Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 8: the median of 2, 5, 6, 9, 11, 14 is ?
Answer
7.5. The middle pair is 6 and 9, and halfway between them is 7.5. - From Lesson 1: in a table the running total of frequencies is 7, 19, 34, 40. Which value is the 20th?
Answer
In the third class. The running total passes 20 there, since 19 is below 20 and 34 is above it. - From Year 7: halfway between 30 and 40 on a number line is ?
Answer
35.
Words used in this lesson
- cumulative frequency
- a running total of the frequencies up to the end of a class. For example: how many values are at or below a boundary.
- upper boundary
- the top end of a class, where its running total is plotted. For example: the right-hand end of the interval.
- lower quartile
- the value with a quarter of the data below it. For example: read at a count of n divided by 4.
- upper quartile
- the value with three quarters of the data below it. For example: read at a count of 3n divided by 4.
- interquartile range
- the upper quartile minus the lower quartile; the spread of the middle half. For example: the width of the box in a box plot.
- interpolate
- find a value between two plotted points in proportion. For example: a count a third of the way up gives a value a third of the way along.
The ferry queue
At a ferry terminal in Istanbul, 40 passengers had their waiting times recorded. The times are grouped in classes 10 minutes wide. Someone asks: “How long did the quarter of passengers who waited least have to wait at most?” The table gives counts for each class, not individual times.
Predict first: do you need the raw times to answer, or can a running total of the counts give it? Where on a graph of running totals would the answer be? Use the first figure to test your ideas, then the second to see what happens to the spread when one class is stretched.
0 values lie below 0 minutes.
Show me the steps
- Slide to 10: a quarter of the 40 values lie below 17.5 minutes. That is the lower quartile, found between the points (10, 4) and (20, 12).
- Slide to 20: half the values lie below 25 minutes, the median, found between (20, 12) and (30, 28).
- Slide to 30: three quarters of the values lie below 32.5 minutes, the upper quartile. The interquartile range is 32.5 − 17.5 = 15.
Plot running totals at the upper boundary, then read across and down.
- Cumulative frequency is the running total: how many values are at or below the end of a class.
- Each total is plotted at the upper boundary of its class, and the points are joined with straight lines.
- For n values, read the lower quartile at n ÷ 4, the median at n ÷ 2 and the upper quartile at 3n ÷ 4.
- The interquartile range is upper quartile − lower quartile: the width of the middle half of the data.
Between two plotted points, the position is found in proportion: a count a quarter of the way up gives a value a quarter of the way along.
The last class ends at 50 minutes: the range can be as large as 50, and the interquartile range is 32.5 − 17.5 = 15.
Show me the steps
- The last class ends at 50: the greatest possible range is 50 − 0 = 50, and the interquartile range is 32.5 − 17.5 = 15.
- Slide to 90: the greatest possible range is now 90, but the lower and upper quartiles are the same, so the interquartile range is still 15.
- The last class holds only 4 of the 40 values, so the extreme values change the range a great deal and the quartiles not at all.
Read the count, not the class. The vertical axis holds counts, the horizontal axis holds values. A common slip is to read a quartile off the vertical axis and call that number the answer. A count of 35 on the vertical axis is never a value; go across to the curve and then down. And plot at the end of the class: the running total has not been reached by the start of its own class.
Worked example: ages of museum visitors
60 visitors to a museum in Mexico City were grouped by age. The running totals are 10 by age 20, 30 by age 40, 50 by age 60 and 60 by age 80. Find the median and the interquartile range.
- Plot (20, 10), (40, 30), (60, 50) and (80, 60), starting from (0, 0).
- The median is at 60 ÷ 2 = 30 values. The curve is at 30 when the age is 40, so the median is 40.
- The lower quartile is at 15 values: between (20, 10) and (40, 30), a quarter of the way along, so the age is 25.
- The upper quartile is at 45 values: between (40, 30) and (60, 50), three quarters of the way, so the age is 55.
- The interquartile range is 55 − 25 = 30.
Your go. The first step is given. 80 lengths of rope offcuts, in cm, have running totals of 6 by 10, 26 by 20, 46 by 30 and 80 by 40. Estimate the median length.
- The median is at 80 ÷ 2 = 40 values. Which two plotted points is a count of 40 between?
- Find how far along that stretch the count 40 is, then use the same fraction of the class width.
Check the answer
The count 40 is between the points (20, 26) and (30, 46), since 46 is the first total above 40. It is 14 of the 20 values along, so the length is 20 + 10 × 14 ÷ 20 = 27 cm. A common slip is to use the next class, 30 to 40, because 40 is the count.
Check before you go on
1. The class 20 to under 30 has frequency 9, and the running total of the classes before it is 15. Which point is plotted for this class on a cumulative frequency graph?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A table has frequencies 6, 9, 11 and 4 for four classes in order. What is the cumulative frequency at the end of the third class?
3. A cumulative frequency curve is drawn for 80 values. At what count on the vertical axis do you read the lower quartile?
Level 2 · Building
4. A cumulative frequency graph is a straight line between (20, 12) and (30, 32), where the first number is the value and the second is the count. At what value is the count 17?
5. From a cumulative frequency curve, the lower quartile is 34, the upper quartile is 51 and the range is 60. What is the interquartile range?
Level 3 · Stretch
6. A cumulative frequency table for 60 values gives: up to 20, 14 values; up to 40, 22; up to 60, 38; up to 80, 52; up to 100, 60. Assume straight lines between points, starting from 0 at value 0. Estimate the interquartile range.
7. Using the same cumulative frequency table, estimate how many of the 60 values are above 90.
Something to think about
Two cumulative frequency curves are drawn on one pair of axes. Curve P rises steeply in the middle and curve Q rises gently all the way. Which group has the larger interquartile range, and what does the steep part of a curve tell you about the data?
One way to think about it
Q has the larger interquartile range. A steep part means many values crowd into a short stretch of the horizontal axis, so the values are packed closely together there. A gentle part means the values are spread out. The quartiles of P are close together; those of Q are far apart.
Exit ticket
8. A cumulative frequency curve for 120 values passes through (10, 18) and (20, 38), straight between them. Estimate the lower quartile.