# Finding the 40th Percentile

Instructor: Sharon Linde
Newspapers and news programs are often citing percentiles when they report on elections, social trends, and other factors that affect the population. This lesson goes over a step by step process for determining the 40th percentile.

## Setting Up the Problem

Margaret is a new research assistant for a market research company and has just been assigned her first task. From a data set containing 11 pieces of information, she is supposed to find the 40th percentile. As we'll see in the last section there are several slightly different ways to define percentile. For now, let's define a percentile as the lowest number in an ordered data set that is greater than or equal to a specific percentage of the other numbers in a set. In Margaret's case, she wants to find which of her 11 data points would be larger than 40% of the numbers in the set. Let's take a closer look at how this works in our small example.

• Step 1 - Get the data

What is the data Margaret is being asked to work on? It's a composite score on a cleaning survey that her company sent out. The scores are:

Data
79 15 22 81 55 43 93 39 45 66 71

The possible scores on this survey were the integers from 0 to 100.

• Step 2 - Count the number of data points

This step is pretty easy for a small data set like this: there are exactly 11 numbers in this sequence. Although this step is simple, it will be important later on.

• Step 3 - Arrange the data points in ascending order

The next step is to arrange our data set from smallest to largest.

Ordered Data
15 22 39 43 45 55 66 71 79 81 93
• Step 4 - Compute the position of the 40th Percentile

To get the 40th percentile position in a data set with 11 data points, just multiply our percentile by the number of points in our data set:

( 0.40 )( 11 ) = 4.4

• Step 5 - Choose the data point that satisfies our definition

We chose to define the 40th percentile as the lowest number that is larger or equal to 40% of the numbers in our data set of 11 numbers. Using this definition, we end up dropping any decimal obtained in this step. So, we'll have to go to the 4th number in our ordered data set to get our answer.

## Solution

Since the fourth number in our ordered data set is 45, our solution for this data set and this definition is 45. 45 is the smallest number in our set of numbers that is larger than or equal to 40% of the numbers in this set.

## Checking Your Work

Generally, when you are computing the 40th percentile of a set of numbers, you would expect to get something a little smaller than the median for that set. In our case, the median number is also 55 - because there are exactly 5 number higher and 5 numbers lower. 45 is lower than this, so that is a good check. However, this check won't always work for small sets of numbers like this.

This is because there is no standard definition of percentile. Three common ways to define the 40th percentile are:

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