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).

  • 4
    $\begingroup$ the whiskers extend to the closest observation not exceeding $\pm$1.5xIQR from the lower/upper quartile. $\endgroup$
    – utobi
    Commented Jan 28, 2023 at 13:21
  • 3
    $\begingroup$ @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. $\endgroup$
    – Glen_b
    Commented Jan 28, 2023 at 16:55

1 Answer 1


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.

enter image description here


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