Quartiles , Percentiles, and Deciles

You take a test and are told that you scored in the 85th percentile . What does this mean?

It does not mean that you must have gotten 85% of the questions correct.

It means that 85% of those taking the test had scores lower than yours .

Definition

A data value x corresponds to the Nth percentile if N% of the data is less than x.

Some special percentiles called quartiles and deciles are listed below.

The first, second, and third quartiles are

= the 25th percentile, = the 50th percentile (or median), = the 75th percentile

The deciles are

= the 10th percentile, = the 20th percentile, ..., = the 90th percentile

Example 1

The sorted weights of 50 high school students are listed below. Find the percentile for 128 pounds.

Solution .

Since 8 of the 50 weights are less than 128, 8/50 or .16 of the data is less than 128. Thus, 128 corresponds to the 16th percentile for this data set.

Example 2

Find the third quartile for the data in example one.

Solution .

Since the third quartile is the 75th percentile and we have 50 pieces of data, we must find the data value in the following position

Rounding this up to 38, we use the 38th piece of ranked data or 183 as the third quartile.

Example 3

Find the sixth decile for the data in example one.

Solution .

Since the sixth decile is the 60th percentile and we have 50 pieces of data, we must calculate

Since this is a whole number we must take the average of the 30th and 31st sorted data values. In this case we find (167+174)/2 = 170.5

The following example shows why we can't just use the 30th value in the last example as the 60th percentile.

Example 4

For the data in example one, find the percentile corresponding to the value 167.

Solution .

Since 29 of the 50 weights are less than 167, 29/50 = .58 of the data is less than 167. Thus 167 corresponds to the 58th percentile.