Choosing and honest charts: Year 7 collecting and presenting data

Warm-up

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

(Unit 2, lesson 5) A price of 80 rises by 25%. What is the new price?
Answer
100. 25% of 80 is 20, and 80 + 20 = 100.
(Lesson 5 of this unit) A pie chart has sectors of 100°, 120° and 80°, and a fourth sector. What angle is the fourth?
Answer
60°. 100 + 120 + 80 = 300, and 360 − 300 = 60.
(Lesson 4 of this unit) Should the bars of a histogram of heights touch, or have gaps?
Answer
Touch. Each class begins exactly where the one before ends.

Words used in this lesson

misleading: giving a false idea, even if the numbers are right. For example: a bar chart whose axis starts at 78.
exaggerate: make a difference look bigger than it is. For example: bars that look three times as tall.
axis start: the value where an axis begins. For example: 0, or 78.
scale: the numbers and steps along an axis. For example: 0, 50, 100.
uneven scale: equal gaps on the page for unequal steps. For example: 1, 2, 3, 5, 10 evenly spaced.
fair: showing the data as it is. For example: bars that match the values.

The numbers are right. Is the chart?

A school poster says, “7C beat 7A hands down!” One bar is three times as tall as the other, and every printed number is correct. Predict first: could the real values behind those two bars be almost the same? Move the start of the axis and compare how tall the bars look with how large the values really are.

Value axis starts at 0

1.17C looks this many times as tall as 7A (to 1 d.p.)

1.17C is this many times 7A (to 1 d.p.)

Axis from 0: the 7C bar looks about 1.1 times as tall as the 7A bar. Its value is about 1.1 times as large.

The figure needs JavaScript. In words: three classes handed in 82, 86 and 90 per cent of their homework. With the axis starting at 0 the tallest bar is about 1.1 times as tall as the shortest. Starting at 40 it looks about 1.2 times, starting at 60 it looks about 1.4 times, and starting at 78 it looks exactly 3.0 times, though the real ratio of 90 to 82 is always about 1.1. From a raised start the axis is marked with a break and the words “axis starts at”.

Show me the steps

Axis starts at 0: the bars are 82, 86 and 90 out of 100 tall. 7C looks about 1.1 times as tall as 7A.
Axis starts at 40: the bars are 42, 46 and 50 above the axis. 7C looks about 1.2 times as tall.
Axis starts at 60: the bars are 22, 26 and 30 above the axis. 7C looks about 1.4 times as tall.
Axis starts at 78: the bars are 4, 8 and 12 above the axis. 7C looks 3.0 times as tall as 7A, but 90 is still only about 1.1 times 82.

The height of a bar should match its value. A bar chart's value axis starts at 0, so a bar twice as tall is twice as large. If the axis starts higher, heights compare the distances above the start, not the values.

Other things that mislead: an uneven scale (equal gaps on the axis for unequal steps), a missing title or unit, and a chart type that does not fit the data.

Back to the poster: with the axis starting at 78, the values 90 and 82 make bars of 12 and 4 units above it, which is three times the height for a difference of only 8 points.

Choosing the chart that fits

Different charts answer different questions. Three questions about the data and the question you want to answer decide which to draw. Send the situations through them.

Situation (1 to 7) 1

Show the chart hidden

not shown yetthe chart that fits

not shown yetbecause it is

Share of a farm's land used for rice, maize, beans and the rest: answer the three questions, then show the chart that fits.

The figure needs JavaScript. In words: when the question is about each group's share of one whole, such as a farm's land shared between crops, a pie chart fits. A change over time, such as a lake's temperature each day, fits a line graph. Counts you want to compare, such as the members of each club, fit a bar chart. Measurements grouped in classes, such as runners' times, fit a histogram. Counts of students in categories can be a pie chart when the question is about shares and a bar chart when it is about comparing sizes.

Show me the steps

Share of a farm's land: each part's share of one whole, not over time, not grouped. Pie chart.
Lake temperature each day for 10 days: a change over time. Line graph.
Comparing the number of members in each of six clubs: separate groups to compare, not shares of one whole, and not over time. Bar chart.
Times of 40 runners in classes of 10 seconds: grouped measurements. Histogram.

Pie chart when the question is about each group's share of one whole; line graph for a change over time; histogram for measurements grouped in classes; bar chart for separate groups or counts you want to compare.

Counts of students in categories can go in a pie chart when the question is about shares, and in a bar chart when it is about comparing sizes.

