Misleading graphs and reports: Year 11 Standard describing data

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

From Lesson 5: group A has IQR 8 and group B has IQR 11. Which group is more consistent?
Answer
A. A smaller IQR means the middle half of the values is less spread out.
From Year 8: what percentage is 15 of 60?
Answer
25%. 15 ÷ 60 = 0.25.
From Lesson 4: a bar is 10 wide and 1.5 tall on a density axis. What is its frequency?
Answer
15. 10 × 1.5.

Words used in this lesson

truncated axis: an axis that starts above zero, cutting off the bottom of the bars. For example: a bar chart whose axis starts at 40.
pictogram: a chart that uses pictures to stand for values. For example: icons of buses for numbers of passengers.
weighted mean: a mean in which each group counts according to its size. For example: the mean of all the plants, not of two group means.
rate: a count compared with the size of the group. For example: goals per match.
limitation: a reason a conclusion may not be fully reliable. For example: the data came from only one week.
impression: the idea a reader gets from looking at a graph. For example: a rise that looks large.

Two headlines

A newspaper in Manila prints two graphs of the same monthly sales figures. Over one graph the headline says sales “soared”. Over the other it says they “barely moved”. Every number on both graphs is correct.

Predict first: how could both headlines be supported by the same figures? What would you check on a graph before you trusted its headline? Use the first figure to move the start of an axis, then the second to regroup a list.

Where the vertical axis starts 0

1.08how many times as tall B looks

1.08how many times as large B really is

Axis from 0: the bars are 52 and 56 tall, B looks 1.08 times as tall as A, and B really is 1.08 times A.

The figure needs JavaScript. In words: two bars stand for 52 and 56. With the axis starting at 0, B looks 1.08 times as tall as A, which is the true ratio. With the axis starting at 40, the bars are 12 and 16 tall and B looks 1.33 times as tall. With the axis starting at 50, the bars are 2 and 6 tall and B looks 3 times as tall, though its value is only 1.08 times A's.

Show me the steps

Axis from 0: the bars are 52 and 56 tall. B looks 56 ÷ 52 = 1.08 times as tall as A, which is also the true ratio.
Slide to 40: the bars are 12 and 16 tall, so B looks 16 ÷ 12 = 1.33 times as tall, though the values have not changed.
Slide to 50: the bars are 2 and 6 tall, so B looks three times as tall as A, but B is still only 1.08 times as large as A.

Check the graph before you read the headline.

The axis: where does a bar chart's vertical axis start? Work out how tall each bar is drawn and compare that with the values.
The scaling: in a pictogram, is the value shown by an icon's height, or by the area the eye sees?
The grouping: a different class width can change the shape and the modal class of the same data.
The measure: an average of averages ignores group sizes, and a count is not a rate.

A short report has a question, how the data were collected, a centre and a spread for each group, a comparison in context, and a conclusion with a limitation.

Class width 10

5number of classes

7highest frequency

Class width 10 gives 5 classes. The tallest holds 7 values: 20 to under 30.

The figure needs JavaScript. In words: 24 values from 12 to 58 are grouped from 10 upward. With width 10 the five classes hold 6, 7, 6, 3 and 2 values, and the tallest is 20 to under 30. With width 5 two classes tie for the most with 4 values each. With width 20 the first class, 10 to under 30, holds 13 values, more than half.

Show me the steps

Width 10: five classes holding 6, 7, 6, 3 and 2 values. The tallest class is 20 to under 30, with 7.
Width 5: ten classes, and the highest frequency is 4, shared by 20 to under 25 and 30 to under 35. The single hump has broken into two.
Width 20: three classes holding 13, 9 and 2 values. The first class holds more than half of all the values, and the data seem to pile up on the left.

An honest number can sit in a dishonest picture. When a graph looks dramatic, work out the real ratio or the real percentage change from the numbers, and compare it with what the picture suggests. And do not accuse a graph of lying when it has not: a truncated axis can be reasonable on a line graph that is meant to show small changes, as long as the axis is labelled and the reader is told.

Worked example: a sales chart

A chart shows sales of 120 in one year and 126 in the next, as two bars on an axis that starts at 100. Describe the impression and the truth.

The bars are 20 and 26 tall, so the second looks 26 ÷ 20 = 1.3 times as tall, a 30% increase.
The real ratio is 126 ÷ 120 = 1.05, so the real increase is 5%.
The chart makes a small rise look six times as large as it is: 30% against 5%.

Your go. The first step is given. Class 1 has 20 students with a mean score of 60. Class 2 has 5 students with a mean score of 80. Find the mean score of all 25 students.

Total for class 1: 20 × 60 = 1200.
Find the total for class 2, add them, and divide by 25.

Check the answer

Class 2's total is 5 × 80 = 400, so the combined total is 1600 and the mean is 1600 ÷ 25 = 64. A common slip is to average 60 and 80 to get 70, which gives the small class the same weight as the large one.

Check before you go on

1. A bar chart's vertical axis starts at 50 instead of 0, and the bars are cut off at the bottom. What goes wrong?

The heights of the bars no longer match the ratio of the values. The values on the chart are wrong. Nothing: the labels give the same information either way. The bars become too wide.

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. Two bars stand for 40 and 50. The vertical axis starts at 30. How many times as tall does the second bar look as the first?

3. A report says: “Route Y's median wait is 3 minutes shorter than route X's, so Y is always faster.” It is based on 10 timed waits for each route. Which is the best limitation to add?

The medians should have been found on a calculator. Only 10 waits were timed on each route, so another sample could give a different result. No limitation is needed, because Y's median is lower. Route X's waits were longer, so X's data must be wrong.

Level 2 · Building

4. A bar chart has an axis that starts at 60. Its bars show 80 and 100. By what percentage is the second value really larger than the first?

5. A bakery's 10 morning customers spend a mean of 70 and its 30 afternoon customers spend a mean of 50. What is the mean spend of all 40 customers?

Level 3 · Stretch

6. A pictogram uses one icon for a value of 30. For a value of 60 the artist draws the icon twice as tall and twice as wide. How many times as large is the area of the 60 icon as the area of the 30 icon?

7. Town A has 1200 people and 90 of them are over 70. Town B has 8000 people and 520 are over 70. By how many percentage points is the share of over-70s in Town A higher than in Town B?

Something to think about

A chart shows one bar for each of five towns, and the reader is told that the biggest bar is the town with the most car thefts. What two pieces of information would you want before agreeing, and why?

One way to think about it

The number of people or cars in each town, because a big town will have more thefts without being less safe; and the time period and the source, because a bar for a short period cannot be compared with a bar for a long one. Counts compare well only when the groups are the same size, so a rate, such as thefts per thousand cars, is the fairer measure.

Exit ticket

8. A bar chart's vertical axis starts at 93. Its two bars show 96 and 108. Which statement is true?

The second value is 5 times the first. The second bar looks 5 times as tall, but the second value is only 12% larger. The second bar looks 5 times as tall, but the second value is only 12.5% larger. The second bar looks 1.125 times as tall, so the chart is fair.