Week 6 · Stem-and-leaf plots: building single and double plots with a key and reading the median, the mode and the range from them · 3 lessons
3
Double (back-to-back) plots to compare two data sets, with the advantages and the disadvantages of the method
Statistics Sec 3 Term 1, Unit 2, Lesson 1
GoalBuild and read a back-to-back stem-and-leaf plot to compare two data sets, and state the advantages and the disadvantage of the method.
New words
back-to-back plot · one column of stems in the middle, with the leaves of one set on the left and of the other set on the rightcompare · put two sets side by side and look at the centre (the median) and the spread (the range)
Explanation
To compare two sets measured in the same way we use one column of stems and two sets of leaves: the leaves of the first set on the left of the stem, the leaves of the second set on the right. On the left the leaves are written outward from the stem, so on both sides the leaf nearest the stem is the smallest. The row “8 4 | 3 | 1 5” holds 34 and 38 on the left and 31 and 35 on the right. The key says that the same stem and leaf mean the same number on both sides. To compare, look at the medians for the centre and at the ranges for the spread. The plot shows the marks out of 50 of two classes in the same quiz (the data are invented). Advantages: no value is lost, and two sets are easy to compare. Disadvantage: with a large amount of data the plot becomes too long to draw and to read.
key: the stem 3 with the leaf 8 is the mark 38, on either side
Class A leaves
Stem
Class B leaves
1
2 8
8 5 1
2
4 7 9
8 5 5 3 2
3
0 3 4 6 6 7 9
9 7 4 2 1
4
5
Worked examples
Reading the plot. Each class has 13 marks, so each median is the 7th value: class A 35, class B 33. The range of A is 49 − 21 = 28, and the range of B is 45 − 12 = 33. Class A has the higher centre; class B is more spread out.
key: the stem 2 with the leaf 4 is 24 °C, on either side
Town 1 leaves
Stem
Town 2 leaves
1
9
9 9 8 7 7 6 4
2
2 5 8
3 1 0
3
1 4 6 7 8
4
0
Two towns. The highest temperatures (°C) over ten days: town 1: 24, 26, 27, 27, 28, 29, 29, 30, 31, 33; town 2: 19, 22, 25, 28, 31, 34, 36, 37, 38, 40. The medians are (28 + 29) ÷ 2 = 28.5 and (31 + 34) ÷ 2 = 32.5. The ranges are 33 − 24 = 9 and 40 − 19 = 21. Town 2 is hotter in the middle and its temperatures are far more spread out.
A warning about the range. In class B the lowest mark 12 is far from the others (the next is 18, then 24). One unusual value makes the range 33, although most marks lie between 24 and 39. The range uses only two values. Next week we meet the quartiles, which use more of the data.
Class activity
Two groups of eight pupils each count how many times they can bounce a ball on the floor in 30 seconds. Write the two sets of counts, build a back-to-back plot, and compare the medians and the ranges.
Practice
1
In a back-to-back plot the left side of the row 4 reads “9 3 1”. Which numbers are these (the stem is 4)?
41, 43 and 49 (written outward from the stem, the leaf next to the stem is 1).
2
Group P: 14, 22, 27, 31, 35, 38, 44. Group Q: 19, 24, 26, 29, 30, 33, 36. Build the back-to-back plot and find the two medians.
Rows 1: P 4, Q 9; 2: P 7 2, Q 4 6 9; 3: P 8 5 1, Q 0 3 6; 4: P 4. Seven values each: the 4th value is 31 for P and 29 for Q.
3
Which of the two groups P and Q has the larger range, and by how much?
P: 44 − 14 = 30. Q: 36 − 19 = 17. P’s range is larger by 13.
4
Name one advantage and one disadvantage of a stem-and-leaf plot.
Advantage: every value is kept (and two sets can be compared). Disadvantage: it is too long for a large amount of data.
5
The two classes have medians 35 and 33. A pupil says “class A is better in every way”. Use the ranges to say what the plot does not show.
The medians only compare the centres. The lowest mark 12 of class B pulls its range to 33, so the ranges alone do not tell who is better; most of B’s marks (24 to 39) are close to A’s.
