What is a Percentile Rank?
A Percentile Rank is the percentage of values in a data set that falls below a given point within a data set when sorted in ascending order.
Percentile Rank vs Percentile
While Percentile Rank and Percentile are related terms, they are not one in the same.
In the case of percentiles, a percentage is given, and a corresponding data value is calculated. Conversely, in the case of percentile ranks, a data value is given, and a percentage is calculated.
To illustrate the difference, suppose a teacher gave test to 10 students that resulted in the following test scores:
{16, 18, 20, 22, 18, 12, 19, 17, 23, 20}
As the teacher, you may want to generate a grade curve based on percentiles (90% = A, 80% = B, etc.), and then use percentile ranks to find the grade of each student based on their scores (score of 23 has a percentile rank of 95% so that student would get an A, score of 22 has a percentile rank of 85% so that student would get a B, etc.).
How to Calculate a Percentile Rank
Since there are several methods for calculating percentile rank, I will explain how to calculate percentile rank using three of the more common formulas.
Percentile Rank Example
Suppose you among a class of 20 students who take a test, which yields the following test scores:
{45, 65, 74, 82, 64, 91, 78, 87, 85, 79, 94, 59, 83, 86, 69, 84, 89, 78, 81, 77}Further, suppose you want to see how your score of 84 compares to the scores achieved by your other 19 classmates.
Below, I have attempted to show my work for all three percentile rank methods used. Note that all inter-formula line results are rounded to 4 decimal places, while the final formula lines are rounded to a maximum of 3 places.
Method #1: For this method, we will need to solve the following percentile formula:
PR = (CFI / N) * 100,
Where CFI is equal to the number of values in the set that are less than or equal to the data value of interest and N is the total number of values in the set.From the chart created earlier, we can see that CFI = 14 and N = 20. Substituting those values for the variables in the first percentile formula gives us the following:
PR = (14 / 20) * 100
PR = 0.7 * 100
PR = 70
So based on Method #1, 70% of the values in the entered data set are less than or equal to 84.
Method #2: For this method, we will need to solve the following percentile formula:
PR = ((CF - (0.5 * F)) / N) * 100,
Where CF is equal to the number of values in the set that are less than or equal to the data value of interest, F is the number of times the value of interest occurs in the data set, and N is the total number of values in the set.From the chart created earlier we can see that CF = 14, F = 1, and N = 20. Substituting those values for the variables in the percentile formula gives us the following:
PR = ((14 - (0.5 * 1)) / 20) * 100
PR = ((14 - 0.5) / 20) * 100
PR = (13.5 / 20) * 100
PR = 0.675 * 100
PR = 67.5
So based on Method #2, 67.5% of the values in the entered data set are less than 84.
Method #3: For this method, we will need to solve the following percentile formula:
PR = (CFE / (CFE + CFG)) * 100,
Where CFE is equal to the number of values in the set that are less than the data value of interest, CFG is the number of values in the set that are greater than the data value of interest. Also, if the value of interest is not found within the data set, this method interpolates a percent rank between its nearest neighbors.From the chart created earlier, we can see that CFE = 13 and CFG = 6. Substituting those values for the variables in the percentile formula gives us the following:
PR = ((13 / (13 + 6)) * 100
PR = (13 / 19) * 100
PR = 0.6842 * 100
PR = 68.421
So based on Method #3, 68.421% of the values in the entered data set are less than 84.