The Five-Number Summary

The terminology for box-and-whisker plots can vary somewhat, and you may encounter some different words, depending on your textbook and instructor.

What other terminology does the five-number summary have?

The top end of your box, at Q₃, may be called the "upper hinge"; the lower end, at Q₁, may be called the "lower hinge".

The lower hinge is also called "the 25th percentile"; the median is "the 50th percentile"; the upper hinge is "the 75th percentile". This means that 25%, 50% and 75% of the data, respectively, is at or below that point.

The distance between the hinges may be referred to as the "H-spread" or, as you will see on the following page, the "Interquartile Range", abbreviated "IQR". ("Hinge" actually has a different technical definition, but the term is sometimes used informally. You should use whichever terminology your book or instructor uses.)

Also, some books and software will include the overall median (that is, Q₂) when computing Q₁ and Q₃ for data sets with an odd number of elements. The Texas Instruments calculators do not include Q₂ in this case, so you may encounter a book answer that doesn't match the calculator answer. And different software packages use all different sorts of formulas. Be careful to use the formula from your book when doing your homework!

Additionally, the box-and-whisker plot may include a cross or an "X" marking the mean value of the data, in addition to the line inside the box that marks the median. The difference between the "X" and the median line can then be used as a measure of "skew".

Please don't ask me to explain "skew".

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• Draw the box-and-whisker plot for the following data set:

77,  79,  80,  86,  87,  87,  94,  99

(Yes, my solution above started with me referring to the listed values as "these data are", rather than (as has become common) "this data is". Technically, "data" is plural; a single value would be a "datum". No, I don't usually use the terms correctly, either.)

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As you can see, you only need the five values listed above (min, Q₁, Q₂, Q₃, and max) in order to draw your box-and-whisker plot. This set of five values has been given the name "the five-number summary".

If you are asked to produce the five-number summary, they're wanting the same values as I found in the example above, but without doing the associated box-and-whisker plot.

• Give the five-number summary of the following data set:

79,  53,  82,  91,  87,  98,  80,  93

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Part of the point of a box-and-whisker plot is to show how spread out your values are. But what if one or another of your values is way out of line? For this, we need to consider "outliers".