Percentile Formula

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Percentile Formula

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

Percentile Definition

Percentile is defined as the value below which a given percentage falls under. For example, in a group of 20 children, Ben is the 4th tallest and 80% of the children are shorter than you. Hence, it means that Ben is at the 80th percentile. It is most commonly used in competitive exams such as SAT, LSAT, etc.

GIA SƯ DẠY SAT

What is Percentile Formula?

The percentile formula is used when we need to compare the exact values or numbers over the other numbers from the given data i.e. the accuracy of the number. Often percentile and percentage are taken as one but both are different concepts. A percentage is where the fraction is considered as one term while percentile is the value below the percentage found from the given data. In our day-to-day life, percentile formulas are usually helpful in finding the test scores or biometric measurements. Hence, the percentile formula is:

Percentile = (n/N) × 100

Or

The percentile of x is the ratio of the number of values below x to the total number of values multiplied by 100. i.e., the percentile formula is

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

Percentile Formula

P = (n/N) × 100

Where,

  • n = ordinal rank of the given value or value below the number
  • N = number of values in the data set
  • P = percentile

Or

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

Steps of Percentile Formula

To find the percentile, here are a few steps to use the percentile formula. If q is any number between zero and hundred, the qth percentile is a value that divides the data into two parts i.e the lowest part contains the q percent of the data and the rest of the data is the upper part.

  • Step 1: Arrange the data set in ascending order
  • Step 2: Count the number of values in the data set and represent it as r
  • Step 3: Calculate the value of q/100
  • Step 4: Multiply q percent by r
  • Step 5: If the answer is not a whole number then rounding the number is required. If it is a whole number, continue to the next step
  • Step 6: Count the values in the data set, find the mean and the next number. The answer is the qth percentile
  • Step 7: Count the value in the data set, once you reach that number according to what we obtained in step 5 that is the qth percentile.

Examples Using Percentile Formula

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 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. Find the percentile for the weight 58 kg.

Solution:

Given:

Scores obtained by students are 35, 41, 42, 56, 58, 62, 70, 71, 77, 90

Number of people with weight below 58 kg = 4

Using percentile formula,

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

Rank = 0.07

Find the percentile using the formula,

Percentile = Rank × Total number of the data set

Percentile = 0.07 × 10

Percentile = 0.7

Since it not a whole number, round the number to the nearest whole number i.e. 0.7 = 1

Count the set to reach the above number. It is 1.

Therefore, the 70th percentile is 45

FAQs on Percentile Formula

What is the Percentile Formula?

The percentile formula determines the performance of a person over others. The percentile formula is used in finding where a student stands in the test compared to other candidates. In our day-to-day life, percentile formulas are usually helpful in finding the test scores or biometric measurements.

What is the Formula to Calculate the Percentile?

The percentile of x is the ratio of the number of values below x to the total number of values multiplied by 100. i.e., the percentile formula is:

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

What is the Formula to Calculate the Percentile By Rank?

The percentile formula where the rank of the number is used is:

P = (n/N) × 100

Where,

n = ordinal rank of the given value or value below the number
N = number of values in the data set
P = percentile

Find the Percentile for 20 from the given set 12, 55, 7, 10, 40.

Arrange it in ascending order – 7, 10, 12, 40, 55

Number of scores below 20 = 3

Using percentile formula,

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

Percentile of 20

= (3/10) × 100

= 0.3 × 100 = 30

Therefore, the percentile for score 20 = 30

Definition of a Percentile in Statistics and How to Calculate It

In statistics, percentiles are used to understand and interpret data. The nth percentile of a set of data is the value at which n percent of the data is below it. In everyday life, percentiles are used to understand values such as test scores, health indicators, and other measurements. For example, an 18-year-old male who is six and a half feet tall is in the 99th percentile for his height. This means that of all the 18-year-old males, 99 percent have a height that is equal to or less than six and a half feet. An 18-year-old male who is only five and a half feet tall, on the other hand, is in the 16th percentile for his height, meaning only 16 percent of males his age are the same height or shorter.

Key Facts: Percentiles

• Percentiles are used to understand and interpret data. They indicate the values below which a certain percentage of the data in a data set is found.

• Percentiles can be calculated using the formula n = (P/100) x N, where P = percentile, N = number of values in a data set (sorted from smallest to largest), and n = ordinal rank of a given value.

• Percentiles are frequently used to understand test scores and biometric measurements.

What Percentile Means

Percentiles should not be confused with percentages. The latter 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. In practical terms, there is a significant difference between the two. For example, a student taking a difficult exam might earn a score of 75 percent. This means that he correctly answered every three out of four questions. A student who scores in the 75th percentile, however, has obtained a different result. This percentile means that the student earned a higher score than 75 percent of the other students who took the exam. In other words, the percentage score reflects how well the student did on the exam itself; the percentile score reflects how well he did in comparison to other students.

