How to Calculate Percentile
- Introduction
- How is the Percentile Formula Defined?
- Percentile Formula
- Calculation of Percentile
- Real-life Applications of Calculating Percentiles
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
In statistics, percentile describes how a data point compares to other data points in a dataset. Percentile is a useful measure in assessing someone's performance compared to others, often used in educational settings like schools and standardized tests. It calculates a person's position relative to a group, indicating what percentage of scores fall below theirs. For instance, if someone's score puts them in the `70^{th}` percentile, it means `70%` of the group scored lower. This formula is versatile and can be applied to various metrics like weight, income, and more. It's a useful way to understand where an individual stands in a group or population.
How is the Percentile Formula Defined?
The percentile formula is a method used to compare specific values within a dataset, focusing on accuracy. It distinguishes between percentiles and percentages: while percentages represent fractions as a whole, percentiles indicate the percentage of values below a given point in the dataset. For example, if a city's temperature for a particular day is in the `95^{th}` percentile compared to historical temperatures for that day, it means that it's hotter than `95` percent of all temperatures recorded for that day in the past. This formula helps understand the relative position of a value in comparison to others within the dataset.
In generic terms, we should understand that if a value is in the `n^{th}` percentile, it is greater than `n` percent of the total values. For example if David is in `85^{th}` percentile in a standardized test, it means that the student has outperformed `85%` of the students who have taken the same test. Only `15%` of students scored better than David.
Percentile Formula
Percentile formulas are essential tools for evaluating various aspects of data, whether it's grading test scores or analyzing biometric measurements. The formula typically involves calculating the position of a value within a dataset. One common way to express this is through the formula:
\(P = \left(\frac{n}{N}\right) \times 100\)
Here,
\(P\) - Percentile
\(n\) - Ordinal rank of a given value or value below the number
\(N\) - Number of values in the dataset
`"Percentile" = "Number of Values Below “x”"/"Total Number of Values" × 100`
This formula helps determine the percentile of a value \(x\) by comparing the number of values below \(x\) to the total number of values, and then multiplying the result by `100`.
Calculation of Percentile
Calculating percentiles involves a systematic process outlined in the percentile formula. When dealing with a value \(q\) between zero and one hundred, the `q^{th}` percentile serves as a dividing point in the data, separating it into two sections. The lower portion comprises \(q\) percent of the data, while the remaining data constitutes the upper part. Here's a simplified breakdown of the steps:
Step `1`: Collect all the relevant values of the dataset.
Step `2`: Organize the dataset in ascending order.
Step `3`: Count how many data points are in the dataset.
Step `4`: Choose the value for which you want to find the percentile.
Step `5`: Calculate how many data points are smaller than the chosen value.
Step `6`: Divide the count of smaller data points by the total number of observations to find the percentile.
This process provides a clear method for understanding and applying percentiles in various contexts.
Real-life Applications of Calculating Percentiles
- Education Assessment: Percentiles are used in educational assessment to rank students' performance relative to their peers. For example, standardized tests often provide percentile rankings to indicate how a student's performance compares to a reference group.
- Healthcare Analysis: In healthcare, percentiles are utilized to assess growth and development in children. Pediatricians use growth charts that display percentiles to track a child's height, weight, and head circumference relative to their age and gender.
- Market Research: Percentiles are employed in market research to analyze income distribution, purchasing power, and consumer behavior. Marketers use percentile rankings to segment consumers based on their spending habits and socioeconomic status.
- Financial Analysis: In finance, percentiles help evaluate investment performance and risk. Analysts use percentiles to assess the distribution of returns, identify outliers, and calculate value at risk (VaR) measures.
- Demographic Studies: Percentiles are applied in demographic studies to examine population characteristics such as income distribution, educational attainment, and housing affordability. Researchers use percentiles to identify socioeconomic disparities and inform policy decisions.
Solved Examples
Example `1`. Consider a dataset of test scores: `90, 72, 100, 85, 65, 95, 98, 80`. What is the percentile for the score `90`?
Solution:
Arrange the data in ascending order - `65, 72, 80, 85, 90, 95, 98, 100`
Number of scores below `90 (n)= \color{#38761d}4`
Total number of scores `(N)= \color{#fb8500}8`
\( P = \left(\frac{n}{N}\right) \times 100 \)
\( P = \left(\frac{\color{#38761d}4}{\color{#fb8500}8}\right) \times 100 \)
\( P = \color{#6F2DBD}{50} \)
Therefore, the percentile for the score `90` is `50`.
This means `50%` of the test scores in the data set are less than than `90`.
Example `2`. The heights of 9 students are as follows (in inches): `75, 57, 83, 68, 70, 63, 76, 80, 50`. Using the percentile formula, find the `50^{th}` percentile value of this dataset.
