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
1
The five-number summary and drawing a box plot from a list and from a stem-and-leaf plot
Statistics Sec 3 Term 1, Unit 2, Lessons 2–3
GoalFind the five numbers of a data set and draw its box plot to scale on a number line.
New words
five-number summary · minimum, Q₁, median Q₂, Q₃ and maximumbox plot · a box from Q₁ to Q₃ with a line at the median, and whiskers to the minimum and the maximumwhisker · the line from the box to the minimum, or to the maximum
Explanation
A box plot shows the shape of the data with only five numbers: the minimum, Q₁, Q₂ (the median), Q₃ and the maximum. To draw it: (1) find the five numbers with the quartile rule of the book; (2) draw a number line with an even scale that covers the data; (3) draw a box from Q₁ to Q₃; (4) draw a line across the box at Q₂; (5) draw a line (a whisker) from the box to the minimum and another to the maximum. Everything is to scale: the length of a box is the real distance Q₃ − Q₁. A quarter of the values lie in each part: from the minimum to Q₁, from Q₁ to Q₂, from Q₂ to Q₃ and from Q₃ to the maximum. Some books mark extreme values separately; in this book the whiskers go all the way to the minimum and the maximum. The picture shows the minutes 11 pupils needed for a puzzle (invented): 12, 16, 22, 30 and 41.
Worked examples
Minimum
Q₁
Q₂ (median)
Q₃
Maximum
12
16
22
30
41
From a list to the five numbers. The 11 times are 18, 25, 12, 30, 22, 16, 41, 20, 27, 14, 35. Ordered: 12, 14, 16, 18, 20, 22, 25, 27, 30, 35, 41. n + 1 = 12, the places are 3, 6 and 9: Q₁ = 16, Q₂ = 22, Q₃ = 30. The minimum is 12 and the maximum 41.
key 4|3 = 43
Stem
Leaves
4
3 7
5
2 5 8
6
0 1 4 6 8
7
0 3 5
8
1 8
From a stem-and-leaf plot. The plot shows 15 invented marks (key 4|3 = 43). It is already ordered, n + 1 = 16, so the places are 4, 8 and 12: Q₁ = 55, Q₂ = 64, Q₃ = 73. The minimum is 43 and the maximum 88. The box runs from 55 to 73 with the line at 64, and the whiskers reach 43 and 88.
Fractional places. Twelve invented marks: 8, 11, 14, 15, 17, 19, 20, 22, 23, 27, 30, 38. The places 3.25, 6.5 and 9.75 give Q₁ = 14 + 0.25 × 1 = 14.25, Q₂ = 19.5 and Q₃ = 23 + 0.75 × 4 = 26. On the scale we draw the box from 14.25 to 26, the line at 19.5, and whiskers to 8 and 38.
Class activity
Measure the arm spans of 11 pupils in centimetres, or count the letters of 11 words in a paragraph. Order the data, find the five numbers and draw the box plot on squared paper with an even scale.
Practice
1
Find the five numbers of 7, 9, 12, 15, 18, 20, 26.