Percentile Formula

Percentiles for the values in a given data set can be calculated using the formula:

n = (P/100) x N

where N = number of values in the data set, P = percentile, and n = ordinal rank of a given value (with the values in the data set sorted from smallest to largest). For example, take a class of 20 students that earned the following scores on their most recent test: 75, 77, 78, 78, 80, 81, 81, 82, 83, 84, 84, 84, 85, 87, 87, 88, 88, 88, 89, 90. These scores can be represented as a data set with 20 values: {75, 77, 78, 78, 80, 81, 81, 82, 83, 84, 84, 84, 85, 87, 87, 88, 88, 88, 89, 90}.

We can find the score that marks the 20th percentile by plugging in known values into the formula and solving for n:

n = (20/100) x 20

n = 4

The fourth value in the data set is the score 78. This means that 78 marks the 20th percentile; of the students in the class, 20 percent earned a score of 78 or lower.

Deciles and Common Percentiles

Given a data set that has been ordered in increasing magnitude, the median, first quartile, and third quartile can be used split the data into four pieces. The first quartile is the point at which one-fourth of the data lies below it. The median is located exactly in the middle of the data set, with half of all the data below it. The third quartile is the place where three-fourths of the data lies below it.

The median, first quartile, and third quartile can all be stated in terms of percentiles. Since half of the data is less than the median, and one-half is equal to 50 percent, the median marks the 50th percentile. One-fourth is equal to 25 percent, so the first quartile marks the 25th percentile. The third quartile marks the 75th percentile.

Besides quartiles, a fairly common way to arrange a set of data is by deciles. Each decile includes 10 percent of the data set. This means that the first decile is the 10th percentile, the second decile is the 20th percentile, etc. Deciles provide a way to split a data set into more pieces than quartiles without splitting the set into 100 pieces as with percentiles.

Applications of Percentiles

Percentile scores have a variety of uses. Anytime that a set of data needs to be broken into digestible chunks, percentiles are helpful. They are often used to interpret test scores—such as SAT scores—so that test-takers can compare their performance to that of other students. For example, a student might earn a score of 90 percent on an exam. That sounds pretty impressive; however, it becomes less so when a score of 90 percent corresponds to the 20th percentile, meaning only 20 percent of the class earned a score of 90 percent or lower.

Another example of percentiles is in children’s growth charts. In addition to giving a physical height or weight measurement, pediatricians typically state this information in terms of a percentile score. A percentile is used in order to compare the height or weight of a child to other children of the same age. This allows for an effective means of comparison so that parents can know if their child’s growth is typical or unusual.

How to Calculate Percentile Step by Step (and Examples)

When scoring tests or other sets of important values, the raw number is not necessarily enough to relate the scores to one another. You might score a 75 on a college exam, but that only tells you your score and not how it compares to the scores of your classmates. This is when percentiles are useful. To get a full understanding of your score, you need to be able to relate it to the others to determine if your score is higher than, lower than, or equal to the average score for that test among your classmates. This article will discuss how percentiles are defined and how to calculate them.

What is a percentile?

A percentile is a term used in statistics to express how a score compares to other scores in the same set. While there is technically no standard definition of percentile, it’s typically communicated as the percentage of values that fall below a particular value in a set of data scores.

Percentiles are commonly used to report values from norm-referenced tests (in which the average is determined by comparing a set of results in the same group) as the percentages of scores that fall below those of the average of the set. For example, a male child age 12 with a weight of 130 pounds is at the 90th percentile of weight for males of that age, which indicates that he weighs more than 90 percent of other 12-year-old boys.

In statistical terms, there are three separate definitions of percentile. They are:

  1. Greater than: The kth percentile is the lowest score in a data set that is greater than a percentage (k) of the scores. For example, if k = .25, you’d be trying to identify the lowest score that is greater than 25% of scores in the data set.
  2. Greater than or equal to: The k**th percentile is the lowest score in the data set that is greater than or equal to a percentage (k) of the scores. For example, if k = .25, you’d be looking for a value that is greater or equal to 25% of the scores.
  3. Weighted average: In this method, the kth percentile is the weighted average of the percentiles calculated in the two definitions above. This method allows for numbers to be more neatly rounded and defines the median of the set as the 50th percentile.

How is percentile helpful?

Percentiles are useful as they can tell you how one value compares to other values in the data set. Typically, if value n is at the kth percentile, then n is greater than k**% of the values in the set. One of the most common examples to express this is by looking at test scores. Calculating where your score lies in the data set (your score and the scores of the other test-takers), can help you figure out how your score compares to the rest of the scores.

