Week 8 · Quartiles from a frequency table: ascending cumulative frequency, the quartile class and the formula, and the graphical method · 3 lessons
1
The ascending cumulative frequency table (Less than …) and the orders n ÷ 4, n ÷ 2 and 3n ÷ 4
Statistics Sec 3 Term 1, Unit 2, Lesson 2
GoalBuild the ascending cumulative frequency table (Less than …) and find the orders n ÷ 4, n ÷ 2 and 3n ÷ 4.
New words
class 5– · from 5 up to but not including 10 (the next class begins at 10)ascending cumulative frequency · how many values are less than a given class boundaryquartile class · the class that contains the place of the quartile
Explanation
When the data are in a frequency table with classes, the single values are lost, so the places (n + 1) ÷ 4 of the list rule cannot be counted one by one. Instead we use the places n ÷ 4, n ÷ 2 and 3n ÷ 4 and find them in the cumulative column. The ascending cumulative table says, for each class boundary, how many values are less than it. The first row is zero, because nothing is smaller than the lowest boundary. Each next row adds the class frequency to the row above. The picture shows the waiting times in minutes of 60 customers at a bakery counter (the data are invented). A class such as 5– means from 5 up to but not including 10. For the quartile class we look in the last column for the first row whose number is at least the place: that row’s class holds the quartile.
Class
Frequency
Less than
Cumulative frequency
0
0
0–
6
5
6
5–
14
10
20
10–
18
15
38
15–
12
20
50
20–
7
25
57
25–30
3
30
60
Worked examples
Building the table. The frequencies 6, 14, 18, 12, 7, 3 add up to 60 = n. The cumulative column starts at 0 and then reads 6, 6 + 14 = 20, 20 + 18 = 38, 38 + 12 = 50, 50 + 7 = 57 and 57 + 3 = 60. The last number must equal n: it is the check.
The orders. With n = 60 the orders are 60 ÷ 4 = 15, 60 ÷ 2 = 30 and 3 × 60 ÷ 4 = 45. Order 15: the first cumulative number at least 15 is 20 (less than 10), so the class 5– holds Q₁. Order 30: the first number at least 30 is 38, so class 10– holds Q₂. Order 45: the first number at least 45 is 50, so class 15– holds Q₃.
Class
Frequency
Less than
Cumulative frequency
0
0
0–
5
10
5
10–
12
20
17
20–
18
30
35
30–
10
40
45
40–50
5
50
50
Another table, with an order that is not a whole number. Fifty pupils’ cycling distances in a week (km): classes 0–, 10–, 20–, 30–, 40–50 with frequencies 5, 12, 18, 10, 5. Cumulative: 0, 5, 17, 35, 45, 50. The orders are 12.5, 25 and 37.5. Keep the decimals: 12.5 is in the class 10– (5 < 12.5 ≤ 17), 25 in 20– (17 < 25 ≤ 35), 37.5 in 30– (35 < 37.5 ≤ 45).
Class activity
Make up a table of 40 pupils’ times to get to school, with six classes of width 5 minutes and frequencies you choose. Build the cumulative column and find the three orders and the class that holds each quartile.
Practice
1
Frequencies 4, 9, 13, 8, 6 (n = 40). Write the cumulative column (with the first zero).
0, 4, 13, 26, 34, 40.
2
For n = 40 write the three orders.
10, 20 and 30.
3
Cumulative column 0, 4, 13, 26, 34, 40 for the classes 10–, 20–, 30–, 40–, 50–60. Which class holds each of the orders 10, 20 and 30?
Order 10: 4 < 10 ≤ 13, the class 20–. Order 20: 13 < 20 ≤ 26, the class 30–. Order 30: 26 < 30 ≤ 34, the class 40–.
4
A pupil finds the cumulative column 0, 6, 20, 38, 50, 57, 59 for frequencies that add up to 60. What does the last number tell you?
It should be 60. Since it is 59 there is an addition mistake somewhere.
5
What does the class 25– mean when the next class is 30–?
