**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 (*Q*_{1}) has 25% of the items below it

• Third quartiles or Upper quartile (*Q*_{3}) has 75% of the items below it

• Second quartiles or median (*Q*_{2}) 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:

Calculate 3rd quartile, 5th deciles & 68th percentile.

Solution:

It is important to arrange the data in ascending order first.

• 3rd quartile

• 5th decile

• 68th percentile

**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 *k*th 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

*k*th percentile

*P*is obtained by

_{k}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 *Q*_{1}, median, *Q*_{3}, *D*_{7}, respectively. Thus

Example 2:

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

(a) Find the value of 80th percentile. Give a brief interpretation of it.

(b) Find

*Q*

_{1}.

(c) Find the percentile rank for the score 82. Give a brief interpretation of it.

Solution:

(a) First arrange the scores in ascending order:

Then the 80th percentile

*P*

_{80}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 *Q*_{1} is equal to *P*_{25}. 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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