# Introduction to Statistics (Online Edition)/Introduction/Percentiles

​Percentiles by David Lane
Prerequisites
Learning Objectives 1. Define percentiles 2. 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.
Two Simple Definitions of Percentile
​A 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 1 for the lowest number to 8 for the highest number.
Table 1. Test Scores.
The first step is to compute the rank (R) of the 25th percentile. This is done using the following formula:
$R={\frac {P}{100}}\times (N+1)$
where P is the desired percentile (25 in this case) and N is the number of numbers (8 in this case). Therefore,
$R={\frac {25}{100}}\times (8+1)={\frac {9}{4}}=2.25$
If R is an integer, the Pth percentile is be the number with rank R. When R is not an integer, we compute the Pth percentile by interpolation as follows:
1. Define IR as the integer portion of R (the number to the left of the decimal point). For this example, IR=2.
​3. Find the scores with Rank I$_R$ and with Rank I$_R$ + 1. For this example, this means the score with Rank 2 and the score with Rank 3. The scores are 5 and 7.
4. Interpolate by multiplying the difference between the scores by F$_R$ 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 5.5. If we had used the first definition (the smallest score greater than 25% of the scores), the 25th percentile would have been 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 5.
For a second example, consider the 20 quiz scores shown in Table 2.
Table 2. 20 Quiz Scores.
We will compute the 25th and the 85th percentiles. For the 25th,
$R={\frac {25}{100}}\times (20+1)={\frac {21}{4}}=5.25$
​$IR=5$ $and$ $FR=0.25.$
Since the score with a rank of IR (which is 5) and the score with a rank of IR + 1 (which are both equal to 5, the 25th percentile is 5. In terms of the formula:
$25th$ $percentile=(.25)\times (5-5)+5=5.$
For the 85th percentile,
$R={\frac {85}{100}}\times (20+1)=17.85$
$IR=17$ $and$ $FR=0.85$
       Caution: FR does not generally equal the percentage to be computed as it does here.
The score with a rank of 17 is 9 and the score with a rank of 18 is 10. Therefore, the 85th percentile is:
$(0.85)(10-9)+9=9.85$
Consider the 50th percentile of the numbers 2, 3, 5, 9.
$R={\frac {50}{100}}\times (4+1)=2.5$
$IR=2$ $and$ $FR=0.5$
The score with a rank of IR is 3 and the score with a rank of IR + 1 is 5. Therefore, the 50th percentile is:
$(0.5)(5-3)+3=4$
Finally, consider the 50th percentile of the numbers 2, 3, 5, 9, 11.
$R={\frac {50}{100}}\times (5+1)=3$
$IR=3$
$and$
$FR=0.$
​Whenever FR = 0, you simply find the number with rank IR. In this case, the third number is equal to 5, so the 50th percentile is 5. You will also get the right answer if you apply the general formula:
$50th$ $percentile=(0.00)(9-5)+5=5.$
This work is in the public domain worldwide because it has been so released by the copyright holder.
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