Five-Number Summary

A five-number summary describes a data set with five values: minimum, first quartile, median, third quartile and maximum. Those are exactly the points you need to draw a box-and-whisker plot.

How the calculation works

Set how many values to use (3 to 10) in the first field and type your numbers into x1, x2 and so on. Only the first N fields are used; the rest are ignored. Entry order does not matter because the data are sorted first.

The median is the middle of the sorted list (the mean of the two middle values when N is even). Quartiles use the “method of halves”: the list is split into a lower and an upper half, and when N is odd the median itself is left out of both halves. Q1 is the median of the lower half and Q3 the median of the upper half. This is the convention taught in many textbooks and used on TI graphing calculators.

The calculator also reports the interquartile range, IQR = Q3 − Q1: the width of the box, which holds the middle half of the data. Unlike the range (max − min), it is not thrown off by a single extreme value.

Worked example

The defaults use N = 9 with 3, 7, 8, 5, 12, 14, 21, 13, 18 (x10 is ignored). Sorted: 3, 5, 7, 8, 12, 13, 14, 18, 21. The median is the 5th value, 12. Lower half 3, 5, 7, 8 → Q1 = (5 + 7)/2 = 6; upper half 13, 14, 18, 21 → Q3 = (14 + 18)/2 = 16. Minimum 3, maximum 21, IQR = 16 − 6 = 10.

Things to keep in mind

• There are many quartile conventions, and on small samples they disagree. For this same data Excel’s QUARTILE.INC returns Q1 = 7 and Q3 = 14, while QUARTILE.EXC returns 6 and 16 like this tool. Check which method your course or software expects.
• To flag outliers with Tukey’s rule, compute the fences Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. Here they are −9 and 31, so there are no outliers. The calculator does not show the fences itself.
• If the median sits much closer to Q1 than to Q3 (or vice versa), the distribution is skewed; comparing it with the mean confirms the direction.
• The tool accepts up to 10 values. For larger data sets use a spreadsheet or statistics package.