Another example is tracking the weight of children compared to other children of the same age. This can help you determine whether your child’s weight falls into an acceptable range for its age, or if you need to take steps to help them gain or lose weight to get closer to an average weight.

Percentile terms

Percentiles can also be used to split your dataset into portions to measure dispersion and identify the average of the values (known as the central tendency). Certain meaningful percentiles are referred to by their own terms. Here are some:

Median

The 50th percentile, in which half the values of a data set are above the 50th percentile, and half are below.

Quartile

Values that split the data set into quarters based on percentiles.

  • The first quartile is referred to as Q1 or the lower quartile. This value is the 25th percentile, in which the lower quarter of the values fall below the 25th percentile while three quarters are above it.
  • The second quartile, or Q2, is the value at the 50th percentile. This is the median of the data set.
  • Q3, the third quartile, is referred to as the ‘upper quartile’ and is the value of the 75th percentile, meaning only 25% of values in the set are above this value.

Interquartile range

Used to measure the dispersion of values, this is meant to show the middle half of the data. One quarter of the data values sit above this number, and one quarter falls below. The IQR is calculated by finding the difference between the first quartile and the third quartile (Q3 – Q1). The larger the IQR, the more spread out the values.

How to calculate percentile

Follow these steps to calculate the kth percentile:

  1. Rank the values in the data set in order from smallest to largest.
  2. Multiply k (percent) by n (total number of values in the data set). This is the index. You’ll refer to this in the next steps as the position of a value in your data set (first, second, third…).
  3. If the index is not a round number, round it up (or down, if it’s closer to the lower number) to the nearest whole number.
  4. Use your ranked data set to find your percentile. Refer to the value that correlates with the index number, as determined in step 3. Since the value for the kth percentile must be greater than the values that precede the index, the next ranked value would be the kth percentile. Similarly, when using the ‘greater than or equal to’ method, steps 1-3 remain the same, but this time we would include the index value. The kth percentile is then calculated by taking the average of the index value in your data set and the next ranked value.

Calculating percentile example

Using the data set below, here’s an example of calculating the 60th percentile:

  1. Rank the values in the data set in order from smallest to largest, as shown below.
  2. Calculate the index. To find the 60th percentile using the data set below, multiply k (.6) by n (8) to reach an index of 4.8.
  3. Round the index to the nearest whole number (5).
  4. To calculate percentile according to the ‘greater than’ method, count the values in your data set from smallest to largest until you reach the number ranked 5th, as determined in step 3. Since the value for the 60th percentile must be greater than the first five values, the 6th ranked value would be the kth (60th) percentile. In this data set, that value is 67.
  5. Using the ‘greater than or equal to’ method, steps 1-3 above remain the same, but this time we would include the fifth-ranked value, or 56, in this data set. The kth (60th) percentile is then calculated by taking the average of that value in your data set (56) and the next ranked value (67). (56 + 67) / 2 = 61.5

There are several methods that can be used to calculate a percentile and these steps demonstrate one of the simplest ways to do so.

Percentile range

A percentile range is expressed as the difference between any two specified percentiles. In this example, the 10-90 percentile range will be used. To find the 10-90 percentile range of the sample data set above, follow these steps:

1. Follow the steps above to calculate the 10th percentile.

  • (.1 x 8)=.8 (round to 1)
  • K=33 (greater than) and k=30 (greater than or equal to)
  • Average. (33 + 30) / 2 = 31.5

2. Find the 90th percentile following the steps above.

  • (.9 x 8)=7.2 (round to 7)
  • K=72 (greater than), k=68 (greater than or equal to)
  • Average (72 + 68) / 2 = 70

3. Subtract the 10th percentile from the 90th percentile.

  • 70 – 31.5 = 38

Formula for Percentile

The Percentile Formula is given as,

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

Another formula to find the percentile is given by:

P = (n/N) × 100 P = (nth percentile/100)  × Total number of values in the list

Here, 

n = Ordinal rank of the given value or value below the number

N = Number of values in the data set 

P = Percentile 

Rank = Percentile/100

Ordinal rank for Percentile value = Rank × Total number of values in the list

Solved Example

Question 1:

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

Solution:

Given,

No. of. scores below 71 = 6

Total no. of. scores = 10

The formula for percentile is given as, Percentile = (Number of Values Below “x” / Total Number of Values) × 100

Percentile of 71

= (6/10) × 100

= 0.6 × 100 = 60

Question 2: 

Consider the list {50, 45, 60, 25, 30}. Find the 5th, 30th, 40th, 50th and 100th percentiles of the list given.

