How to Calculate Percentiles

You do know what a percentage is, right? A percentage is an expression that is used to define a number in terms of a fraction of 100. For example, if we say 80% of the students are present in a class, this means that if the class has 100 students, 80 of them are present. The terms 'percentage' and 'percentile' have one thing in common... the term 'percent'. 'Per cent' is a Latin term which means 'per hundred'. We have already discussed what a percentage is, but what exactly is a percentile? A percentile, also known as a centile, is a number or measure that expresses the number of frequencies that are below that particular measure. I am not surprised if you are confused, but here is an example that will clear the confusion. Suppose you scored 99 percentile in an exam and there are about 100 people who gave that exam. Here, 100 are the total number of frequencies or observations. A percentile expresses the number of frequencies that are below your measure. If your measure is 99, it means that there are 98 students that you have outnumbered in the exam. Did you understand the difference between a percentage and percentile? If you have scored 99 percentile, it doesn't mean that you have scored 99 percent, it means that you have outnumbered 98 others who are below you, and only one student ahead of you. Keep reading to understand more on how to calculate percentiles and help yourself understand the concept in a better way.

How to go About Calculating Percentiles?

Calculating percentiles is definitely different from calculating percentages. Percentages have a standard definition which is accepted universally. However, this is not the case with percentiles. A percentile has no standard definition, which is why various institutions use different approaches in calculating percentiles, though the end result is more or less the same. The only difference is on how they consider the end result when it is not a whole number, as in when the end result is a fraction. This will be cleared when this article proceeds further.

Formula to Calculate Percentiles

As mentioned earlier, there are various different approaches to calculate a percentile. This approach is the simplest one which will give you the basic idea of how you need to calculate your percentile rank or your percentile score.

p_i = 100 (i - 0.5) / n

Let us take an example so that you understand which value refers to which variable in the above formula. Say for instance, we have a set of values or observations and we need to calculate the percentiles of the given observations. These values are:

1

3

5

7

9

i would be the sorted order of this data. So, in this case the value of i would be,

xi	1	3	5	7	9
i	1	2	3	4	5

n is the total number of values or observations given to you, which in this case is 5. So, now if we need to calculate the percentiles, the approach to the same would be, p_i = 100 (i - 0.5) / n.

• p₁ = 100 (1 - 0.5) / 5 = 10
• p₂ = 100 (2 - 0.5) / 5 = 30
• p₃ = 100 (3 - 0.5) / 5 = 50
• p₄ = 100 (4 - 0.5) / 5 = 70
• p₅ = 100 (5 - 0.5) / 5 = 90

Now let us combine these results that we have in a table format, so that you understand it in a better way. If you look at the table below, we have x_i, we have i and we have the p_i.

xi	1	3	5	7	9
i	1	2	3	4	5
pi	10	30	50	70	90

Now if I ask you to look at the aforementioned table and tell me which is the 50th percentile, what would you answer? Yes, 50 is the percentile of the measurement (x) 5. So, the 50th percentile is 5. Similarly, if you need to calculate the 20th percentile, then applying the formula, the calculation would be something like,

p_i = 100 (i - 0.5) / n

Here, we have p_i = 20 and n = 5, so the expression can be represented as follows,

20 = 100 (x - 0.5) / 5 = 1.5

So, this expression gives that the 20th percentile (what we started with) corresponds to x = 1.5, which we calculated.

There are many other approaches and complex strategies when it comes to calculating percentiles. However, this was just a basic introduction for you to have an idea of what exactly are percentiles, and what you should do to calculate them. This basic approach would definitely help you in determining the percentile positions, especially in case of competitive exams. I hope this article proved to be of help. Happy calculating... :-)