Calculation of Quartiles, Deciles & Percentiles for Ungrouped Data

Calculation of Quartiles, Deciles & Percentiles for Ungrouped Data

Quartiles
• Divides an array into four equal parts
• Each portion contains equal number of items
• First quartiles or lower quartile (Q1) has 25% of the items below it
• Third quartiles or Upper quartile (Q3) has 75% of the items below it
• Second quartiles or median (Q2) has 50% of the items below it

Deciles & Percentiles
• Divides an array into 10 & 100 parts respectively
• Each portion contains equal number of items
• General formula for Quartiles, Deciles & Percentiles for ungrouped data


Where, P= is the desired percentile

Example 1:
The Quick oil company has a number of outlets in the metropolitan Seattle area. The numbers of changes at the Oak Street outlet in the past 20 days are:

65, 98, 55, 62, 79, 79, 59, 51, 90, 72, 56, 70, 62, 66, 80, 94, 63, 73, 71, 85.

Calculate 3rd quartile, 5th deciles & 68th percentile.
Solution:
It is important to arrange the data in ascending order first.
51, 55, 56, 59, 62, 62, 63, 65, 66, 70, 71, 72, 73, 79, 79, 80, 85 90, 94, 98.

• 3rd quartile

• 5th decile

• 68th percentile

= 14th item+(15th–14th)×.28=79

Calculation of percentiles, deciles and quartiles:
The finding of percentiles, deciles and quartiles using graphs is not always accurate. They can found more accurately by using the following formulas:

(i) Ranked raw data:
The (approximate) value of the kth percentile Pk is


where k is the number of percentile and n is the sample size.

The percentile rank of a value xi is obtained by


(ii) Grouped data:
The kth percentile Pk is obtained by

where = the lower boundary of the percentile class,

C= the preceding cumulative frequency to the percentile class,

f= the frequency of the percentile class,

w= the width of the percentile class.

Note:
If, for example, k=25, 50, 75, 70, the above formula reduces to Q1, median, Q3, D7, respectively. Thus

Example 2:
The following are the test scores of 12 students in a statistics class:

70, 77, 65, 56, 99, 62, 79, 73, 85, 87, 92, 82.

(a) Find the value of 80th percentile. Give a brief interpretation of it.
(b) Find Q1.
(c) Find the percentile rank for the score 82. Give a brief interpretation of it.
Solution:
(a) First arrange the scores in ascending order:
56, 62, 65, 70, 73, 77, 79, 82, 85, 87, 92, 99.

Then the 80th percentile P80 is obtained by

The value of 9.6th term can be approximated by the average of 9th and 10th terms in the ranked data. Therefore,

Thus approximately 80% of the scores are less than 86 and 20% are greater than 86 in the given data.

(b) The Q1 is equal to P25. Therefore,


(c) Percentile rank of 82,

From this result we can interpret that about 58% of the students scored less than 82.

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