Key takeaway: A box plot draws a data set as a rectangle with two whiskers over a number line, marking the smallest value, the three quartile checkpoints and the largest value.
Grade 66.SP.B.4Statistics and Data
Definition of Box Plot
A box plot pictures how a set of numbers is spread out by using only five of them. A rectangle stretches from the first quartile to the third quartile, a line cuts it at the median, and two whiskers run out to the smallest and largest values.
The data are divided into four sections that each contain roughly a quarter of the values. A long section means those values are thinly scattered; a short one means they are bunched together. Individual values disappear, which is the price paid for a clean summary. Several box plots drawn on one scale make comparing groups quick, for example boys against girls, or this year against last.
Formula and notation
The summary has five numbers in order:
\[\text{min} \le Q_1 \le \text{median} \le Q_3 \le \text{max}\]
The width of the box is the interquartile range, \(\text{IQR} = Q_3 – Q_1\), and the full length from end to end is the range.
Key facts
- The rectangle spans the central half of the data, from Q1 up to Q3.
- Whiskers and box halves each carry roughly one fourth of the entries.
- The median line need not sit in the center of the box.
- A longer whisker shows more spread among the top or bottom quarter.
Box Plot example
Ten students tracked their weekly biking time in minutes and sorted it: 12, 15, 17, 20, 22, 25, 28, 31, 35, 44. List the minimum, the three quartile checkpoints and the maximum, then give the length of the box.
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The minimum is 12 and the maximum is 44.
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Ten values means the median sits between the fifth and sixth: \(\frac{22 + 25}{2} = 23.5\).
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The five smallest values have middle 17, so \(Q_1 = 17\). The five largest have middle 31, so \(Q_3 = 31\).
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Box length: \(31 – 17 = 14\).
The summary is 12, 17, 23.5, 31, 44, and the box is 14 units long.
Common mistake
Watch out
Pia looks at a box plot with long whiskers and concludes that half of the data lies in each whisker.
Why it is wrong
Length does not equal count. A whisker holds about a quarter of the values, exactly like each half of the box. A stretched whisker only says those values are scattered over a wider range.
Related terms and lessons
- Grade 6 · Statistical Questions and Data DisplaysLessonWorksheetQuiz
- Algebra 1 · Statistics: Center, Spread and Data DisplaysLessonWorksheetQuiz
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