Statistics is a polar branch of mathematics that helps us understand data. Ever wondered how well you stack up against others? Percentiles are the key! This article will break down this statistical concept and show you how to find percentiles with ease.
Table of Content
What are Percentiles?
Imagine a bunch of data points, like test scores. Percentiles help us see how scores are spread out.
In statistics, a percentile is a measure used to indicate the value below which a given percentage of observations in a group of observations fall.
General Formula of Percentile
For calculating the percentile of ‘x’ in the data,
Percentile = (Number of values below ‘x’/Total number of values) × 100
Percentiles are a type of quartile obtained by adopting a subdivision into 100 groups. The 25th percentile is also known as the first quartile (Q1), the 50th percentile as the median or second quartile (Q2), and the 75th percentile as the third quartile (Q4). For example, the 50th percentile (median) is the score below (or at or below, depending on the definition) which 50% of the scores in the distribution are found.
Note: To find the rank or the number of percentile we can modify this formulas as Rank = (Desired Percentile/100) × (n+1).
How to Calculate Percentile?
There are different ways by which we can calculate percentile but the basic idea is to find where that number stands if the total no. of observations were 100.
Here are steps by which we can calculate the value at a given percentile in given data.
Step 1 : Arrange Data
Firstly, the data needs to be arranged in an order, preferably increasing order of values.
Step 2 : Calculate Rank
After arranging the data in order, we need to calculate the rank. The formula for rank is given as
Rank = (Desired Percentile/100) × (n+1)
Where n is the number of observations.
Step 3 : Find the Value
- Case 1: If the rank is a whole number, the value at that position in the ordered dataset is the desired percentile.
- Case 2: If the rank is a decimal, interpolate to the nearest whole number to find the percentile value.
Interpolation
Interpolation helps us estimate the exact value when the desired percentile falls between two data points. Here’s how it works:
Identify the Surrounding Data Points: Locate the two data points surrounding the desired percentile.
Calculate the Weighted Average: Take a weighted average of these two data points based on how close the desired percentile is to each point. The closer it is, the more weight it gets.
Calculate the Interpolated Value: To find the interpolated value, we use the formula:
Interpolated Value= Value at Lower Position + (Difference between Higher and Lower Values) × Fractional Rank
Where, “Fractional Rank” is the fractional part of the rank of the desired percentile, indicating its position between the two data points.
How is Percentile Useful?
Percentiles are useful because they provide a way to compare individual values to a larger group. There are multiple examples where this is put to use.
- Competitive Exam : Percentile score are used in competitive exams to access students as compared to a large pool of students who attempted the exam. For example a 75 percentile indicates that the student scored better than 75% of the students who appeared in the exam.
- Healthcare : Physicians also use percentile for a child’s height and weight to compare with a group of children with similar age and gender. This way doctors can access how low or high the child’s growth is.
- Traffic Management : Percentile is also used to calculate the speed limit to be set. Usually, 85th percentile speed of traffic on a road is utilized as a guideline for setting speed limits.
Conclusion
Finding percentiles in statistics is a useful way to understand the distribution of your data. It helps us analyze data and understand where a particular observation stands among a pool of observation in the data.
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Solved Questions on Percentile
Q1: Find the 75th percentile of the following dataset: 18, 15, 12, 20, 25, 22, 28, 30.
Solution:
Arrange the data: 12, 15, 18, 20, 22, 25, 28, 30.
Calculate rank:
Rank = (desired percentile/100) × (n+1)
(75/100) × (8+1) = 6.75
Interpolating between the 6th and 7th values due to the rank falling between them.
25 + (28 − 25) × 0.75
= 25 + 3 × 0.75
= 25 + 2.25
= 27.25
The 75th percentile is 27.25.
Q2: In a class of 50 students, Sarah scored 85 out of 100 on her math test. If Sarah’s score is at the 80th percentile, how many students scored lower than her?
Solution:
Step 1: Calculate the number of students represented by the 80th percentile.
Number of students = 80/100 × 50 = 40
Step 2: Subtract the number of students represented by the 80th percentile from the total number of students to find the number of students who scored lower than Sarah.
Students scored lower than Sarah = 50 − 40 = 10
Therefore, 10 students scored lower than Sarah.
Q3: If the 90th percentile of a dataset is 75, what does it indicate?
Solution:
It indicates that 90% of the observations in the dataset are below the value of 75.
Practice Questions on Percentile
P1: Calculate the 90th percentile for the following dataset of ages: 25, 28, 30, 32, 35, 38, 40, 42, 45, 50.
P2: In a study of test scores, the following scores were recorded: 75, 80, 85, 90, 95, 100. Determine the 25th percentile of these scores.
P3: A survey collected data on the monthly incomes of individuals in a small town: $1000, $1200, $1500, $1800, $2000, $2200, $2500. What is the 60th percentile income?
P4: The weights of a sample of students were measured in kilograms: 50, 52, 55, 58, 60, 62, 65, 68, 70, 72. Find the 80th percentile weight of these students.
Percentile in Statistics : FAQs
What are percentile ?
Percentiles are scores that divide your data set into 100 equal parts. They tell you what percentage of data falls below a certain value. For example, the 35th percentile means 35% of your data has values lower than that specific percentile.
What is the symbol for percentile?
It has no such specific symbol but you can use %ile as short form for percentile.
How are percentiles used in real life?
Percentiles have a wide range of applications. Here are a few examples : Education, Market Research, Sports, Social Science Research.
What is the difference between percentage and percentile ?
Percentiles and percentages are related but distinct concepts:
Percentile: Tells you where a certain % of data falls below a value (divides data into 100 sections).
Percentage: A general way to say what part of something is out of 100 (e.g., 70% passed).
Why are percentiles important?
Percentiles offer a clearer picture of how your data is spread out compared to just using the average. They help you identify: data distribution, outliers, comparison.