Week 7 · Quartiles of single data and of stem-and-leaf plots: Q1, Q2 and Q3 by their order, with the rule for going between two values · 3 lessons
3
Quartiles from a stem-and-leaf plot, and finding the median in two ways
Statistics Sec 3 Term 1, Unit 2, Lesson 2
GoalFind the quartiles from a stem-and-leaf plot, and find the median in two ways.
New words
ordered leaves · the leaves of every row in increasing order, so the plot reads like an ordered listmedian in two ways · the mean of the two middle values, or the quartile rule at place (n + 1) ÷ 2
Explanation
A stem-and-leaf plot with ordered leaves is an ordered list, so quartiles are found in the same way: count the places from the first leaf of the top row, row after row. First check that the leaves are in order; if not, order them. Then find n and the three places, count to them in the plot and use the rule between two values. The plot below shows 16 invented marks of a quiz, with n + 1 = 17, so the places are 4.25, 8.5 and 12.75. The median can also be found in two ways, and they must agree: (1) the mean of the two middle values, here the 8th and the 9th; (2) the rule with the place (n + 1) ÷ 2 = 8.5, which is half way between the 8th and the 9th value, the same number.
key 2|3 = 23 marks
Stem
Leaves
1
4 5 7 9
2
2 3 3 6 8
3
1 1 4 7 8
4
2 5
Worked examples
Counting places in the plot of 16 marks. The rows hold 4, 5, 5 and 2 values: row 1 is 14, 15, 17, 19, row 2 is 22, 23, 23, 26, 28, row 3 is 31, 31, 34, 37, 38 and row 4 is 42, 45. The 4th value is 19 and the 5th is 22, so Q₁ = 19 + 0.25 × (22 − 19) = 19.75.
The median in two ways. The 8th value is 26 and the 9th is 28. (1) The mean: (26 + 28) ÷ 2 = 27. (2) The rule at place 8.5: 26 + 0.5 × (28 − 26) = 27. Both give Q₂ = 27. For Q₃ the place is 12.75: the 12th value is 34 and the 13th is 37, so Q₃ = 34 + 0.75 × (37 − 34) = 36.25.
key 1|0 = 10
Stem
Leaves
0
1 1 1 2 2 3 3
1
0 1 1 1 4
2
1 2 2
A plot with 15 values (key 1|0 = 10): rows 0 | 1 1 1 2 2 3 3, 1 | 0 1 1 1 4 and 2 | 1 2 2. Here n + 1 = 16, the places are 4, 8 and 12, which are whole numbers, so no rule between values is needed: Q₁ = 2 (the 4th value), Q₂ = 10 (the 8th), Q₃ = 14 (the 12th).
Class activity
Take the 16 marks of the plot and add the mark 50. How many values are there now? Find the new places of the quartiles and the new Q₁, Q₂ and Q₃. Which of the three changed most?
Practice
1
How many values are in the plot with rows 3 | 0 4 7, 4 | 1 1 5 8, 5 | 2 6? What are the three places?
3 + 4 + 2 = 9 values; n + 1 = 10: places 2.5, 5 and 7.5.
