Week 9 · Box plots and the semi-interquartile range: the five numbers, drawing and reading boxes, and (Q3 − Q1) ÷ 2 from data, stem-and-leaf plots and frequency tables · 3 lessons
2
Reading and comparing box plots: what the box and the whiskers show, and 50% of the data inside the box
Statistics Sec 3 Term 1, Unit 2, Lessons 2–3
GoalRead a box plot: the centre, the spread and the 50% of the data inside the box, and compare two box plots.
New words
box length · Q₃ − Q₁, the interquartile range; the middle half of the data lies in the boxcompare · compare the medians for the centre, the lengths of the boxes and of the whole plots for the spread
Explanation
A box plot is read from left to right like a scale. The line inside the box is the median: it tells where the middle of the data is. The box holds the middle half of all values (50%), so a long box means that the middle half is spread out and a short box means that it is packed together. Each whisker holds one quarter of the values. If the median line is nearer to one end of the box, the values on that side are crowded and the values on the other side more spread. To compare two data sets drawn on the same scale, compare first the medians (the centres), then the lengths of the boxes (the middle spread) and the whole lengths of the plots (the range). The picture shows the daily sales of two shops (invented): the lower box is shop 1 and the upper one is shop 2.
Worked examples
Reading one box (5, 12, 15, 25, 40). The median is 15. The box goes from 12 to 25, so its length is 25 − 12 = 13 and half of the values lie between 12 and 25. The line is close to the left end of the box and the right whisker is long: the lower values are crowded and the upper ones are spread out.
Comparing two shops. Shop 1: 20, 34, 42, 50, 66. Shop 2: 25, 36, 40, 46, 55. The medians 42 and 40 are close, so the centres are about the same. The box of shop 1 is 50 − 34 = 16 long and that of shop 2 is 46 − 36 = 10 long; the ranges are 46 and 30. So shop 1’s sales are more spread out, and shop 2 sells a steadier amount.
Counting from a box. A data set has 80 values. The box holds half of them, 40 values; each whisker holds a quarter, 20 values; and 20 values lie between the minimum and Q₁. If a pupil has a mark above Q₃, she is in the top quarter of the class.
Class activity
Draw on the same scale the box plots with the five numbers 30, 40, 50, 60, 70 and 35, 45, 50, 55, 65. They have the same median. Write two sentences to say how they differ.
Practice
1
Read the five numbers from the box plot. What is the range and what is the length of the box?
Five numbers 10, 20, 25, 40, 50. Range 50 − 10 = 40. Box length 40 − 20 = 20.
2
What percentage of the data lies between the minimum and Q₁? And between Q₁ and Q₃?
25% and 50%.
3
Class X has the five numbers 12, 20, 30, 44, 60 and class Y has 15, 24, 30, 38, 50. Which class has the more spread out middle half?
Class X: its box is 44 − 20 = 24 long and class Y’s is 38 − 24 = 14 long. The medians are equal (30).
4
A data set has 60 values. How many lie inside the box?
Half of them: 30.
5
In a box plot the median line is at the very right end of the box. What does this say about the values between Q₂ and Q₃?
Q₂ equals Q₃, so a quarter of the values are all packed at (or very near) the same number.
