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Box plot example with an outlier

A box plot of 20 values with Tukey’s 1.5 × IQR whiskers, fences and one outlier labeled.

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Box plot of a small dataset (n = 20) with whiskers drawn by Tukey's rule: each whisker runs to the most extreme data value within 1.5 × IQR of the box, and values beyond that are plotted as outliers. The outlier is drawn as a filled point on the center line of the box. Each whisker end is labeled as the largest or smallest value inside the fences. The 20 data values are drawn as dots beside the box, with guides at the values that set each part of the plot. Dashed lines mark the fences, each labeled with the formula and value that set it: Q3 + 1.5 × IQR = 51 and Q1 − 1.5 × IQR = 3.
Box plot with interquartile range and outliers. Derek Jansen, Grad Coach, CC BY 4.0.

What it shows

A box plot of 20 values with whiskers drawn by Tukey’s rule, used by most statistics packages. The box runs from the first quartile to the third and the line inside it is the median. Each whisker stops at the most extreme value within 1.5 times the interquartile range (IQR) of the box, and a value beyond that is plotted on its own as an outlier.

The data points are drawn as dots beside the box, with guides to the values that set each part of the plot. The two fences are dashed lines, each labeled with the calculation that places it.

How to read it

  1. Start with the box. Its bottom edge is Q1 (21), its top edge is Q3 (33), and the line across it is the median (27).

  2. The height of the box is the IQR: 33 − 21 = 12.

  3. Multiply the IQR by 1.5 to get 18. Add that to Q3 for the upper fence (51), or take it from Q1 for the lower fence (3).

  4. Each whisker runs to the last data value inside its fence: 12 at the bottom, 41 at the top. The value 58 is past the upper fence, so it is drawn as a separate point.

The numbers behind it

  • Data (n = 20): 12, 15, 17, 18, 21, 21, 22, 24, 25, 26, 28, 29, 30, 31, 33, 33, 35, 38, 41, 58.
  • Q1 = 21, median = 27, Q3 = 33, IQR = 12, 1.5 × IQR = 18.
  • Lower fence = 3, upper fence = 51. Lower whisker = 12, upper whisker = 41, one outlier = 58.
  • The common quartile methods agree on these data: Tukey’s hinges, R’s default (Hyndman and Fan type 7) and the type 6 method used by Minitab and SPSS all give Q1 = 21 and Q3 = 33.

Ways to use it in teaching

  • Hand out the data list without the plot and have students find the quartiles, fences and outlier, then compare their answers with the figure.

  • Pair it with the minimum-to-maximum version of the same data to discuss when an unusual value should be flagged.

  • In a data-cleaning session, note that the plot flags 58 but cannot say whether 58 should be removed.

Deciding what to do with an outlier is a separate question, covered in our guide to removing outliers from a dataset.

Sources

Tukey, J. W. (1977), Exploratory Data Analysis, Addison-Wesley (the 1.5 × IQR rule); Hyndman, R. J. and Fan, Y. (1996), "Sample quantiles in statistical packages", The American Statistician 50(4): 361–365 (the quantile types).

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