Percentile Formula: A Complete Guide for Calculation and Application
The percentile formula helps determine an individual's performance relative to others. A percentile indicates the percentage of scores that fall below a specific value. This formula is commonly used to evaluate how a student performs on a test in comparison to other candidates. Essentially, the percentile shows where a person stands in relation to others. The percentile of a given value 'x' is calculated by dividing the number of values below 'x' by the total number of values in the dataset. Percentile calculations can be applied to various aspects, such as weight, income, and other metrics.
What is Percentile?
The percentile of a data value in a dataset is a statistical measure that indicates the percentage of data points that fall below a specific value. For instance, in a group of 20 children, if Ben is the 4th tallest and 80% of the children are shorter than him, then Ben is at the 80th percentile. This concept is commonly used in competitive exams like the SAT, LSAT, and others.
Formula for Percentile
The Percentile Formula can be expressed in the following ways:
1. First Formula:
$\text{Percentile} = \left( \dfrac{\text{Number of Values Below "x"}}{\text{Total Number of Values}} \right) \times 100$
This Percentile Calculation Formula based on how many values fall below the given value "x" in a dataset, expressed as a percentage of the total number of values.
2. Second Formula:
$P = \left( \dfrac{n}{N} \right) \times 100$
Where:
P = Percentile
n = The number of values below the specified value
N = Total number of values in the dataset
This Formula Of Percentile helps calculate the percentile rank by dividing the number of values below the given value by the total values and then multiplying by 100.
3. Third Formula:
$P = \left( \dfrac{\text{nth Percentile}}{100} \right) \times \text{Total Number of Values in the List}$
This formula is used to find the exact position of the nth percentile in a dataset. The nth percentile is divided by 100 and then multiplied by the total number of values in the list to determine the corresponding position.
Note: Also you can checkout Percentage formula.
Simple Steps to Calculate the Kth Percentile
The kth percentile is a number that separates your data into two parts:
The lower part contains k percent of the data.
The upper part contains the rest of the data.
Here’s how you can calculate it:
Sort the Data: Arrange all the numbers in your dataset in ascending order (smallest to largest).
Count the Numbers: Find out how many numbers are in your dataset. Call this n.
Calculate k/100: Divide your desired percentile (k) by 100. Remember, k is a number between 0 and 100.
Find the Index: Multiply the result from Step 3 by n. This gives you an index.
Check the Index
If the index is a whole number, go to Step 6.
If the index is not a whole number, round it to the nearest whole number and go to Step 7.
For Whole Number Index
Count the numbers in your dataset from left to right until you reach the index.
Take the average of this number and the next one. This average is the kth percentile.
For Rounded Index
Count the numbers in your dataset from left to right until you reach the rounded index.
The number you land on is the kth percentile.
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 the number is not a whole number, round it to the nearest whole number.
So, 9.6 becomes 10.
Now, count the numbers from left to right in the data set until you reach the 10th number.
The 10th number in the data set is 85.
Thus, the 80th percentile of the 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,
\[\text{Percentile (P)} = \begin{pmatrix} \dfrac{\text{Number of Values Below “x”}}{\text{Total Number of Values}} \end{pmatrix} \times 100 = \dfrac{n}{N} \times 100\]
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.
FAQs on Percentile Formula with Solved Examples
1. What is the Percentile Rank Formula?
Percentile Rank is a term used in statistics that comes from Percentiles. A Percentile is the percentage of scores lower than or equal to a specific value in a given set. Percentiles break a distribution into 100 equal parts. This is often used to understand scores from different tests. The formula for Percentile Rank (PR) is based on the total number of ranks and how many are below or above the percentile.
The Percentile Rank formula is:
Percentile Rank \[(PR) = \frac{(M + (0.5 \times R))}{Y}\]
Where:
M = Number of Ranks below the score
R = Number of Ranks equal to the score
Y = Total Number of Ranks
2. What is the Percentile Range?
The Percentile Range represents the gap between two designated percentiles. While you can select any pair, the 10th and 90th percentiles are the most frequently utilized. To determine the 10-90 percentile range:
Calculate the 10th percentile with the percentile formula.
Calculate the 90th percentile using the same formula.
Subtract the 10th percentile value from the 90th percentile value.
3. Why is the Percentile Calculation Formula Used?
The Percentile Calculation Formula is a simple way to find the score corresponding to a particular rank. It helps you determine the position of a score relative to others, making it easy to understand where a score stands.
4. What is the Difference Between Percentile and Percentile Rank?
Percentile and Percentile Rank are closely related but have different meanings. Percentile refers to the value below which a given percentage of scores fall. For example, the 90th percentile is the score below which 90% of the scores lie. On the other hand, Percentile Rank tells you the percentage of scores that are below or equal to a specific score. It indicates the relative position of a score within a set of data. While Percentile shows the score itself, Percentile Rank tells you how a score compares to others in a distribution.
5. What is Percentile Used For?
Percentiles help compare values with others. They give you an idea of how a value stands relative to other values. Percentiles are also useful for solving mathematical problems by showing how different methods can be applied to equations.
6. How is percentile calculated?
Percentile Formula For Grouped Data is calculated by using the following formula:
$\text{Percentile} = \left( \frac{\text{Number of Values Below "x"}}{\text{Total Number of Values}} \right) \times 100$
Where:
x is the data point of interest.
Number of Values Below "x" is the number of data points smaller than the given value.
Total Number of Values is the total number of data points in the dataset.
This formula tells you what percentage of the data falls below the given value.
7. What is the 95% percentile value?
The 95% percentile value is the data point below which 95% of the data in a dataset falls. This means that 95% of the data points are smaller than the 95th percentile value, and only 5% of the data points are greater. In practical terms, if you score in the 95th percentile of a test, you have performed better than 95% of the other test-takers.
8. What is a 75% percentile?
The 75% percentile, also known as the third quartile (Q3), is the value below which 75% of the data points in a dataset fall. It represents the point where 75% of the data is less than this value, and 25% of the data is greater than it. In a ranked dataset, the 75th percentile is the data point that divides the lowest 75% from the top 25%.
9. How do you find the 75th percentile score?
To find the 75th percentile score (also known as Q3), follow these steps:
Arrange the data in ascending order.
Calculate the position of the 75th percentile using the formula:
$P = \dfrac{75}{100} \times (N + 1)$
Where:
P = Position of the 75th percentile in the ordered list.
N = Total number of data points.
Locate the value at this position in the ordered list. If the position is not an integer, average the values at the two closest positions.
For example, in a dataset of 10 values, the position of the 75th percentile would be:
$P = \dfrac{75}{100} \times (10 + 1) = 8.25$
This means the 75th percentile score is between the 8th and 9th values in the ordered list.
10. What is the 50th percentile?
The 50th percentile, commonly referred to as the median, is the value that separates a dataset into two equal portions. Half of the data lies below this point, while the other half is above it. Essentially, it represents the central value in a sorted dataset.
11. What is the Percentile Deviation Formula?
The Percentile Deviation Formula is used to measure the deviation or difference between a particular value and its percentile in a dataset. While there isn't a universally fixed formula for percentile deviation, it is generally calculated by finding the difference between the given data point and its corresponding percentile rank, giving an idea of how much the value deviates from the expected percentile range.