A line graph may start its axis above 0 if the axis is marked, because the question is about change. A bar chart should not, because its bars are compared by height.

A common mistake: “Every number on the chart is correct, so the chart is honest.” The numbers can all be right and the picture can still give a false idea of how big a difference is. Check where the axis starts and whether the steps are equal.

Worked example: undoing a misleading chart

Three classes spend 50, 52 and 54 minutes reading each week. A bar chart starts its axis at 48, with 1 cm for each minute above 48, so the bars are 2 cm, 4 cm and 6 cm. How does the chart mislead, and how should it be drawn?

The third bar is 6 ÷ 2 = 3 times as tall as the first, though its value is only 54 ÷ 50 = 1.08 times as large.
Start at 0 with 1 cm for each 10 minutes: the bars are 50 ÷ 10 = 5 cm, 5.2 cm and 5.4 cm.
Now the bars are almost the same height, as the values are. 5 cm, 5.2 cm and 5.4 cm.

Your go. The first step is given. A newspaper chart has its axis start at 150 visitors. The bar for 200 visitors is 5 cm tall. How tall is the bar for 250 visitors, and how many times as tall does it look as the first?

The 200 bar is 50 above the axis start and 5 cm tall, so 1 cm stands for 10 visitors.
Now find the height of the 250 bar and compare, then check.

Check the answer

250 is 100 above the start, so 10 cm tall. It looks 10 ÷ 5 = 2 times as tall, but 250 ÷ 200 = 1.25 times as large.

Check before you go on

1. A bar chart's vertical axis runs from 70 to 100, and one bar looks three times as tall as another. What is the likely problem?

The bars are too wide The axis does not start at 0, so heights no longer match the values A pie chart was needed The printed values must be wrong

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A teacher wants to show what share of a class's homework time goes on maths, science, languages and art. Which chart fits best?

Bar chart Pie chart Line graph Histogram

3. On a bar chart the value axis starts at 30. A bar for 90 and a bar for 60 are drawn. How many times as tall does the 90 bar look as the 60 bar?

Level 2 · Building

4. The horizontal axis of a line graph is labelled 1, 2, 3, 5, 10 with equal space between every label. What is wrong?

Nothing is wrong, because labels on an axis should be whole numbers. The steps are unequal (1, 1, 2, 5) but drawn as equal gaps. The axis should show only even numbers, such as 2, 4, 6 and 10. The graph should start at 0, because every graph must start at 0.

5. A shop's sales rose from 60 to 90 units. On an honest bar chart, starting at 0, the 90 bar is 9 cm tall. How tall should the 60 bar be, in centimetres?

6. Zainab draws a bar chart of three test scores, 71, 74 and 77, with the axis running from 70 to 80. She says, “The last score is seven times the first.” What would you tell her?

She is right, because the last bar is seven times as tall as the first, so the score is too. The axis starts at 70, so heights are distances above 70: 77 is only about 1.1 times 71. She should draw a pie chart instead, because the scores are parts of one whole. Three bars is too few for a chart, so she needs many more scores before drawing one.

Level 3 · Stretch

7. A bar chart's value axis starts at 80 and uses the same scale all the way up. The bar for 84 is 4 cm tall. How tall is the bar for 92, in centimetres?

8. Two charts show the same four values, which lie between 92 and 97. Chart X has a value axis from 0 to 100. Chart Y has a value axis from 90 to 100. Which statement is best?

Both are equally fair, because the numbers in both charts are the same. Chart X is fair; chart Y exaggerates the differences. Chart Y is better, because its bars are taller and clearer to see. Neither can be compared with the other, because the axes are different.

Something to think about

A bar chart of daily temperatures in a city shows values between 21 and 24 degrees. A student starts the axis at 20 to make the differences clear. Is that dishonest? What would you add so a reader is not misled?

One way to think about it

It depends on whether the reader is told. A chart that starts at 20 shows the small differences clearly, but it makes 24 look like four times 21's height above the axis (4 against 1). Label the axis start clearly, say that it does not start at 0, or use a line graph where a higher start is common and expected.

Exit ticket

9. A bar chart's value axis starts at 40. The bars for 45 and 55 are drawn. Which statement is true?

The chart is fair, because the numbers are printed on the bars. The 55 bar is 3 times as large as the 45 bar, because it is 3 times as tall. The 55 bar looks 3 times as tall, but 55 is only about 1.2 times 45. The 55 bar looks about 1.2 times as tall as the 45 bar, as the values do.