From a grouped table to an ascending cumulative table
Mathematics Prep 2 Term 1, Unit 4, Lesson 1
GoalRead a grouped frequency table, and build the ascending cumulative table by adding the frequencies as a running total.
New words
class · a group of values; the class 5– means from 5 up to but not including 10class limits · the lower limit starts the class and the next lower limit is its upper limitfrequency f · how many values are in a classcumulative frequency · a running total: how many values are less than a limit
Explanation
A grouped table sorts data into classes of equal length. The waiting times of 40 pupils at a school canteen are in the classes 0–, 5–, 10–, 15– and 20–25 minutes. The class 5– means times from 5 up to but not including 10, so a wait of exactly 10 minutes is in the class 10–. The table tells you how many values are in each class, but not exactly how many are less than 12 minutes, because we do not know where in the class 10– the values lie. The ascending cumulative table answers ‘how many are less than…’. Its rows are ‘less than 0’, ‘less than 5’, ‘less than 10’ and so on, the upper limits. The first row is 0 (nothing is less than the smallest limit). Then you add the frequencies as you go down: 0, 0 + 4 = 4, 4 + 9 = 13, 13 + 13 = 26, 26 + 10 = 36, 36 + 4 = 40. The last entry must equal the total, 40. Add the frequencies, not the limits.
Class
f
Upper limit
Cumulative f
less than 0
0
0–
4
less than 5
4
5–
9
less than 10
13
10–
13
less than 15
26
15–
10
less than 20
36
20–
4
less than 25
40
Worked examples
Upper limit
Cumulative f
less than 0
0
less than 10
3
less than 20
10
less than 30
22
less than 40
30
Build the cumulative table for the frequencies 3, 7, 12, 8 in the classes 0–, 10–, 20–, 30–40. Running totals: 0; 3; 3 + 7 = 10; 10 + 12 = 22; 22 + 8 = 30. So: less than 10 → 3, less than 20 → 10, less than 30 → 22, less than 40 → 30 (the total).
What the grouped table can tell. From the canteen table: the number of pupils who waited less than 15 minutes is 4 + 9 + 13 = 26, which is the cumulative entry. But the number who waited less than 12 minutes cannot be found exactly, because the 13 pupils of the class 10– are somewhere between 10 and 15.
Upper limit (kg)
Cumulative f
Class
f
less than 10
0
less than 20
5
10–
5
less than 30
17
20–
12
less than 40
41
30–
24
less than 50
48
40–
7
less than 60
50
50–
2
Rebuilding the frequencies. The masses of 50 boxes give the cumulative values 0, 5, 17, 41, 48, 50 for less than 10, 20, 30, 40, 50, 60 kg. Subtract each entry from the next: 5 − 0 = 5, 17 − 5 = 12, 41 − 17 = 24, 48 − 41 = 7, 50 − 48 = 2. So the frequencies of the classes 10–, 20–, 30–, 40–, 50–60 are 5, 12, 24, 7, 2.
Class activity
Class survey. Ask 20 classmates how many minutes they take to get to school, and put the answers into classes of 10 minutes (0–, 10–, 20–, …). Make the frequency table and then the cumulative table. Check that the last cumulative entry is 20.
Practice
1
Build the cumulative table for the frequencies 4, 6, 11, 9 in the classes 0–, 5–, 10–, 15–20.
0, 4, 10, 21, 30.
2
In the canteen table, how many pupils waited less than 20 minutes?
4 + 9 + 13 + 10 = 36.
3
The cumulative entries are 0, 6, 15, 29, 32 for less than 10, 20, 30, 40, 50. Find the frequencies.
6, 9, 14, 3.
4
The frequencies in a table are 5, 8, 12 and 5. What is the last cumulative entry?
The total: 30.
5
Omar says a wait of exactly 10 minutes is in the class 5–. Is he right?
No. The class 5– ends before 10, so 10 minutes is in the class 10–.