Solution:

Given list – 50, 45, 60, 25, 30

Ordered list – 25, 30, 45, 50, 60

N = 5

Percentile (P)Ordinal rankPercentile value
5th(5/100) × 5 = [0.25] = 11st number in the ordered list = 25
30th(30/100) × 5 = [1.5] = 22nd number in the ordered list = 30
40th(40/100) × 5 = 22nd number in the ordered list = 30
50th(50/100) × 5 = [2.5] = 33rd number in the ordered list = 45
100th(100/100) × 5 = 55th number in the ordered list = 60

What is Percentile?

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,

k = 0.80

Step 3:

Find 80th percentile,

80th percentile = 0.80 ×

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,

= 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 ×0.60=300

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

FAQs (Frequently Asked Questions)

1. What is the Percentile Rank Formula?   

Percentile Rank is a usual term mainly used in statistics which is arrived from Percentile. Percentile (also referred to as Centile) is the percentage of scores that range between 0 and 100 which is less than or equal to the given set of distribution. Percentiles divide any distribution into 100 equal parts. This is predominantly used for the interpretation of scores with different ranges across various tests. The percentile Rank formula (PR) is arrived at based on the total number of ranks and the number of ranks below and above percentile.

A Percentile Rank formula is given by:

Where,

  • M = Number of Ranks below x
  • R = Number of Ranks equals x
  • Y = Total Number of Ranks

2. What is the Percentile Range?

A percentile range is a difference between two specified percentiles. These could theoretically be any two percentiles, but the 10-90 percentile range is the most common. To find the 10-90 percentile range:

  1. Calculate the 10th percentile using the percentile formula.
  2. Calculate the 90th percentile using the percentile formula.
  3. Subtract the 10th percentile from the 90th percentile.

3.  What is the Formula for Percentile?

In most common terms it is defined as the simplest and straightforward way of calculating percentile. With the help of the formula one can easily calculate the score of rank which can be equal to, or less than the given score. It can easily give you an idea of a perfect score.

4. What is the Percentile Rank Formula?

In general terms, percentile ranks are frequently expressed as a number between 1 and 99, with 50 being the average. Thus to calculate the Percentile Rank the formula which is discovered is  R = P / 100 (N + 1) where; R represents the rank order of the score, P represents the percentile rank and N represents the number of scores that is in the distribution.

5.  What is percentile used for?

Surprising the Percentile will tell you how to compare values with other values. It will give you an understanding of relative standing value. Besides, it will also give you an idea of calculating different mathematical problems by providing you with the concept of using different methods to solve the equations. 

What Are And How To Calculate Find Percentiles – Percentile Rank Formula Calculation Explained

Calculating percentile

Percentiles – How to calculate Percentiles

Formula for Percentile

Solved Examples for Percentile Formula

Q.1: The scores for some candidates in a test are 40, 45, 49, 53, 61, 65, 71, 79, 85, 91. What will be the percentile for the score 71? Use the percentile formula.

Solution: Given parameters:

No. of. Scores below the value “71” i.e. n = 6

Total no. of. Scores i.e. N = 10

The formula for percentile is given as below:

Thus, Percentile of 71 will be 60.

Q.2: The scores for some candidates in a test are 40, 45, 49, 53, 61, 65, 71, 79, 85, 91. What will be the score with a percentile value of 90?

Solution:  It is obvious that we need to find the score “x” with percentile 90 for the given data set.

Thus, No. of. Scores below the value “x” i.e. n =?

Total no. of. Scores i.e. N = 10

Given percentile value, P = 90

The formula for percentile is given as below:

Thus 9th value in the data set i.e. 85 will be the score with 90 percentile.

Example 1:

If the scores of a set of students in a math test are 20 , 30 , 15 and 75 what is the percentile rank of the score 30?

Arrange the numbers in ascending order and give the rank ranging from 1 to the lowest to 4 to the highest.

Note that, if the percentile rank R is an integer, the P th percentile would be the score with rank R

when the data points are arranged in ascending order.

If R is not an integer, then the P

th percentile is calculated as shown.

Let I be the integer part and be the decimal part of D of R . Calculate the scores with the ranks I and I+1 . Multiply the difference of the scores by the decimal part of R . The P th percentile is the sum of the product and the score with the rank I

.Example 2:

Determine the 35 th percentile of the scores 7,3,12,15,14,4 and 20

.Arrange the numbers in ascending order and give the rank ranging from 1 to the lowest to 7 to the highest.

The integer part of R is 2 , calculate the score corresponding to the ranks 2 and 3 . They are 4 and 7 . The product of the difference and the decimal part is 0.45(7−4)=1.35

.

Therefore, the 35 th  percentile is 2+1.35=3.35 .

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