Using the box plot maker
Type or paste at least three numbers, separated by commas, spaces or new lines. The five-number summary appears in the box, and the box plot is drawn below it with the data as dots above. The working finds the median, then the quartiles, the interquartile range and the outlier fences.
Quartiles can be worked out in more than one way, so choose the method your class or software uses. The answers often differ a little, and for some lists they agree.
How to read a box plot
The box runs from the lower quartile, Q1, to the upper quartile, Q3, so it holds the middle half of the data. The line inside it is the median. Its length is the interquartile range, IQR = Q3 − Q1, a measure of spread that a few extreme values don't affect.
The whiskers reach out to the smallest and largest values that aren't outliers. A value counts as an outlier if it is more than 1.5 × IQR below Q1 or above Q3, and it is drawn on its own as a circle. Without outliers, the whiskers run to the minimum and maximum, and the five numbers the plot shows are the five-number summary.
Wikipedia traces the plot to Mary Eleanor Spear's range bar in Charting Statistics (1952). Its history section dates John Tukey's box-and-whisker plot to 1970, and he published it in his book Exploratory Data Analysis in 1977.
The three quartile methods
Leave the median out: split the ordered list at the median and, when the count is odd, leave the median out of both halves. Q1 and Q3 are the medians of the halves. The TI-83 and other TI-8X calculators work this way in 1-Var Stats, and so does this page by default.
Keep the median in: the same, but an odd median goes into both halves. These are Tukey's hinges, which R's fivenum function gives. For an even count the two methods agree.
Interpolate: Q1 is the value at position (n + 1) ÷ 4 in the ordered list and Q3 at 3(n + 1) ÷ 4, going part way between two values when the position isn't whole. Excel's QUARTILE.EXC and R's quantile function use this by default, according to Wikipedia's table. For 7, 15, 36, 39, 40, 41 it gives Q1 = 13 at position 1.75, where the other methods give 15.
Questions
Why does my textbook get a different Q1?
It is probably using another method. Try the other settings; one of them should match. Wikipedia lists several methods in use, and none is the single correct one.
What is the five-number summary?
The minimum, Q1, the median, Q3 and the maximum. Together they show where the data sits and how it spreads.
Can a box plot have no whisker on one side?
Yes. If Q1 equals the minimum, for example, there is nothing between them, so the lower whisker has no length.
How do I copy the plot?
Right-click or long-press the plot to copy or save it, or print the page.
Are my numbers sent anywhere?
No. The plot is drawn in your browser.
Sources
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