Hi is it normal for the upper fence to be greater than max? If not, what might have gone wrong? I am using Empirical Rule and doing calculation mean +3 * IQR


  • $\begingroup$ Perhaps unusual, but not 'illegal'. In quartile-based boxplot, Fences depend on IQR and either Q1 or Q3, but the max or min might be close to Q3 or Q1, respectively. Certainly, if this happens in the upper tail, then there is no outlier in the upper tail. // This can happen easily with samples from short-tailed distn's such as uniform. In R, try code boxplot(runif(20)) several times. $\endgroup$
    – BruceET
    Commented May 1, 2022 at 3:26
  • $\begingroup$ Your last sentence doesn't make sense to me. Empirical Rule is not directly related to boxplots. // Technically speaking, Tukey's boxplots use 'fourths'. If boxplot uses quartiles instead of 'fourths', then upper fence is usually Q3 + 1.5(IQR).// Usually, Empirical Rule refers to 'mound shaped' samples which may be nearly normal. // Large normal samples often have a few outliers at each end, so fence outside max in upper tail wouldn't happen. $\endgroup$
    – BruceET
    Commented May 1, 2022 at 3:39

1 Answer 1


Yes, it is fairly normal for the upper fence to be greater than the maximum. That's the point of the fences -- with a Normal or light-tailed distribution, there will often be no points beyond the fences. Any that are beyond the fences get marked as outliers.

Note that (according to Tukey) the end of the whisker of the boxplot shouldn't be the fence, it should be the last observation before the fence

  • $\begingroup$ Thank you Thomas. $\endgroup$
    – Nathan
    Commented May 1, 2022 at 4:04
  • $\begingroup$ With a small sample from a normal distribution and there will often be no points beyond the fences. With a large sample there will usually be some points beyond the fences, and there is not a good reason to mark them as outliers. The switch seems be around $83-92$ $\endgroup$
    – Henry
    Commented May 1, 2022 at 23:17
  • $\begingroup$ Sure. I didn't claim there was a good reason apart from "Tukey did it that way" $\endgroup$ Commented May 2, 2022 at 1:59

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