, 29, 30, 32, 33, 34, 35, 36, 37, 39, 40, 42, 45

Second Quartile Q2

Since the data given in the problem is ungrouped therefore following formula will be used for the calculation of second quartile.

5th Decile

The calculation of 5th decile for ungrouped data is given below:

50th Percentile (P50)

The calculation of 50th percentile for ungrouped data is given below:

Median

If the observations are Even in numbers, then median is the arithmetic mean of two central values. These two central values are calculated as: