Week 13 · Ascending cumulative frequency tables and their graphs · 3 lessons
2
Drawing the ascending cumulative curve
Mathematics Prep 2 Term 1, Unit 4, Lesson 1
GoalDraw the ascending cumulative curve: plot each upper limit with its cumulative frequency, starting at height 0, and join the points.
New words
axes · limits on the horizontal axis, cumulative frequency on the vertical axispoint to plot · (upper limit, cumulative frequency)starting point · (first lower limit, 0)scale · equal steps on each axis, for example 10 minutes per square and 5 pupils per square
Explanation
The ascending cumulative curve shows the cumulative table as a picture. Time to cycle to school was recorded for 40 pupils: the classes 0–, 10–, 20–, 30–, 40–50 minutes have frequencies 4, 8, 14, 10 and 4, so the cumulative values are 0, 4, 12, 26, 36, 40. Step 1: draw the axes and choose an equal scale, here 10 minutes per square on the horizontal axis and 5 pupils per square on the vertical axis. Step 2: plot the points (upper limit, cumulative frequency): (10, 4), (20, 12), (30, 26), (40, 36), (50, 40). Plot at the UPPER limit, not the middle of the class, because the number 4 counts everything less than 10. Step 3: add the starting point (0, 0) at the first lower limit. Step 4: join the points in order with straight segments (or a smooth curve). The curve starts at 0, ends at the total and never goes down.
Worked examples
Plot the table for the times of 30 pupils in the classes 0–, 5–, 10–, 15–20 with the frequencies 3, 9, 12, 6. The cumulative values are 0, 3, 12, 24, 30, so the points are (0, 0), (5, 3), (10, 12), (15, 24), (20, 30). The last point has height 30, the total.
Common mistake. For the class 10–20 with cumulative value 12, a pupil plots (15, 12) at the middle of the class. That is wrong, because 12 pupils are less than 20, not less than 15. The correct point is (20, 12). Another mistake is to plot the frequency 8 instead of the cumulative value 12.
Checking a curve. The points (0, 0), (10, 6), (20, 5), (30, 14) cannot be an ascending cumulative curve: it goes down from 6 to 5. A cumulative frequency never decreases, so at least one value is a mistake. Also, the last height must equal the total number of values.
Class activity
Draw it. Use the table you made from your class survey. Draw the axes on squared paper, plot the points at the upper limits, start at 0, and join them. Does your curve go up steeply where the most pupils are?
Practice
1
The cumulative table has less than 10 → 5, less than 20 → 14, less than 30 → 27, less than 40 → 30 (and 0 at 0). List the points to plot.
(0, 0), (10, 5), (20, 14), (30, 27), (40, 30).
2
A pupil plots (5, 4) for the class 0–10 with cumulative frequency 4. What is the correct point?
(10, 4): plot at the upper limit.
3
Frequencies 6, 10, 14, 10 in the classes 0–, 5–, 10–, 15–20. Give the points of the curve.
(0, 0), (5, 6), (10, 16), (15, 30), (20, 40).
4
The curve of 40 values ends at height 38. What went wrong?
The last point must be at the total, 40, so a frequency was lost in the running total.
5
Can a cumulative curve have the points (10, 8) and (20, 6)?
No. It would go down, but a running total never decreases.
