*Sharon Linde*Show bio

Sharon has an Masters of Science in Mathematics and a Masters in Education

Lesson Transcript

Instructor:
*Sharon Linde*
Show bio

Sharon has an Masters of Science in Mathematics and a Masters in Education

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.
Updated: 05/03/2021

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. 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.

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.

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.

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 |

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

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 fourth number in our ordered data set to get our answer, which 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.

Generally, when you're 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 numbers 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. The other ways to define the 40th percentile are:

- The smallest number that is greater than 40% of the numbers in the set.
- If the number in Step 5 is not an integer, then use a number between the two integer positions on either side using the process of interpolation.

**Interpolation** is a mathematical process to get a number a specified distance between two other numbers. Since Step 5 gave us 4.4, we'd want to find how much that last 0.4 adds to the fourth position of 45.

The math to do that looks like this:

= (fourth position) + (the decimal greater than fourth position) (the difference between fifth position and fourth position)

= 45 + (0.4) (50 - 45)

= 45 + (0.4) (5)

= 45 + 2

= 47

Using these different definitions would yield different answers. Using our first definition gives us 45, the smallest number that is greater than 40% of the numbers in the set is 50. Interpolation gives 47.

The reason that there is no standard definition is because for large data sets there is essentially no difference which definition is used. Think of an example with 11,000 data points for this same kind of survey. Is the difference between the 4,400th and 4,401st position likely to be a gap of 10 on a 100 point survey? What do you think would happen for a million data sets?

To find the percentile of a data set:

- Get the data
- Count the number of data points
- Arrange the data points in ascending order
- Compute the position of the 40th percentile
- Choose the data point that satisfies the definition

Three definitions of **percentile** are:

- 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.
- The smallest number that is greater than 40% of the numbers in the set.
- If the number in Step 5 is not an integer, then use a number between the two integer positions on either side using the process of interpolation.

**Interpolation** is a mathematical process to get a number a specified distance between two other numbers.

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