Percentiles are values dividing an AOS into 100 equal groups. So `p^(th)` percentile of an AOS, symbolized as `P_p`, is a value that (Fig. 1):

• `p%` elements of this series are less than or equal to `P_p`,
• `(100–p)%` elements of this series are greater than or equal to `P_p`
• `P_p` can belong to this series or not.

Fig. 1 `p^(th)` percentile of an ascending ordered series

We can notice that median is the same as 50th percentile.

In practice, to determine `p^(th)` percentile of an AOS, we:

• calculate `k=(np)/100`,
• if `k` is an integer, `P_p` is the average of values `k^(th)` and `(k + 1)^(th)` of series,
• if `k` is not an integer, `P_p` is the `m^(th)` value of series; `m` is the integer next to `k`.

Example : What is the `80^(th)` percentile of an AOS consisting of 247 numbers.

• `k=(247xx80)/100=197,6`
• Because `k` is not an integer, so `80^(th)` percentile of this AOS is the `198^(th)` number.

Quartiles are the values, symbolized as `Q_1`, `Q_2`, `Q_3`, dividing an AOS into 4 equal groups (Fig. 2).

Fig. 2 Quartiles

Hence `Q_1` is the same as `25^(th)` percentile, `Q_2` is the same as `50^(th)` percentile or median, and `Q_3` is the same as `75^(th)` percentile.

So for the AOS consisting of 247 number in the previous example

• `Q_1` is the `62^(th)` number,
• `Q_2` or median is the `124^(th)` number,
• `Q_3` is the `186^(th)` number,

Quartiles can express, at the same time, the central tendency and degree of dispersion. So they are widely used in data description and data analysis.

Box plot is the visualization of quartiles, it is extensively used to illustrate the distribution of a variable. This plot can be horizontal or vertical. Fig. 3 is the horizontal box plot of 150 numbers taken randomly in the range [0 - 1000].

Fig. 3 Components of a box plot

Box plot consists of :

• a rectangular (box), starts at `Q_1`, ends at `Q_3`. The distance between these two values is interquartile range (`IQR`),
• in this rectangular, there is a line at median `M`,
• two lines, frequently named as whiskers, start at two ends of this rectangular,
• values outside whiskers (if exist) are considered as outliers.

There are several options for the other ends of whiskers, the most popular are:

• lowest value `x_min` and highest value `x_max`, (in this case, there is not outlier),
• values in the interval of `R` and farthest from corresponding quartile (as in Fig. 3).

`R` is the maximal length of whiskers. It is the product of `IQR` and whisker coefficient (common value is 1,5).