Box Plot
In short: Box plot pictures a data set with only five landmarks: the smallest value, three cut points and the largest, drawn as a rectangle with a divider and two tails.
6th GradeStatistics6.SP.B.4
What is Box Plot?
A box plot squeezes a whole data set into five numbers drawn over a number line. The rectangle covers the central half of the values, the divider inside marks the median, and the tails reach the smallest and largest values.
Precisely, the five numbers are the minimum, the first quartile \(Q_1\), the median, the third quartile \(Q_3\) and the maximum. Each quarter of the data lies in one section of the plot, even though the sections may be different lengths.
How to spot it
- A graph shows a rectangle with a line inside and two lines extending left and right from it.
- A question gives data and asks for the median, quartiles, range or spread.
- Two groups are compared by stacking their plots on one number line.
- The box is short on one side of the median and long on the other, hinting the data are lopsided.
The method
- Sort the values from smallest to largest, then locate the middle one.
- Take the middle of the bottom half as \(Q_1\) and the middle of the top half as \(Q_3\).
- Mark all five numbers on one scale, then outline a rectangle spanning \(Q_1\) to \(Q_3\) and cut it at the median.
- Extend tails to the minimum and to the maximum, and read the spread off the picture.
Box Plot: solved example
Problem
A theater sold the following numbers of tickets over nine nights: 8, 11, 13, 13, 16, 17, 20, 22, 25. Find the five numbers for a box plot.
Solution
The middle is the 5th value: 16. Lower half 8, 11, 13, 13 has median \(\frac{11 + 13}{2} = 12\), so \(Q_1 = 12\). Upper half 17, 20, 22, 25 has median \(\frac{20 + 22}{2} = 21\), so \(Q_3 = 21\).
The five numbers are 8, 12, 16, 21, 25.
Try it
Question 1
A plot has minimum 3, \(Q_1 = 7\), median 10, \(Q_3 = 18\), maximum 31. What is the range?
Show the answer
\(31 – 3 = 28\).
Question 2
In that plot, what fraction of the data lies between 7 and 18?
Show the answer
The box holds the middle half, \(\frac{1}{2}\) of the values.
Box Plot vs. Histogram
A box plot shows five summary numbers and nothing about individual values or the exact count in a bin. A histogram shows the count in each interval but not the quartiles. The first compares groups well; the second shows shape and peaks.
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Whole dictionary: A–Z · 6th Grade · Statistics