Percentile Formula: Meaning, Calculation Procedure and Solved Examples

Percentile Formula Overview

The percentile formula determines the performance of a person over others. A percentile is a number that tells the percentage of scores that fall below the given number. Let's discuss this percentile formula in detail and solve a few examples. The percentile is used to show where a student stands on a test compared to other candidates.

What is Percentile?

A percentile can generally be defined as a numeric value representing the position below which a certain percentage of scores in a dataset lie. It is important to distinguish between percentiles and percentages, as they are separate concepts. Percentages are used to express fractions of a whole, whereas percentiles indicate the values below which a specific percentage of data points can be found within a dataset. If you wish to understand your relative position compared to others, you require a statistical measure known as a percentile.

For instance, suppose you are the fourth tallest person among a group of individuals, with 80% of people being shorter than you. This implies that you are in the 80th percentile.

If your height is 5.4 inches, then "5.4 inches" represents the height at the 80th percentile within that group.

What is Percentile Formula?

The percentile formula is employed when comparing specific values or numbers to other numbers within a given dataset, measuring their precision. Percentile and percentage are often conflated, despite being distinct concepts. A percentage represents a fraction as a single term, whereas a value's percentile is the percentage of values that fall below the given value out of the entire dataset. The following example illustrates the meaning and disparity between percentiles and percentages.

Consider an exam conducted with a total of 100 marks:

• A student is said to have scored 100 "percent" if their score is 100 out of 100.
• A student is said to have scored in the 100 "percentile" if all the students (100% of students) scored lower than them.

Percentile Formula

In mathematics, the term "percentile" is used to analyze and interpret data, providing insights into values such as test scores, health indicators, and various measurements. Informally, it signifies the percentage of data points that fall below a specific percentile value. For example, if we score in the 25th percentile, it means that 25% of test-takers have scores lower than ours. This 25 is referred to as the percentile rank.

Percentiles are instrumental in our daily lives, aiding in the understanding of data. They divide a dataset into 100 equal parts, representing the percentage of the total frequency of the dataset that is at or below a given measure. Let's consider a student's percentile in some exams as an example.

Suppose a student scores in the 60th percentile on the quantitative section of a test. This indicates that the student has performed as well as or better than 60% of the other students. If a total of 500 students took the test, it means that the student has outperformed 300 out of 500 students (500 * 0.60 = 300). In other words, 200 students have scored higher than this particular student.

Therefore, percentiles are utilized to understand and interpret data, specifically in the context of test scores and biometric measurements. They indicate the values below which a certain percentage of the data in a dataset can be found.

To calculate a percentile, one can use the following formulas:

Another formula to find the percentile is:

Where,

• n represents the ordinal rank of the given value or the value below it
• N is the total number of values in the dataset
• P denotes the percentile 
• Rank is the ratio of the percentile to 100, and the ordinal rank for the percentile value is given by Rank multiplied by the total number of values in the list.

Percentile Formula Calculation

To calculate the percentile using the percentile formula, follow these steps:

• Step 1: Gather the dataset. 
• Step 2: Arrange the dataset in ascending order. 
• Step 3: Determine the total number of observations in the dataset. 
• Step 4: Identify the specific data value for which you want to find the percentile. 
• Step 5: Count the number of data values that are smaller than the identified value. 
• Step 6: Divide the count from Step 5 by the total number of observations from Step 3. This will give you the percentile of the given data value.

In essence, if q represents any number between zero and one hundred, the qth percentile is the value that divides the dataset into two parts: the lower part containing q percent of the data and the upper part containing the remaining data. By following these steps, you can apply the percentile formula to calculate the desired percentile of a data value.

Procedure to Calculate K^th Percentile 

The kth percentile represents a value within a dataset that divides the data into two sections: the lower part containing k percent of the data and the upper part containing the remaining data.

Here 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 dataset in ascending order.
• Step 2: Count the number of values in the dataset and denote it as 'n'.
• Step 3: Calculate the value of k/100, where k represents the desired percentile.
• Step 4: Multiply the 'k' percent by 'n'. The resulting number is called the index.
• Step 5: If the index is not a whole number, round it to the nearest whole number, and proceed to Step 7. If the index is a whole number, proceed to Step 6.
• Step 6: Count the values in your dataset from left to right until you reach the corresponding index. Find the mean (average) of that value and the next value. The resulting value represents the kth percentile of your dataset.
• Step 7: Count the values in your dataset from left to right until you reach the corresponding index. The obtained value will be the kth percentile of your dataset.

By following these steps, you can determine the kth percentile of a given dataset, where k is any number between zero and one hundred.

Percentile Formula: Solved Example

Example 1: The scores obtained by 10 students are 38, 47, 49, 58, 60, 65, 70, 79, 80, 92. Using the percentile formula, calculate the percentile for score 70?

Solution:

Given:

Scores obtained by students are 38, 47, 49, 58, 60, 65, 70, 79, 80, 92

Number of scores below 70 = 6

Using the percentile formula,

Percentile = (Number of Values Below “x” / Total Number of Values) × 100

Percentile of 70

= (6/10) × 100

= 0.6 × 100 = 60

Therefore, the percentile for score 70 = 60%

Example 2: The weights of 10 people were recorded in kg as 35, 41, 42, 56, 58, 62, 70, 71, 90, 77. How to find percentile for the weight 58 kg?

Solution:

Given:

Weight of the people are 35, 41, 42, 56, 58, 62, 70, 71, 77, 90

Number of people with weight below 58 kg = 4

Using the formula for percentile,

Percentile = (Number of Values Below “x” / Total Number of Values) × 100

Percentile for weight 58 kg

= (4/10) × 100

= 0.4 × 100 = 40%

Therefore, the percentile for weight 58 kg = 40%

Example 3: In a college, a list of scores of 10 students is announced. The scores are 56, 45, 69, 78, 72, 94, 82, 80, 63, 59. Using the percentile formula, find the 70th percentile.

Solution: Arrange the data in ascending order - 45, 56, 59, 63, 69, 72, 78, 80, 82, 94

Find the rank,

Rank = Percentile ÷ 100

Rank = 70 ÷ 100 = 0.7

So, the rank is 0.7

Using the formula to calculate the percentile,

Percentile = Rank × Total number of the data set

Percentile = 0.7 × 10

Percentile = 7

Now, counting 7 values from left to right we reach 80, and we can say that all the values below 80 will come under the 70th percentile. In other words, 70% of the values are below 80.

Therefore, the 70th percentile is 80.