Percentiles

Site:	Saylor Academy
Course:	MA121: Introduction to Statistics
Book:	Percentiles

Printed by:	Guest user
Date:	Friday, April 26, 2024, 1:38 AM

Description

This section discusses percentiles, which are useful for describing relative standings of observations in a dataset.

Introduction
Two Simple Definitions of Percentile
Third Definition
Questions
Answers

Introduction

Learning Objectives

Define percentiles
Use three formulas for computing percentiles

A test score in and of itself is usually difficult to interpret. For example, if you learned that your score on a measure of shyness was 35 out of a possible 50, you would have little idea how shy you are compared to other people. More relevant is the percentage of people with lower shyness scores than yours. This percentage is called a percentile. If 65% of the scores were below yours, then your score would be the 65th percentile.

Source: David M. Lane, https://onlinestatbook.com/2/introduction/percentiles.html
This work is in the Public Domain.

Two Simple Definitions of Percentile

There is no universally accepted definition of a percentile. Using the 65th percentile as an example, the 65th percentile can be defined as the lowest score that is greater than 65% of the scores. This is the way we defined it above and we will call this "Definition 1". The 65th percentile can also be defined as the smallest score that is greater than or equal to 65% of the scores. This we will call "Definition 2". Unfortunately, these two definitions can lead to dramatically different results, especially when there is relatively little data. Moreover, neither of these definitions is explicit about how to handle rounding. For instance, what rank is required to be higher than 65% of the scores when the total number of scores is 50? This is tricky because 65% of 50 is 32.5. How do we find the lowest number that is higher than 32.5 of the scores? A third way to compute percentiles (presented below) is a weighted average of the percentiles computed according to the first two definitions. This third definition handles rounding more gracefully than the other two and has the advantage that it allows the median to be defined conveniently as the 50th percentile.

Third Definition

Unless otherwise specified, when we refer to "percentile," we will be referring to this third definition of percentiles. Let's begin with an example. Consider the 25th percentile for the 8 numbers in Table 1. Notice the numbers are given ranks ranging from $\mathrm{1}$ for the lowest number to $\mathrm{8}$ for the highest number.

Table 1. Test Scores.

Number	Rank
3 5 7 8 9 11 13 15	1 2 3 4 5 6 7 8

Number

Rank

The first step is to compute the rank ( $\mathrm{R}$ ) of the 25th percentile. This is done using the following formula:

$\mathrm{R}=\mathrm{P} / 100 \mathrm{x}(\mathrm{N}+1)$

where $\mathrm{P}$ is the desired percentile ( $\mathrm{25}$ in this case) and $\mathrm{N}$ is the number of numbers ( $\mathrm{8}$ in this case). Therefore,

$\mathrm{R=25 / 100} \times \mathrm{(8+1)=9 / 4=2.25}$

If $\mathrm{R}$ is an integer, the $\mathrm{Pth}$ percentile is the number with rank $\mathrm{R}$ . When $\mathrm{R}$ is not an integer, we compute the $\mathrm{Pth}$ percentile by interpolation as follows:

1. Define $\mathrm{IR}$ as the integer portion of $\mathrm{R}$ (the number to the left of the decimal point). For this example, $\mathrm{IR=2}$ .

2. Define $\mathrm{FR}$ as the fractional portion of $\mathrm{R}$ . For this example, $\mathrm{FR}=0.25$ .

3. Find the scores with Rank $\mathrm{IR}$ and with Rank $\mathrm{IR \, + \, 1}$ . For this example, this means the score with Rank 2 and the score with Rank 3. The scores are $\mathrm{5}$ and $\mathrm{7}$ .

4. Interpolate by multiplying the difference between the scores by $\mathrm{F}_{\mathrm{R}}$ and add the result to the lower score. For these data, this is $(0.25)(7-5)+5=5.5$ .

Therefore, the 25th percentile is $\mathrm{5.5}$ . If we had used the first definition (the smallest score greater than 25% of the scores), the 25th percentile would have been $\mathrm{7}$ . If we had used the second definition (the smallest score greater than or equal to 25% of the scores), the 25th percentile would have been $\mathrm{5}$ .

For a second example, consider the 20 quiz scores shown in Table 2.

Table 2. 20 Quiz Scores.

