Percentile Formula

Mostly we can define percentile as a number where a certain percentage of scores fall below that given number. Percentile and percentage are often confused, but both are different concepts. The percentage is used to express fractions of a whole, while percentiles are the values below which a certain percentage of the data in a data set is found. If you want to know where you stand compared to the rest of the crowd, you need a statistic that reports relative standing, and that statistic is called a percentile.

For example, you are the fourth tallest person in a group of 20.80% of people who are shorter than you. That means you are at the 80th percentile.

If your height is 5.4inch then "5.4 inch" is the 80th percentile height in that group.

Percentile Formula is given as -

Where, n =ordinal rank of a given value 

 N = number of values in the data set,

 P = Percentile

Formula for Percentile

In mathematics, we use the term percentile to understand and interpret data. It is also used informally to indicate that a certain percentage falls below that percentile value. Percentiles play a vital role in everyday life in understanding values such as test scores, health indicators, and other measurements. For example, if we score in the 25th percentile, then 25% of test-takers are below this score. Here 25 is called the percentile rank. Percentile divides a data set into 100 equal parts. A percentile is a measurement that tells us what percent of the total frequency of a data set was at or below that measure. As an example, let us consider a student’s percentile in some exams.

If on this test, a given student scored in the 60th percentile on the quantitative section, she scored at or better than 60% of the other students. Further, if a total of 500 students took the test, then this student scored at or better than 500 × 0.60 = 300 students out of 500 students. In other words, 200 students scored better than that particular student.

Therefore, percentiles are used to understand and interpret the given data. They also indicate the values below which a certain percentage of the data in a data set is found. Percentiles are frequently useful to understand the test scores and biometric measurements in our day-to-day routine.

Percentile is calculated by the ratio of the number of values below ‘x’ to the total number of values.

The Percentile Formula is given as,

Procedure to Calculate Kth Percentile 

The kth percentile is a value in a data set that divides the data into two parts. The lower part contains k percent of the data, and the upper part contains the rest of the data.

The following are the steps to calculate the kth percentile (where k is any number between zero and one hundred).

Step 1: Arrange all data values in the data set in ascending order.

Step 2: Count the number of values in the data set where it is represented as 'n'.

Step 3: calculate the value of k/100, where k = any number between zero and one hundred.

Step 4: Multiply 'k' percent by 'n'.The resultant number is called an index.

Step 5: If the resultant index is not a whole number then round to the nearest whole number, then go to Step 7. If the index is a whole number, then go to Step 6. 

Step 6: Count the values in your data set from left to right until you reach the number. Then find the mean for that corresponding number and the next number. The resultant value is the kth percentile of your data set. 

Step 7: Count the values in your data set from left to right until you reach the number. The obtained value will be the kth percentile of your data set.

Solved Examples

Percentile formula example.

Example 1: Let us consider the percentile example problem: In a college, a list of grades of 15 students has been declared. Their grades are given as: 85, 34, 42, 51, 84, 86, 78, 85, 87, 69, 74, 65. Find the 80th percentile?

Solution:

Step 1:

Arrange the data in ascending order. 

Let us arrange in Ascending Order = 34, 42, 51, 65, 69, 74, 78, 84, 85, 85, 86, 87.

Step 2:

Find Rank,

\[\text{Rank} = \frac{Percentile}{100}\] 

\[= \frac{80}{100}\]

k = 0.80

Step 3:

Find 80th percentile,

80th percentile = 0.80 \[\times\] 12 

= 9.6

Step 4:

Since it is not a whole number, round to the nearest whole number. 

Therefore, 9.6 is rounded to 10. 

Now, Count the values from left to right in the given data set until you reach the number 10.

From the given data set, the 10th number is 85.

Therefore, the 80th percentile of the given data set is 85

Example 2: The scores for students are 40, 45, 49, 53, 61, 65, 71, 79, 85, 91. What is the percentile for score 71?

Solution: We have,

No. of. scores below 71 = 6 Total no. of. scores = 10 

The formula for percentile is given as,

\[\text{Percentile (P)} = \begin{pmatrix} \frac{\text{Number of Values Below “x”}}{\text{Total Number of Values}} \end{pmatrix} \times 100  = \frac{n}{N} \times 100\]

Percentile of 71= \[\frac{6}{10} \times 100\]

= 0.6 × 100 = 60

Example 3: If in the test, a student scored 60th percentile on the quantitative section and if 500 students took the test, then this student scores at or better than will be?

Solution: 500 \[\times\] 0.60=300

Thus Percentile also indicates the values below which a certain percentage of the data in a data set is found.