Chapter 8 of 8

All 8 chapters
  1. 1Samples and bias, readunderstood
  2. 2Averages from frequency tables, readunderstood
  3. 3Cumulative frequency, readunderstood
  4. 4Box plots, readunderstood
  5. 5Histograms with unequal widths, readunderstood
  6. 6Scatter graphs and correlation, readunderstood
  7. 7Data over time, readunderstood
  8. 8Telling the truth with data, readunderstood

M4.7.8 Chapter 8

Telling the truth with data

A chart shows one tall bar and one short bar, but the two numbers are almost the same. Is the chart telling the truth?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Spot a misleading graph or a misleading average
  • Write a short report that compares two groups

Numbers do not lie, but the way we show them can mislead. A reader who knows the common tricks is hard to fool. This chapter shows three: a cut axis, an average that is not typical, and a comparison that tells only half the story.

Amina measures the mean height of the bean plants in two beds. Bed A has a mean of 52 cm and Bed B has a mean of 56 cm. Here is one chart of these numbers.

A bar chart with the y axis mean height (cm) from 50 to 60. Two bars: Bed A reaches 52 and Bed B reaches 56. The bars stand on a floor at 50, not at 0.505254565860Bed ABed Bmean height (cm)
Bed B looks three times as tall as Bed A. The axis starts at 50.

The bar for Bed A is only 2 units tall above the floor. The bar for Bed B is 6 units tall. So Bed B looks three times as tall. But 56 is only a little more than 52. The difference is 4 cm, which is about 8% of 52.

A bar chart of the same numbers with the y axis mean height (cm) from 0 to 60. Bed A reaches 52 and Bed B reaches 56. The bars are nearly the same height.0102030405060Bed ABed Bmean height (cm)
The same numbers, with the axis from 0. The bars are nearly equal.

In a bar chart the length of a bar shows the size. A cut axis breaks that link. A fair bar chart starts at 0. If an axis has to be cut, a zigzag break mark on the axis warns the reader.

The bar chart with the y axis mean height (cm) cut: 0 at the foot, then a zigzag break mark, then 50 to 60. Bed A reaches 52 and Bed B reaches 56.5052545658600Bed ABed Bmean height (cm)
A cut axis with a zigzag break. The break says that this axis does not start at 0.

Other tricks are common. Two charts side by side may use different scales. A chart may show a short stretch of time, to hide a longer story. A bar chart with classes of different widths, drawn with height as the frequency, makes wide classes look big. Always ask: what is the scale, and what is left out?

Now a trick with an average. A bus company times 7 journeys in minutes: 12, 13, 13, 14, 15, 16 and 85. The 85 minute journey was held up by a flood. A leaflet says: "the average journey is 24 minutes".

A bar chart with the y axis minutes and seven bars labelled 1 to 7, one for each journey. The first six bars are low: 12, 13, 13, 14, 15 and 16. Bar 7 is far taller, at 85.0204060801001234567minutes
One journey took 85 minutes. The other six took 16 minutes or less. The mean is 24 minutes.

Worked example

Find the mean, the median and the mode of the journey times 12, 13, 13, 14, 15, 16 and 85.

  1. Mean: 12 + 13 + 13 + 14 + 15 + 16 + 85 = 168. Then 168 ÷ 7 = 24.
  2. Median: the values are in order. The 4th of the 7 values is 14.
  3. Mode: 13 is the only value that appears twice.

Answer: The mean is 24 minutes, the median is 14 minutes and the mode is 13 minutes

A value far from the rest of the data is an outlier. The 85 is an outlier. It pulls the mean up. The median does not care how far away the largest value is.

Worked example

Which average best describes a typical journey: the mean or the median?

  1. Count the journeys shorter than the mean of 24 minutes. There are 6.
  2. So the mean is higher than six of the seven journeys. It is not typical.
  3. The 85 minute journey is an outlier. The median, the 4th value, is 14 minutes.
  4. Here, with an outlier in the data, the median describes a typical journey better.

Answer: The median, 14 minutes

Now the second half of the story. To compare two groups, one average is not enough. We need a number for the middle, the median, and a number for the spread, the IQR or the range. The two bus routes from the box plots chapter show this.

Two box plots on one scale, time (min), from 0 to 50. Route A has whiskers from 10 to 45 and a box from 18 to 32. Route B, drawn with hatching, has whiskers from 15 to 40 and a box from 22 to 28. Both have the median at 25.Route ARoute B01020304050time (min)
The same median. A much smaller box for Route B.

