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7. Quantiles
Standard Deviation and Variance


Percentiles and quartiles are specific types of quantiles. More generally, quantiles are values that partition (or split) a set of values into approximately equal-sized subsets. Common quantiles have special names:

  • The 2-quantile is called the median.
  • The 4-quantiles are called quartiles.
  • The 10-quantiles are called deciles.
  • The 100-quantiles are called percentiles.
  • The 1,000-quantiles are called permilles.

The formula for p-th q-quantile of an n-element set is analogous to the one presented for percentiles:


Let's calculate the deciles of the following set:

1, 4, 6, 8, 10, 12, 14, 16, 18, 20

There are exactly 10 elements and 10 deciles, so each decile corresponds to a single element.

Decile d Position Decile value
1^\text{st} \lceil\frac{1}{10}10\rceil=\lceil1\rceil=1 1
2^\text{nd} \lceil\frac{2}{10}10\rceil=\lceil2\rceil=2 4
3^{rd} \lceil\frac{3}{10}10\rceil=\lceil3\rceil=3 6
4^\text{th} \lceil\frac{4}{10}10\rceil=\lceil4\rceil=4 8
5^\text{th} \lceil\frac{5}{10}10\rceil=\lceil5\rceil=5 10
6^\text{th} \lceil\frac{6}{10}10\rceil=\lceil6\rceil=6 12
7^\text{th} \lceil\frac{7}{10}10\rceil=\lceil7\rceil=7 14
8^\text{th} \lceil\frac{8}{10}10\rceil=\lceil8\rceil=8 16
9^\text{th} \lceil\frac{9}{10}10\rceil=\lceil9\rceil=9 18
10^\text{th} \lceil\frac{10}{10}10\rceil=\lceil10\rceil=10 20