Statistics Definitions > Percentiles, Percentile Rank & Percentile Range

**Contents**:

## 1. What are Percentiles?

Watch the video or read on below:

Percentiles are commonly used to report scores in tests, like the SAT, GRE and LSAT. for example, the 70th percentile on the 2013 GRE 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

The word “percentile” is used informally in the above definition. In common use, the percentile usually indicates that a certain percentage falls below that percentile. For example, if you score in the 25th percentile, then 25% of test takers are below your score. The “25” is called the **percentile rank**. In statistics, it can get a little more complicated as there are actually three definitions of “percentile.” Here are the first two (see below for definition 3), based on an arbitrary “25th percentile”:

**Definition 1**: The* n*th percentile is the lowest score that is greater than a certain percentage (“n”) of the scores. In this example, our n is 25, so we’re looking for the lowest score that is greater than 25%.

**Definition 2:** The *n*th percentile is the smallest score that is greater than **or equal to** a certain percentage of the scores. To rephrase this, it’s the percentage of data that falls at or below a certain observation. **This is the definition used in AP statistics.** In this example, the 25th percentile is the score that’s greater or equal to 25% of the scores.

They may seem very 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 |

## 3. How to Find a Percentile

**Sample question: **Find out where the 25th percentile is in the above list.

Step 1: Calculate what rank is at the 25th percentile. Use the following formula:

Rank = Percentile / 100 * (number of items + 1)

Rank = 25 / 100 * (8 + 1) = 0.25 * 9 = 2.25.

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.

Step 2: Choose either definition 1 or 2:

**Definition 1**: 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).

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you’re are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

A note on the math calculations:

43-33=10

10*.25 or 10/4 = 2.5

so the weighted mean would be 33+2.5=35.5

Thanks for spotting that! It’s been updated.

Absolutely confusing. How do the percentile calculated. Give a list of attributes and then arrive it. From an example

I’m not sure what you mean by “how do[sic] the percentile calculated”. There could be many different ways to approach this, depending on what definition of percentile you’re using. Can you give me a sample question of what you’re looking for?

The definition was good and understandable, thanks for your help.

that was really interesting and its also the first time ive had to learn percentile.

thank you.

very useful for me because I don’t know how to calculate percentile. the web site here explanation is very useful and precisely interpretation.

Please tell me the percentile score of the following data

14/40

24/40

84/120

Please reply

Mrs Shariq,

I’m not sure what data you are giving? Is this “14 out of 40” “24 out of 40” and “84 out of 120”? If so, you’d need to know the number of items in the set for the percentile rank. Did you mean “percentage” instead? As in 14/40 = .35 = 35%.

Which percentile calculation method is true, z score (mean+ sd*zscore) or the method described above?

Z-scores only apply to normally distributed data. Where did you find that formula?

Mean+sd*zscore formule is used usually for anthropometrics data. If you search for percentile and standard deviation at google you can see this formule.

I think this formule can be used for normally distrubed population percentile, isnot it? For a sample data taken from a population this formule must change as i know, like: mean+ zscore*standard error

Standar error=sample standard deviation/sqrt(sample size)

Good presentation

Please batao is m k ya Karna h

Q. explain P60=75. ???????

Please

explain p60=75.

Ans me

I’m not sure as you don’t give any context. But it sounds like “the 60th percentile score is 75.”

How can i find out coefficent of percentile?

That seems to be an informal term that one or two authors use. I’m not sure exactly what you’re trying to calculate (coefficient of percentile deviation?), but does this book help? It basically says it’s P

_{90}_{ – P}_{10}/ P_{90}_{ + P}_{10}.