A short report has three parts. First, compare the middles with the medians. Second, compare the spreads with the IQRs or the ranges. Third, say what it means in plain words, with the numbers and the units.

Worked example

Write a short report comparing the journey times on Route A and Route B.

  1. The medians: both routes have a median of 25 minutes.
  2. The IQRs: Route A has 32 − 18 = 14 minutes. Route B has 28 − 22 = 6 minutes.
  3. The ranges: Route A has 45 − 10 = 35 minutes. Route B has 40 − 15 = 25 minutes.
  4. Report: both routes have a median of 25 minutes. Route B is more consistent, with an IQR of 6 minutes against 14.

Answer: Both routes have a median of 25 minutes, but Route B is more consistent: its IQR is 6 minutes against 14

Try it

Make one chart tell two stories

You need: squared paper, a ruler, a pencil.

  1. Take the counts of visitors in 5 weeks: 50, 51, 53, 52 and 55. Draw a bar chart with the axis from 0 to 60.
  2. Draw the same bar chart again with the axis from 48 to 56. Make both charts the same height on the page.
  3. Compare the two charts. Which one would you use to say "visitors are soaring"? Which is the fair one?

What you will see: The cut chart makes week 5 look more than three times week 1. The fair chart shows a gentle rise of 5 visitors in 4 weeks.

If you cannot try it, this is what happens.
A bar chart with the bars labelled Wk 1, Wk 2, Wk 3, Wk 4 and Wk 5, and the y axis visitors from 48 to 56 with no break mark. The bars stand at 50, 51, 53, 52 and 55. The y axis does not start at 0.4850525456Wk 1Wk 2Wk 3Wk 4Wk 5visitors

A wrong idea many people have

The mean is always the best average.

Why it feels right: The mean uses every value, so it seems the fairest.

The test that breaks it: In the journey times, the mean is 24 minutes, though six of the seven journeys took 16 minutes or less.

What is true: With an outlier, the median usually describes a typical value better. Say which average you used, and why.

A wrong idea many people have

Any chart whose axis does not start at 0 is dishonest.

Why it feels right: A cut axis is a favourite trick, so it is tempting to blame it every time.

The test that breaks it: A line graph of the temperature of a room, from 18 °C to 24 °C, needs no 0. Readers look at the changes along the line, not at the size of a bar.

What is true: A cut axis misleads most in a bar chart, where length means size. A cut axis should be marked with a break, and the numbers on it read.

Key words

outlier
a value that is far from the rest of the data.
misleading
giving a wrong idea, even if the numbers are true.

Back to your guess

The answer is A: Not always. A chart can cut off its axis, which makes a small difference look big.

A chart can be drawn from correct numbers and still mislead. If the axis starts above zero, a small difference can look large. Always read the scale. Charts can be trusted when they are drawn fairly, and we can learn to check.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The chart shows the mean jump heights of two classes. What is wrong with it?

    A bar chart with the y axis mean jump height (cm) from 40 to 44. Two bars: Class A reaches 41 and Class B reaches 43. The bars stand on a floor at 40, not at 0.4041424344Class AClass Bmean jump height (cm)
    Tap your answer.

    1. Not this one. That is what the chart seems to say. Read the scale: 43 cm is only 2 cm more than 41 cm. Go to M4.7.8. Try another.

    2. Yes, that is right. The bars start at 40 and not at 0. Class B is only a little higher than Class A, but its bar looks three times as tall.

    3. Not this one. The mean is a fine average for jump heights. What misleads here is the scale on the vertical axis. Go to M4.7.8. Try another.

    See the answer

    The axis is cut, so a small difference looks big.

    The bars start at 40 and not at 0. Class B is only a little higher than Class A, but its bar looks three times as tall.

    • If you chose A: That is what the chart seems to say. Read the scale: 43 cm is only 2 cm more than 41 cm. Go to M4.7.8.
    • If you chose C: The mean is a fine average for jump heights. What misleads here is the scale on the vertical axis. Go to M4.7.8.

  2. Waiting times in minutes are 4, 5, 5, 6, 7, 9 and 48. Which average best describes a typical wait?

    Tap your answer.

    1. Yes, that is right. The median is the 4th of the 7 values, 6. It is not pulled up by the outlier, 48.

    2. Not this one. The one wait of 48 minutes pulls the mean up. Six of the seven waits are under 12 minutes. Go to M4.7.8. Try another.

    3. Not this one. The range measures spread, not the middle. It is the largest value minus the smallest, 48 − 4. Go to M4.7.8. Try another.

    See the answer

    The median, 6 minutes

    The median is the 4th of the 7 values, 6. It is not pulled up by the outlier, 48.

