Quantile is a collective term for percentiles, deciles, and quartiles. It is a measure that divides a distribution into equal percentile distances.

Percentiles

Percentiles divide a distribution of scores into 100 parts, i.e. each part contains 1% of the scores in the distribution. Percentiles can be any whole number from 1 to 100. A given percentile corresponds to a point on the distribution that gives the corresponding percentage of scores up to and including that point. For example: if someone has an IQ of 100, they are in the 50th percentile, meaning that 50% of the population has the same IQ or lower.

In the literature, the term percentile rank is often used synonymously with percentiles.

Since the median is the middle score in a distribution, it is equal to the 50th percentile.

Percentiles are not the same as percents. Percentiles represent a certain percentage of scores in a distribution, arranged from highest to lowest. For example, if someone's test score is in the 60th percentile, it means that 60% of the test takers performed poorly on that test, not that the test taker answered 60% of the questions correctly.

To determine which score is in a certain percentile, we need to calculate the position of that percentile in a given distribution. The formula for calculating the position of the i-th percentile is

i=P/100*n

where

  • i - the position of the desired percentile in the distribution,
  • P - the desired percentile,
  • n - the total number of results in the distribution.

First, the results need to be sorted from smallest to largest. Then, the desired percentile is divided by one hundred and this quotient is multiplied by the total number of results in the distribution. If the result is an integer, then the position of the desired percentile is in the middle between the result obtained and the first larger result. If the result obtained is not a whole number, then it is rounded to the first larger whole number and it represents the ordinal position of the desired percentile.

Example: Calculating percentiles

Let's assume that we want to calculate the 33rd percentile using the example of the height of girls in the 52 class of the "X" primary school. The measurements gave us the following results in centimeters: 140, 141, 138, 140, 122, 160, 154, 132, 148, 135, 140.

The first step in calculating percentiles is to arrange the results by size, and then they would look like this: 122, 132, 135, 138, 140, 140, 140, 141, 148, 154, and 160. We have 11 measurements in total. When we insert the data into the formula for calculating the percentile position, we get

p33=33/100*11, or
p33=3.63.

Since the result obtained is not a whole number, we round it to the first larger whole number, which is 4. The desired percentile is in the 4th position of the ordinally arranged results and is 138.

Answer: The score corresponding to the 33rd percentile is 138, which means that 33% of girls in the 52 grade of the "X" elementary school are shorter than 138 cm or of the same height, and 67% are taller or of the same height.

Deciles

Deciles divide the distribution of scores into 10 parts, each part containing 10% of the distribution scores. Deciles can be any whole number from 1 to 10. The score corresponding to the tenth percentile falls on the upper limit of the first decile of the distribution, etc.

Deciles are a special case of percentiles. Therefore, the decile position is calculated according to the formula for the position of the corresponding percentile. For example:

D1=P10=10/100*n
D2=P20=20/100*n

Example: Calculating deciles

Measuring the height of girls 52 in the "X" class of primary school we have obtained the following data expressed in cm: 140, 141, 138, 140, 122, 160, 154, 132, 148, 135, 140. Suppose we want to find out which result corresponds to the 9th decile.

The first step in calculating deciles is to arrange the results by size, and then they would look like this: 122, 132, 135, 138, 140, 140, 140, 141, 148, 154 and 160. We have a total of 11 results for measuring the height of the girls.

To calculate the 9th decile, we multiply 11 by 90% or by 0.90, 11*0.90=9.9. Since the product is not a whole number, we round it up to the next higher number, which is ten. The ninth decile position is at the tenth position of the distribution of scores arranged from highest to lowest, which is 154.

Answer: The score corresponding to the 9th decile is 154, which means that 90% of the girls in class 52 are 154cm or less tall.

The interdecile rank is the difference between the score that falls in the 9th decile and the score that falls in the 1st decile. The interdecile rank covers the middle 80% of the results.

Quartiles

Quartiles divide the distribution into four equal parts, each part containing 25% of the results of the distribution.

The result corresponding to the 50th percentile falls on the upper limit of the 2nd quartile and is also the median of the distribution. The first quartile is denoted by Q1, and the third by Q3. The position of the median, i.e. the second quartile, is calculated according to the formula described in the text about the median.

Quartiles are a special case of percentiles. Therefore, the quartile position is calculated according to the same formula as for calculating the percentile position:

Q1=P25=25/100*n
Q3=P75=75/100*n

Calculating the quartile position is explained through the following example.

Example: Calculating the quartile

Measuring the height of girls 52 in the "X" class we got the following data expressed in cm: 140, 141, 138, 140, 122, 160, 154, 132, 148, 135, 140.

The first step in calculating the quartiles is to arrange the results by size and then they would look like this: 122, 132, 135, 138, 140, 140, 140, 141, 148, 154 and 160. We have a total of 11 results of measuring the girls' heights.

To calculate the 1st quartile, we multiply 11 by ¼ or by 0.25, 11*0.25=2.75. Since the product is not a whole number, we round it to the first higher number, which is three. The position of the first quartile is in the third position and is 135.

To calculate the 3rd quartile, we multiply 11 by ¾ or by 0.75; 11*0.75=8.25. Since the product is not a whole number, we round it up to the first higher number, which is nine. The position of the third quartile is in the ninth position of the ordinally arranged scores, which is 148.

Answer: The score corresponding to the 1st quartile is 135, which means that 25% of the girls in class 52 are 135 or less tall. The score corresponding to the 3rd quartile is 148, which means that 75% of the girls in class 52 are 148 or less tall.

The interquartile range is the difference between the score that falls in the 3rd quartile and the score that falls in the 1st quartile. It is reported next to the median, includes the middle 50% of the scores, and ignores values ​​below the 1st quartile and above the 3rd quartile. The interquartile range is a useful measure of variability if there are extreme values ​​in a series of scores and if the distribution is suspected to be skewed. In such distributions, it is a better measure of variability than the standard deviation.

The semi-interquartile range is obtained by dividing the interquartile range by two. This gives 25% of the results below and above the median, i.e. below and above the 2nd quartile. The semi-interquartile range is particularly useful when working with ordinal variables or when there are extreme values ​​on either side of the median. The semi-interquartile range is usually reported alongside the median.