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



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:





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




Hence it is proved that:


Second Quartile= Fifth Decile = 50th Percentile = Median = 27.50


In order to see the relationship between Quartiles, Deciles and Percentiles in case of frequency distribution click here.

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50 responses to Quartiles, Deciles and Percentiles for Ungrouped Data

@Yoly, Just replace the 5 by 1 in both formula for decile and percentile.

For decile, i = 1, 2, 3, 4, 5, 6, 7, 8, 9
For percentile i = 1, 2, 3, . . . . . . . 99
For Quartile i = 1, 2, 3

thank you so much for giving the formulas of quartile, decile and percentile i was looking all over the web but no one was giving this much clearly the formulas with example so once again thank you so much

Why is it some formula for percentile just use n not n+1 ?. What is the difference?

Hmm it appears like your website ate my first comment (it was extremely long) so I guess I’ll just sum it up what I submitted and say, I’m thoroughly enjoying your blog. I as well am an aspiring blog writer but I’m still new to everything. Do you have any helpful hints for first-time blog writers? I’d certainly appreciate it.

This is just so wonderful..i couldn’t understand what my lecturer was explaining until i got it here..Am in kenya!

how to solve this decile for ungroup data?

anthony is a secretary in one big company in metro manila.his salary is in the 7th decile.should anthony be glad about his salary or not? explain your answer?

solution: 70% of the employeed received a salary that is less than or equal to his salaryu and 30% of the employees receive a salary that is greater than his salary.Anthony should beplease with his salary.

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