    • If you chose B: The one wait of 48 minutes pulls the mean up. Six of the seven waits are under 12 minutes. Go to M4.7.8.
    • If you chose C: The range measures spread, not the middle. It is the largest value minus the smallest, 48 − 4. Go to M4.7.8.

  3. A leaflet says: "Route A has the lower mean time, so it is more reliable." Why is this not enough?

    Tap your answer.

    1. Not this one. The mean can be used for times. The problem is that one number cannot show how much the times vary. Go to M4.7.8. Try another.

    2. Not this one. A mean is one number. Some journeys on Route A can still be slow if its times are spread out. Go to M4.7.8. Try another.

    3. Yes, that is right. A mean says nothing about spread. Two routes with the same mean can have very different spreads.

    See the answer

    Reliable means little spread, so the IQRs or ranges must be compared too.

    A mean says nothing about spread. Two routes with the same mean can have very different spreads.

    • If you chose A: The mean can be used for times. The problem is that one number cannot show how much the times vary. Go to M4.7.8.
    • If you chose B: A mean is one number. Some journeys on Route A can still be slow if its times are spread out. Go to M4.7.8.

  4. The box plots show the heights of plants in two beds. Which statement is true?

    Two box plots on one scale, height (cm), from 0 to 70. Bed X has whiskers from 20 to 50 and a box from 28 to 42, with the median at 35. Bed Y, drawn with hatching, has whiskers from 25 to 60 and a box from 33 to 52, with the median at 45.Bed XBed Y010203040506070height (cm)
    Tap your answer.

    1. Not this one. The thick line of Bed X is further left than that of Bed Y. A higher median is further right on the scale. Go to M4.7.8. Try another.

    2. Not this one. The box of Bed Y is longer than the box of Bed X, so its IQR is bigger. The spreads are not the same. Go to M4.7.8. Try another.

    3. Yes, that is right. Bed Y has the median further right, and its box is longer. Its IQR is 52 − 33 = 19 against 42 − 28 = 14.

    See the answer

    Bed Y has the higher median and the larger spread.

    Bed Y has the median further right, and its box is longer. Its IQR is 52 − 33 = 19 against 42 − 28 = 14.

    • If you chose A: The thick line of Bed X is further left than that of Bed Y. A higher median is further right on the scale. Go to M4.7.8.
    • If you chose B: The box of Bed Y is longer than the box of Bed X, so its IQR is bigger. The spreads are not the same. Go to M4.7.8.

  5. Which of these can make a bar chart misleading?

    Tap your answer.

    1. Not this one. Labels and units help the reader. They do not mislead. Go to M4.7.8. Try another.

    2. Yes, that is right. A cut axis makes small differences look large, because bar length no longer shows the size.

    3. Not this one. A common scale is what makes bars fair to compare. Go to M4.7.8. Try another.

    See the answer

    Starting the vertical axis above zero.

    A cut axis makes small differences look large, because bar length no longer shows the size.

    • If you chose A: Labels and units help the reader. They do not mislead. Go to M4.7.8.
    • If you chose C: A common scale is what makes bars fair to compare. Go to M4.7.8.

  6. What is an outlier?

    Tap your answer.

    1. Yes, that is right. An outlier stands apart from the other values. It can pull a mean away from the typical value.

    2. Not this one. That is the mode. An outlier is not common. It stands apart from the others. Go to M4.7.8. Try another.

    3. Not this one. That is the median. An outlier is far from the middle. Go to M4.7.8. Try another.

    See the answer

    A value that is far from the rest of the data.

    An outlier stands apart from the other values. It can pull a mean away from the typical value.

    • If you chose B: That is the mode. An outlier is not common. It stands apart from the others. Go to M4.7.8.
    • If you chose C: That is the median. An outlier is far from the middle. Go to M4.7.8.

  7. From an earlier chapter

    The four quarterly readings of a year are 14, 26, 18, 10. What is the mean for the year?

    See the answer

    17

    14 + 26 + 18 + 10 = 68. Then 68 ÷ 4 = 17.

  8. Find the median of these 7 journey times, in minutes: 9, 10, 11, 12, 14, 15, 60.

    See the answer

    12

    The values are in order. The 4th of the 7 values is 12. The outlier, 60, does not change the median.

Keep it

Copy this into your exercise book.

Check the scale: a cut axis misleads. An outlier pulls the mean, so use the median. To compare groups, give the median and the IQR, with the units.

Draw: Two bar charts of the same two numbers, one from 0 and one cut, with the words "same numbers" between.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (1.1 MB). It opens on any phone.