Week 13 · Ascending cumulative frequency tables and their graphs · 3 lessons
3
Reading the curve: how many below, how many at least, what percent
Mathematics Prep 2 Term 1, Unit 4, Lesson 1
GoalRead the curve for the number below a value, the number at least a value and a percentage, and rebuild the grouped table from it.
New words
number below a value · go up from the value to the curve, then across to the vertical axisat least a value · the total minus the number below itpercentage · the number divided by the total, times 100estimate · between the plotted points we assume the values are spread evenly
Explanation
The curve of the cycling times of 40 pupils lets you answer questions quickly. To find how many took less than 25 minutes: go up from 25 on the horizontal axis to the curve, then straight across to the vertical axis. The curve is a straight line between (20, 12) and (30, 26), so at 25, half way, the height is 12 + 14 ÷ 2 = 19: about 19 pupils (an estimate, since we assume the pupils are spread evenly inside the class). At a plotted limit the answer is exact: less than 30 minutes, 26 pupils. To find how many took at least 30 minutes, subtract: 40 − 26 = 14. For a percentage divide by the total: less than 20 minutes is 12 ÷ 40 × 100 = 30%. The number between two limits is the difference of two readings: between 10 and 30 minutes is 26 − 4 = 22 pupils. Careful: ‘at least 30’ is not 26; 26 is the number below 30.
Worked examples
From the curve: how many pupils took less than 40 minutes? The point (40, 36) gives 36 pupils. How many took at least 40 minutes? 40 − 36 = 4 pupils. What percent is that? 4 ÷ 40 × 100 = 10%.
Between two limits. How many pupils took from 10 minutes up to less than 30? Read 26 at 30 and 4 at 10, and subtract: 26 − 4 = 22 pupils. This equals the frequencies of the two classes: 8 + 14 = 22.
Between plotted points. How many took less than 35 minutes? The curve is straight from (30, 26) to (40, 36): at 35 the height is 26 + 10 ÷ 2 = 31, about 31 pupils. This is an estimate; we cannot know the exact number from the grouped data.
Class activity
Ask the curve. Using your own class-survey curve, write three questions (how many less than…, how many at least…, what percent…) and answer them from the curve. Swap with a partner and check each other’s readings.
Practice
1
Use the cycling curve. How many pupils took less than 20 minutes? How many took at least 20 minutes?
12 pupils; 40 − 12 = 28 pupils.
2
What percent of the pupils took less than 30 minutes?
26 ÷ 40 × 100 = 65%.
3
How many pupils took from 20 minutes up to less than 40?
36 − 12 = 24 pupils.
4
Estimate how many pupils took less than 45 minutes.
Half way between 36 and 40: 38 pupils.
5
Dalia says: ‘26 pupils took at least 30 minutes.’ Correct her.
26 is the number below 30. At least 30 is 40 − 26 = 14.
