Percentiles: what are they and how are they calculated?

Measures of position or non-central tendency help to know the position of a series of values that do not have a central location within the distribution. The objective of these measures is to divide a set of data or observations into equal parts. Among these measures we can find quartiles, deciles and percentiles or centiles. In this article we will only focus on percentiles.

In statistics, percentile is understood as the non-central position measure that divides an ordered distribution of data into one hundred equal parts. In the words of Salazar and Del Castillo (2018) “they are certain numbers that divide the sequence of ordered data into one hundred equal percentage parts.” Once these have been organized from lowest to highest, the percentile will indicate the value below which a certain percentage of observations can be found. Let’s look at some examples.

A 4-year-old girl, with a height of 105 cm, is in the 80th percentile of height for girls of that age. This means that 80% of girls of that age are below that height. Likewise, we can affirm that this girl’s height is surpassed by 20% of girls her age. A university student obtained a grade of 9 on a Biology exam and was in the 90th percentile, which means that 90% of the students have a score less than 9 and that 10% of the students are above it. The percentiles They mainly indicate the percentage that is below a certain value, but they also inform us of those that are above. In this way, the 3rd percentile indicates that it exceeds 3% of the values ​​and is exceeded by the remaining 97%. The 50th percentile is that value of the variable that exceeds 50% of the observations and is exceeded by the other 50% of the observations. The 99th percentile exceeds 99% of the data and is in turn exceeded by the remaining 1%.

What are percentiles used for?

Percentiles are used to locate the data of a sample. They allow you to position a series of data ordered in such a way that you can determine the percentage that is below and above a certain position. Knowing the location of a piece of data in relation to others can be valuable information.

Thus, for example, knowing that 70% of the observations are located below the 70th percentile tells us that the distribution of the data, for the most part, is below said percentile.

The percentile also allows you to manipulate a series of data in an easy and understandable way and assign a position to them.. On a practical level, it can help check if the element being analyzed is within normal ranges or if it is below or above the average in a variable.

For example, when psychometric tests or neuropsychological batteries are applied, percentiles can determine whether the results obtained are within or outside the normal range.

How to calculate percentiles?

To calculate a percentile we must take into account the following steps:

Sort the data from lowest to highest. Calculate the position of the percentile by applying the following formula:

i= k(n+1)/100

Where:

i= Position within the ordered sample that represents the percentile.

K= percentile number to find.

n= Sample size.

Yes i is an integer, then Pk = Yi (value of where it is located Yo).

Yes i is not an integer, then Pk = (1-d) Yi + (d) Yi+1 (Yi+1 represents the value that is in the position next to Yi).

d= decimal values.

Let’s look at everything in an example: we want to know the 25th percentile (P25) of ages for a group of male and female employees who are part of a pharmaceutical industry.

EmployeesAgesSample sizeMen18, 19, 20, 22, 24, 26, 29, 31, 35, 45, 6111Women19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 3514

To find the 25th percentile of the group of men in the company, we substitute the values ​​into the formula:

i= k(n+1)/100

i= 25(11+1)/100=3

As it is an integer, the age value found in position number 3 is taken as the value of the 25th percentile, in this case the age of the group of men that occupies position 3 is 20 years. This means that 25% of the sample of men is below the age of 20.

For the group of women in the company, we also apply the same procedure.

i=k(n+1)/100

i= 25(14+1)/100= 3.75

However, since it has decimals, we cannot take the position 3.75, so we must interpolate the results, taking i= 3 and d= 0.75. Replacing the data in the formula would look like this:

Pk = (1-d) Yi + (d) Yi+1

P25 = (1-0.75)21 + (0.75) 22

P25=21.75

So the 25th percentile for the female group is 21.75 years. This means that 25% of the sample of women is below the age of 21.75 years. Likewise, we can affirm that 75% of women are above the age of 21.75 years.

Like any measure of non-central position, percentiles allow us to know the location of certain values ​​throughout a set of ordered data and provides information on the percentage of data that can be found below a certain positionallowing us to directly infer those that are above it.

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