A Formula for Finding the kth Percentile

Learning Outcomes

Given a data set, find the $k^{th}$ percentile

A Formula for Finding the $k$ th Percentile

If you were to do a little research, you would find several formulas for calculating the $k$ th percentile. Here is one of them.

$k$ = the $k$ th percentile. It may or may not be part of the data.

$i$ = the index (ranking or position of a data value)

$n$ = the total number of data

Order the data from smallest to largest
Calculate $\displaystyle{i}=\frac{{k}}{{100}}{({n}+{1})}$
If $i$ is an integer, then the $k$ th percentile is the data value in the $i$ th position in the ordered set of data.
If $i$ is not an integer, then round $i$ up and round $i$ down to the nearest integers. Average the two data values in these two positions in the ordered data set. This is easier to understand in an example.

Example

Listed are 29 ages for trees found in the Saint Louis Botanical Garden in order from smallest to largest.

$18$ ; $21$ ; $22$ ; $25$ ; $26$ ; $27$ ; $29$ ; $30$ ; $31$ ; $33$ ; $36$ ; $37$ ; $41$ ; $42$ ; $47$ ; $52$ ; $55$ ; $57$ ; $58$ ; $62$ ; $64$ ; $67$ ; $69$ ; $71$ ; $72$ ; $73$ ; $74$ ; $76$ ; $77$

Find the $70$ th percentile.
Find the $83$ rd percentile.

Show Solution

1. - $k$ = $70$
  - $i$ = the index
  - $n$ = $29$

$\displaystyle{i}=\frac{{k}}{{100}}{({n}+{1})}={(\frac{{70}}{{100}})}{({29}+{1})}={21}$ . Twenty-one is an integer, and the data value in the $21$ st position in the ordered data set is $64$ . The $70$ th percentile is 64 years.

1. - $k$ = $83$ rd percentile
  - $i$ = the index
  - $n$ = $29$

$\displaystyle{i}=\frac{{k}}{{100}}{({n}+{1})}={(\frac{{83}}{{100}})}{({29}+{1})}={24.9}$ , which is NOT an integer. Round it down to $24$ and up to $25$ . The age in the $24$ th position is $71$ and the age in the $25$ th position is $72$ . Average $71$ and $72$ . The $83$ rd percentile is $71.5$ years.

Try It

Listed are $29$ ages for Academy Award winning best actors in order from smallest to largest.

Calculate the $20$ th percentile and the $55$ th percentile.

Show Solution

$k$ = $20$ . Index = $\displaystyle{i}=\frac{{k}}{{100}}{({n}+{1})}=\frac{{20}}{{100}}{({29}+{1})}={6}$ The age in the sixth position is $27$ . The $20$ th percentile is $27$ years.

$k$ = $55$ . Index = $\displaystyle{i}=\frac{{k}}{{100}}{({n}+{1})}=\frac{{55}}{{100}}{({29}+{1})}={16.5}$ . Round down to $16$ and up to $17$ . The age in the $16$ th position is $52$ and the age in the $17$ th position is $55$ . The average of $52$ and $55$ is $53.5$ . The $55$ th percentile is $53.5$ years.

NOTE

You can calculate percentiles using calculators and computers. There are a variety of online calculators.

A Formula for Finding the Percentile of a Value in a Data Set

Order the data from smallest to largest
$x$ = the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile
$y$ = the number of data values equal to the data value for which you want to find the percentile
$n$ = the total number of data
Calculate $\displaystyle\frac{{{x}+{0.5}{y}}}{{n}}{({100})}$ . Then round to the nearest integer.

Example

Listed are $29$ ages for Academy Award winning best actors in order from smallest to largest.

Find the percentile for $58$ .
Find the percentile for $25$ .

Show Solution

a. Counting from the bottom of the list, there are $18$ data values less than $58$ . There is one value of $58$ .

$x=18\quad\text{and}\quad{y=1}$

$\dfrac{x+0.5y}{n}(100)=\dfrac{18+0.5(1)}{29}(100)=63.80$

$58$ is the $64$ th percentile.

b. Counting from the bottom of the list, there are three data values less than $25$ . There is one value of $25$ .

$x=3\quad\text{and}\quad{y=1}$

$\dfrac{x+0.5y}{n}(100)=\dfrac{3+0.5(1)}{29}(100)=12.07$

$25$ is the $12$ th percentile.

Try It

Listed are $30$ ages for New York Times published columnists in order from smallest to largest.

$18$ ; $21$ ; $22$ ; $25$ ; $26$ ; $27$ ; $29$ ; $30$ ; $31$ , $31$ ; $33$ ; $36$ ; $37$ ; $41$ ; $42$ ; $47$ ; $52$ ; $55$ ; $57$ ; $58$ ; $62$ ; $64$ ; $67$ ; $69$ ; $71$ ; $72$ ; $73$ ; $74$ ; $76$ ; $77$

Find the percentiles for $47$ and $31$ .

Show Solution

Percentile for $47$ : Counting from the bottom of the list, there are $15$ data values less than $47$ . There is one value of $47$ .

$x=15\quad\text{and}\quad{y=1}$

$\dfrac{x+0.5y}{n}(100)=\dfrac{15+0.5(1)}{30}(100)=51.67$

$47$ is the $52$ nd percentile.

Percentile for $31$ : Counting from the bottom of the list, there are eight data values less than $31$ . There are $two$ values of $31$ .

$x=8\quad\text{and}\quad{y=2}$

$\dfrac{x+0.5y}{n}(100)=\dfrac{8+0.5(2)}{30}(100)=30$

$31$ is the $30$ th percentile.

Module 2: Descriptive Statistics

Learning Outcomes