# Boxplot | 5-Number-Summary

I have a question regarding the boxplot. On some web pages, the Minimum and the Maximum of the 5-Number-Summary correspond to the whiskers. However, regarding this definition, my question is:

how is it possible to illustrate outliers?

In fact, no data points can be lower than the minimum (lower whisker) or higher than the maximum (upper whisker).

• the whiskers extend to the closest observation not exceeding $\pm$1.5xIQR from the lower/upper quartile. Commented Jan 28, 2023 at 13:21
• @Made The usual Tukey boxplot is not identical to a five number summary; its whiskers don't necessarily get drawn all the way to the extremes. Commented Jan 28, 2023 at 16:55

To clarify your doubt, consider the following example using the standard definition of the boxplot.

Suppose we have the following observations $$x = (-40,0, 2, 3, 4,10, 40)$$. The median is 3, the first quartile is $$Q_1 = 1$$, and the third quartile is $$Q_3 = 7$$, thus $$\text{IQR} = 8$$. Let $$u = Q_3+1.5\times \text{IQR} =16$$ and $$l = Q_1-1.5\times \text{IQR}=-8$$.

The upper whisker would then be $$\max_{x_i\leq u} x,$$

which equals 10. The lower whisker would be $$\min_{x_i\geq l} x,$$ which equals 0.

Therefore, observations -40 and 40 fall outside the whiskers, and are thus "outlying" observations.

The conclusion is thus: the maximum and the minimum observed values may or may not correspond to the whiskers, depending on the distribution of observations.

Note: There are many ways to compute sample quantiles. In this example, I calculated them in R by the quantile function and using the default method.