Score	Rank
4 4 5 5 5 5 6 6 6 7 7 7 8 8 9 9 9 10 10 10	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Score

Rank

We will compute the 25th and the 85th percentiles. For the 25th,

$\mathrm{R}=25 / 100 \mathrm{x}(20+1)=21 / 4=5.25$

$\mathrm{IR}=5$ and $\mathrm{FR}=0.25$

Since the score with a rank of

$\mathrm{IR}$ (which is

$\mathrm{5}$ ) and the score with a rank of

$\mathrm{IR+1}$ (which is

$\mathrm{6}$ are both equal to

$\mathrm{5}$ , the 25th percentile is 5. In terms of the formula:

$\mathrm{25th \, percentile =(.25) x (5-5)+5=5 }$

For the 85th percentile,

$\mathrm{R}=85 / 100 \times(20+1)=17.85 .$

$\mathrm{IR}=17$ and $\mathrm{FR}=0.85$

Caution: $\mathrm{FR}$ does not generally equal the percentile to be computed as it does here.

The score with a rank of $\mathrm{17}$ is $\mathrm{9}$ and the score with a rank of $\mathrm{18}$ is $\mathrm{10}$ . Therefore, the 85th percentile is:

$\mathrm{(0.85)(10-9) \, + \, 9 \, = \, 9.85}$

Consider the 50th percentile of the numbers $\mathrm{2, \, 3, \, 5, \, 9}$ .

$\mathrm{R}=50 / 100 \times(4+1)=2.5$

$\mathrm{IR}=2$ and $\mathrm{FR}=0.5$

The score with a rank of IR is $\mathrm{3}$ and the score with a rank of $\mathrm{IR + 1}$ is $\mathrm{5}$ . Therefore, the 50th percentile is:

$\mathrm{(0.5)(5 - 3) + 3 = 4}$ .

Finally, consider the 50th percentile of the numbers $\mathrm{2, \, 3, \, 5, \, 9, \, 11}$ .

$\mathrm{R}=50 / 100 \times(5+1)=3$

$\mathrm{IR}=3$ and $\mathrm{FR}=0$

Whenever $\mathrm{FR \, = \, 0}$ , you simply find the number with rank $\mathrm{IR}$ . In this case, the third number is equal to $\mathrm{5}$ , so the 50th percentile is $\mathrm{5}$ . You will also get the right answer if you apply the general formula:

$\mathrm{50th \, percentile \, = \, (0.00) \, (9 \, - \, 5) \, + \, 5 \, = \, 5}$ .

Questions

Question 1 out of 6.

For the scores 3, 5, 7, 9, 12, 21, 25, 30, calculate the 25th percentile based on "Definition 1".

_______

Question 2 out of 6.

For the scores 3, 5, 7, 9, 12, 21, 25, 30, calculate the 25th percentile based on "Definition 2".

_______

Question 3 out of 6.

For the scores 3, 5, 7, 9, 12, 21, 25, 30, calculate the 25th percentile based on "Definition 3".

_______

Question 4 out of 6.

For the scores 3, 5, 7, 9, 12, 21, 25, 30, calculate the 80th percentile based on "Definition 3".

_______

Question 5 out of 6.

What is the 75th percentile of $\mathrm{Y}$ based on "Definition 3"?

$\mathrm{Y}$

9.99
6.43
8.04
10.18
12.37
11.58
7.04
12.23
11.65
11.32
14.16
7.85
8.41
13.28
12.36
15.64
8.85
7.82
8.17
9.25
9.61
8.63
10.81
5.40
8.96
10.94
8.94
6.52
7.74
10.97
7.53
9.17
10.19
10.73
10.17
11.19
8.97
7.87
11.75
5.00
10.71
9.00
10.94

Question 6 out of 6.

What is the 25th percentile of $\mathrm{Y}$ based on "Definition 3"?

$\mathrm{Y}$

9.05
9.38
9.81
11.16
9.46
10.00
10.85
9.86
9.29
11.04
10.21
8.83
11.38
14.33
9.05
7.88
10.66
10.90
10.74
9.78
11.79
11.41
12.55
13.59
9.99
12.63
12.64
7.63
9.79
7.00
7.91
9.85
13.31
8.99
10.13
12.57
12.91
6.59
12.39
9.50
8.72
8.14
13.79

Answers

1. According to Definition 1, the 25th percentile is the lowest score higher than 25% of the scores. Since there are $\mathrm{8}$ scores, this would be the lowest score higher than $(0.25 \times 8 = 2 \, \mathrm{scores}$ . The score $\mathrm{7}$ is higher than the scores $\mathrm{3}$ and $\mathrm{5}$ .