Statistics Definitions > Percentiles, Percentile Rank & Percentile Range Contents:
1. What are Percentiles?
Watch the video for an overview of percentiles and a couple of examples:
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While the term “percentile” lacks a universal definition, the most widely accepted definition of a percentile is a value that has a certain percentage of scores below it [1]. For example, let’s say the 70th percentile on an exam was 156. That means if you scored 156 on the exam, your score was better than 70 percent of test takers.
- The 25th percentile is also called the first quartile.
- The 50th percentile is generally the median (if you’re using the third definition—see below).
- The 75th percentile is also called the third quartile.
- The difference between the third and first quartiles is the interquartile range.
2. Percentile Rank
Percentile rank refers to data point’s position in a set of data, when the data points are ordered from smallest to largest.
For example, if you score in the 75th percentile, then 75% of exam results are below your score. The “75” is called the percentile rank, but — like most things in statistics — that definition isn’t set in stone. the terms “percentile” and “percentile rank” are, unfortunately, often used synonymously. For example, percentile ranks are often reported in exams such as the SAT, GRE, and LSAT. The College Board defines a student’s percentile rank as follows:
“A student’s percentile rank represents the percentage of students whose score is equal to or lower than their score. For example, if a student’s score is in the 75th percentile, 75% of a comparison group achieved scores at or below that student’s score.” The College Board [4].
This definition implies that the 75th percentile is the same as the 75th percentile rank, which is not quite correct: “75th” is the percentile and “75” is the percentile rank.
3. How to Find a Percentile
The above definitions may seem similar, but they can lead to big differences in results, although they are both the 25th percentile rank. Take the following list of test scores, ordered by rank:
| Score | Rank |
|---|---|
| 30 | 1 |
| 33 | 2 |
| 43 | 3 |
| 53 | 4 |
| 56 | 5 |
| 67 | 6 |
| 68 | 7 |
| 72 | 8 |
Example question: Find out where the 25th percentile is in the above list.
- Calculate which rank is at the 25th percentile with the following formula: Rank = Percentile / 100 * (number of items + 1) = 25 / 100 * (8 + 1) = 0.25 * 9 = 2.25.
- Round: A rank of 2.25 is at the 25th percentile. However, there isn’t a rank of 2.25 (ever heard of a high school rank of 2.25? I haven’t!), so you must either round up, or round down. As 2.25 is closer to 2 than 3, I’m going to round down to a rank of 2.
- Choose a definition to find the 25th percentile.
- Definition 1 (the most common): The lowest score that is greater than 25% of the scores. That equals a score of 43 on this list (a rank of 3).
- Definition 2: The smallest score that is greater than or equal to 25% of the scores. That equals a score of 33 on this list (a rank of 2).
Depending on which definition you use, the 25th percentile could be reported at 33 or 43! A third definition attempts to correct this possible misinterpretation:
Definition 3: A weighted mean of the percentiles from the first two definitions. In the above example, here’s how the percentile would be worked out using the weighted mean:
- Multiply the difference between the scores by 0.25 (the fraction of the rank we calculated above). The scores were 43 and 33, giving us a difference of 10: (0.25)(43 – 33) = 2.5
- Add the result to the lower score. 2.5 + 33 = 35.5
In this case, the 25th percentile score is 35.5, which makes more sense as it’s in the middle of 43 and 33. In most cases, the percentile is usually definition #1. However, it would be wise to double check that any statistics about percentiles are created using that first definition.
4. Percentile Range
A percentile range is the 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:
- Calculate the 10th percentile using the above steps.
- Calculate the 90th percentile using the above steps.
- Subtract Step 1 (the 10th percentile) from Step 2 (the 90th percentile).
Is the 95th percentile the top 5%?
Yes, the 95th percentile represents the top 5% of a dataset. This signifies that 5% of all the data points within the set fall above this 95th percentile value.
On the other hand, 95% of all the data points are below this particular percentile. If you were to rank all the data points from lowest to highest, the 95th percentile would be the value below which 95% of the data falls.
When used in this way, percentiles are an effective way to understand data distribution, especially when dealing with large datasets, as it gives a clear indication of where the majority of values (in this case, 95%) lie. That said, it’s probably more intuitive to say “Top 5%” than “95th percentile”!
Is it better to be in the 1st percentile or 100th?
The 100th percentile is defined as the largest entry in an ordered list [5]. So, being in the 100th percentile is generally better than being in the 1st percentile — especially when it comes to test scores. This is because being in the 100th percentile signifies that you have a better score than all of the people within the dataset. If you were in the 1st percentile, that means you outperformed just 1% of the people in the dataset.
But, there are circumstances where being in the 1st percentile could be better. For example, if the dataset were measuring which countries had the most diseases per capita, you would want to be in the bottom 1% (the countries with the least disease).
Can percentiles be zero?
No. Percentiles measure of the percentage of data points that are lower than a particular data point. As there are no data points below zero percent, this statistic doesn’t exist. It’s similar to division by zero: if you try and calculate 0% of something on a calculator, it would be undefined.
What percentile rank is average?
The average percentile rank is 50% as it falls in the middle of a dataset. In other words, a percentile rank of 50 is the median of a set of data.
References
- Boston University. Finding Percentiles with the Normal Distribution. Retrieved July 2, 2023 from: https://sphweb.bumc.bu.edu/otlt/MPH-Modules/PH717-QuantCore/PH717-Module6-RandomError/PH717-Module6-RandomError7.html
- AP Statistics – Chapter 2 Notes §2.1 Describing Location in a Distribution
- DavidCShannon, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons
- SAT: Understanding scores.
- Naval Postgraduate school. Module 2: Descriptive Statistics (and a bit about R) Statistics (OA3102)