Quartiles and the Five-Number Summary
The five-number summary is minimum, , median, , maximum. is the median of the lower half and the median of the upper half. A box plot draws a box from to with a line at the median and whiskers out to the extremes (or to the last non-outlier).
A box plot throws away most of the data on purpose. What survives is exactly what you need to compare groups side by side: where the middle is, how wide the middle half is, and whether the tails are lopsided. Each of the four pieces — left whisker, left box, right box, right whisker — holds about a quarter of the data, so a long piece means the values there are spread thin, not that there are more of them.
Data (sorted): . With the median is the th value, , and it belongs to neither half. Lower half : . Upper half : . Summary: , and . The long right whisker shows the upper quarter is spread out.
Twelve values: . The median is the average of the th and th, . Now the halves are the first six and the last six values: and . Summary ; .
Two classes take the same test. Class A: (five-number summary). Class B: . A has the higher median ( against ), but B is far more consistent: its middle half spans points against A's . Side-by-side boxes make both facts visible at once.
A value is an outlier if it lies below or above . Outliers are plotted as separate dots beyond the whiskers.
Including the median in both halves when is odd, or reading a longer section of the box as "more data". For , putting in both halves would give and — a different (and nonstandard) summary. And the long right whisker holds the same quarter of the data as the short left one.
For the data (, ), is an outlier?
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, so the upper fence is . Since , it is not an outlier (barely), and the whisker correctly runs all the way to .
Outliers and the Fences
The fences are and . Values outside them are flagged as outliers and drawn as separate dots; the whiskers stop at the most extreme values still inside the fences. The mean and standard deviation are sensitive to outliers; the median and IQR are resistant.
The fences are measured in units of the IQR so that they adapt to each data set's own spread: a value is unusual relative to how spread out the middle half is. The factor is a convention chosen so that, for bell-shaped data, fewer than of values are flagged (see "Why It Works"). An outlier is a flag, not a verdict — it may be a typo, or the most interesting point in the data.
Data: . Median ; lower half gives ; upper half gives ; . Fences: and . So and are outliers, and the whiskers run only to and .
For the same data the mean is and the median . Delete the outlier : the mean drops to , while the median only moves to . One value shifted the mean by more than , the median by — which is why skewed data are summarized by median and IQR.
Drawing the whiskers to the minimum and maximum when those are outliers. For the data above, whiskers from to would hide exactly the information the plot is meant to show; the whiskers end at and and the outliers get their own dots.
A data set has and . Which of the values are outliers?
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, fences and . Outliers: (below ) and (above ); and are inside.
Percentiles
The th percentile is the value below which of the data falls. The median is the th percentile; and are the th and th. A score at the th percentile beat of the scores — it says nothing about the score itself. A cumulative frequency graph (ogive) plots, for each value, how many data lie below it, so percentiles can be read straight off it.
Percentiles convert a position in the data into a fraction of the data, so they compare fairly across groups of different sizes: the th percentile of a class of and of a state of mean the same thing. On an ogive, the height at is the number of values below ; reading across from of the total and down to the axis gives the th percentile.
In a class of , a student has the th highest score, so scored higher. She is above of her classmates: , about the rd percentile.
Scores of students: –: , –: , –: , –: , –: , –: . Running totals at : . The median (the th student) is where the graph reaches : at . The th percentile is where it reaches , between (at ) and (at ): , assuming the scores spread evenly within the bin.
Reading "90th percentile" as "scored 90%". Percentiles rank; percents measure. A score of on a brutal exam can be at the th percentile.
A box plot shows , median , . What percent of the data lies between and ? Above ?
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Between and lies the middle half: . Above , the th percentile: .
Why It Works
For a normal distribution, and sit standard deviations either side of the mean, so . The upper fence is then above the mean, and the area beyond is about . So in well-behaved bell-shaped data the rule flags roughly values in — rare enough to deserve a look, common enough that a flag alone proves nothing. In skewed data it flags many more on the long side, which is informative in its own right.