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 P_k is

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

The percentile rank of a value x_i is obtained by

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

where ℓ= 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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