7 – Measures of Location

In a previous blog post we looked at the measures of Central Tendency. These measures like Mean, Median and Mode give us an idea about the mid point around which data is spread.

As an extension to this, there are other Measures of Location that give us an idea about different points in the spread of data apart from just the central location. They are called Percentiles and Quartiles.

Percentile:

Percentile is a measure that divides given data into 100 parts. A p^th percentile implies that there are at least p percentage of observations less than or equal to this value and at least 100 – p observations greater than or equal this value.

To illustrate with an example, look at the table below for marks scored by students in a Statistics class.

The given data items need to be arranged in sorted order (which in this case is already done).  Then the p^th  percentile is calculated as

where

i denotes the index for given percentile

p is the percentile of interest

N is the number of items in data set

If “i” is an integer, p^th  percentile is the average of values occurring in i^th position and (i+1)^st position.

If “i” is not an integer, p^th  percentile is the value occurring in i^th position after rounding off “i” to next higher value.

So, to get the 30^th percentile in above example,

i = (30/100) * 12 = 3.6

As “i” is not an integer, we round off 3.6 to 4. Hence, 30th percentile in given data is the observation in 4th position in sorted order, which is 60. There are at least 30% of observations less than or equal to 60 (4 out of 12 values) and at least 70% of observations greater than or equal to 60 (9 out of 12 values).

This is mostly used in exam scores scenario. When we say someone scored 75% in an exam, it doesn’t tell us how well he did relative to the other students in his class. Instead, if it converts as 90 percentile for instance, it tells us that he stands above 90% of the students in his class.

Quartile:

Quartiles are nothing but percentiles taken in multiples of 25.

25^th percentile is Quartile 1

50^th percentile is Quartile 2

75^th percentile is Quartile 3

So, the calculation approach for quartiles remain exactly the same as percentiles illustrated above.

Percentiles calculated in multiples of 10 are also called Deciles (10th, 20th, 30th, etc. percentiles).

50th percentile (Quartile 2) also happens to be the Median by definition, as Median is the central value by location.