Solution:
Arrange the data in ascending order - `50, 57, 63, 68, 70, 75, 76, 80, 83`
Percentile `(P)=` `50`
Total number of student heights `(N)=` `9`
We have percentile formula,
\( P = \left(\frac{n}{N}\right) \times 100 \)
To find the rank `n` the formula becomes,
\( n = \left(\frac{P \times N}{100}\right)\)
\( n = \left(\frac{\color{#6F2DBD}{50} \times \color{#fb8500}9}{100}\right)\)
\( n = 4.5 \)
The value of `n` can be rounded off to `5`.
And the `5^{th}` term in the sorted data is `70`.
The `50^{th}` percentile value is `70`.
This means height of `50%` of the students is less than than `70` inches.
Example `3`. Consider a dataset representing ages (in years) of individuals: `50, 25, 55, 35, 40, 65, 30, 45, 60`. How to find percentile for the age `45`?
Solution:
Arrange the data in ascending order - `25, 30, 35, 40, 45, 50, 55, 60, 65`
Number of individuals of age below `45 (n)=` `4`
Total number of individuals `(N)=` `9`
\( P = \left(\frac{n}{N}\right) \times 100 \)
\( P = \left(\frac{\color{#38761d}4}{\color{#fb8500}9}\right) \times 100 \)
\( P = \color{#6F2DBD}{44.44} \)
Therefore, the percentile for the age `45` is `44.44`.
This means the age `44.44%` of the individuals is less than than `45` years.
Example `4`. Consider a dataset representing temperatures (in degrees Celsius): `40, 55, 20, 15, 10, 35, 30, 45, 50, 25`. How to calculate percentile for the temperature `40`?
Solution:
Arrange the data in ascending order - `10, 15, 20, 25, 30, 35, 40, 45, 50, 55`
Number of temperatures below `40 (n)=` `6`
Total number of temperatures `(N)=` `10`
\( P = \left(\frac{n}{N}\right) \times 100 \)
\( P = \left(\frac{\color{#38761d}6}{\color{#fb8500}{10}}\right) \times 100 \)
\( P = \color{#6F2DBD}{60} \)
Therefore, the percentile for the temperature `40` is `60`.
This means `60%` of the temperature values in the data set are less than than `40` degrees celcius.
Example `5`. Suppose you have a dataset of reaction times (in milliseconds): `136, 170, 125, 100, 161, 159, 110`. Find the `25^{th}` percentile.
Solution:
Arrange the data in ascending order - `100, 110, 125, 136, 159, 161, 170`
Percentile `(P)=` `25`
Total number of reaction times `(N)=` `7`
We have percentile formula,
\( P = \left(\frac{n}{N}\right) \times 100 \)
To find the rank `n` the formula becomes,
\( n = \left(\frac{P \times N}{100}\right)\)
\( n = \left(\frac{\color{#6F2DBD}{25} \times \color{#fb8500}{7}}{100}\right)\)
\( n = 1.75 \)
The value of `n`, `1.75` can be rounded off to `2`.
And the `2^{nd}` term in the sorted data is `110`.
The `25^{th}` percentile value is `110`.
Practice Problems
Q`1`. A dataset of `11` exam scores (out of `100`) is given: `85, 70, 60, 98, 75, 95, 55, 90, 100, 65, 80`. What is the `90^{th}` percentile?
- `90`
- `95`
- `98`
- `100`
Answer: c
Q`2`. Consider a dataset of ages (in years) of individuals: `55, 35, 40, 25, 20, 45, 60, 65, 50, 30`. What is the percentile for the age of `35`?
- `35`
- `40`
- `45`
- `30`
Answer: d
Q`3`. Suppose you have a dataset of weights (in kilograms) of individuals: `90, 70, 55, 85, 75, 50, 60, 95, 80`. What is the `80^{th}` percentile?
- `75`
- `80`
- `85`
- `90`
Answer: c
Q`4`. In a dataset of temperatures (in degrees Celsius): `45, 50, 23, 10, 35, 41, 59, 60`. What is the percentile for the temperature `50`?
- `62.5`
- `31.2`
- `76.9`
- `53.8`
Answer: a
Q`5`. Suppose you have a dataset of `10` exam scores (out of `100`): `85, 95, 75, 65, 99, 98, 70, 80, 90, 100`. How to calculate percentile for the score `90`?
- `95`
- `60`
- `75`
- `50`
Answer: d
Frequently Asked Questions
Q`1`. What is a percentile?
Answer: A percentile is a measure indicating the value below which a given percentage of observations in a group fall.
Q`2`. How is the percentile calculated?
Answer: Percentile is calculated using the formula: \( P = \left(\frac{n}{N}\right) \times 100 \), where \( n \) is the number of values below the given value and \( N \) is the total number of values in the dataset.
Q`3`. Why are percentiles useful?
Answer: Percentiles help understand where a particular value stands in relation to others in a dataset, providing insights into distributions and relative performance.
Q`4`. What is the difference between percentile and percentage?
Answer: Percentile indicates a position within a dataset, while percentage represents a proportion out of `100`.
Q`5`. What is the median percentile?
Answer: The median percentile, also known as the `50^{th}` percentile, divides a dataset into two equal parts, with half of the values falling below and half above it.