Chapter 4 of 8

All 8 chapters
  1. 1Collecting data, readunderstood
  2. 2Bar charts and pie charts, readunderstood
  3. 3Mean, median and mode, readunderstood
  4. 4Comparing two groups, readunderstood
  5. 5Line graphs and honest charts, readunderstood
  6. 6The language of chance, readunderstood
  7. 7Working out chances, readunderstood
  8. 8Experiments and fair games, readunderstood

M3.7.4 Chapter 4

Comparing two groups

Two teams both score 4 goals a game on average. Are the teams the same?

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

  • Compare two sets of data using an average and the range
  • Choose the best average for a set of data

To compare two groups, use two numbers. An average tells you what is typical. The range tells you how spread out the data is. A small range means the data is steady.

Teams Red and Blue each played 7 matches. We put their goals on the same scale, so the two pictures can be compared.

Draw it first

Compare team Red and team Blue

  1. Step 1: a dot plot for team Red on a scale from 0 to 9, with dots at 3, 3, 4, 4, 4, 5, 5. Step 2: a dot plot for team Blue on the same scale, with dots at 0, 1, 2, 4, 5, 7, 9. Step 3: both plots show the mean and the range: Red has mean 4 goals and range 2 goals, and Blue has mean 4 goals and range 9 goals. The axis label is goals.Red0123456789goals

    1 Draw a dot plot for team Red. Put one dot for each match.

  2. Step 1: a dot plot for team Red on a scale from 0 to 9, with dots at 3, 3, 4, 4, 4, 5, 5. Step 2: a dot plot for team Blue on the same scale, with dots at 0, 1, 2, 4, 5, 7, 9. Step 3: both plots show the mean and the range: Red has mean 4 goals and range 2 goals, and Blue has mean 4 goals and range 9 goals. The axis label is goals.Red0123456789goalsBlue0123456789goals

    2 Draw team Blue on the same scale from 0 to 9.

  3. Step 1: a dot plot for team Red on a scale from 0 to 9, with dots at 3, 3, 4, 4, 4, 5, 5. Step 2: a dot plot for team Blue on the same scale, with dots at 0, 1, 2, 4, 5, 7, 9. Step 3: both plots show the mean and the range: Red has mean 4 goals and range 2 goals, and Blue has mean 4 goals and range 9 goals. The axis label is goals.Redmean 4 goals0123456789range 2 goalsgoalsBluemean 4 goals0123456789range 9 goalsgoals

    3 Mark the mean and the range on each plot.

Worked example

Red scored 3, 3, 4, 4, 4, 5 and 5 goals. Blue scored 0, 1, 2, 4, 5, 7 and 9 goals. Compare the two teams.

  1. Red: 3 + 3 + 4 + 4 + 4 + 5 + 5 = 28. The mean is 28 ÷ 7 = 4.
  2. Blue: 0 + 1 + 2 + 4 + 5 + 7 + 9 = 28. The mean is 28 ÷ 7 = 4.
  3. The range of Red is 5 − 3 = 2.
  4. The range of Blue is 9 − 0 = 9.

Answer: The means are equal, but Red is steadier. The ranges are 2 and 9.

Two dot plots on the same scale from 0 to 9. The axis label is goals. Red has dots at 3, 3, 4, 4, 4, 5, 5. Blue has dots at 0, 1, 2, 4, 5, 7, 9. Red: mean 4 goals, range 2 goals. Blue: mean 4 goals, range 9 goals.Redmean 4 goals0123456789range 2 goalsgoalsBluemean 4 goals0123456789range 9 goalsgoals
Same mean, different range. Red is steady. Blue is up and down.

Which average should you use? The mean uses every value, so one very big or very small value can pull it. An odd value like that is called an outlier.

Worked example

Seven children walk to school. Their times in minutes are 4, 5, 6, 6, 6, 7 and 22. Which average describes a typical walk best?

  1. Add the times: 4 + 5 + 6 + 6 + 6 + 7 + 22 = 56.
  2. Mean: 56 ÷ 7 = 8.
  3. Median: in order, the middle (4th) time is 6.
  4. The walk of 22 minutes is an outlier. It pulls the mean up. The median stays close to most children.

Answer: The median, 6 minutes, describes a typical walk better.

A dot plot of walking times with the scale from 0 to 22 minutes. The axis label is minutes. Lines mark mean 8 minutes and median 6 minutes. One dot at 22 stands far from the others.median 6 minutesmean 8 minutes0246810121416182022minutes
Six dots are close together. One dot, the outlier, is far away. The mean is dragged towards it.
A table with two columns, Average and Best when. mean: no outliers. median: an outlier. mode: most common.AverageBest whenmeanno outliersmedianan outliermodemost common
Mode is the only average that works for words, such as the colour chosen most often.

See it three ways

With things

Two groups of three jars hold beans. Group A has 4, 5 and 6 beans. Group B has 1, 5 and 9 beans.

As a picture

Two rows of towers of cubes, one cube for each bean. Group A has jar 1: 4, jar 2: 5, jar 3: 6. Group B has jar 1: 1, jar 2: 5, jar 3: 9.Group A4jar 15jar 26jar 3Group B1jar 15jar 29jar 3

In symbols

Both means are 15 ÷ 3 = 5. The ranges are 6 − 4 = 2 and 9 − 1 = 8.

Try it

Names against words

You need: paper and a pencil, a page of writing.

  1. Write the first names of 7 people. Count the letters in each name.
  2. Pick 7 words from a page. Count the letters in each word.
  3. Find the mean and the range of each group.
  4. Say which group is more spread out.

What you will see: Take seven first names with 3, 4, 4, 4, 4, 4 and 5 letters. The mean is 4 and the range is 2. Seven words with 2, 2, 3, 4, 4, 6 and 7 letters also have a mean of 4. Their range is 5.

If you cannot try it, this is what happens.
Two dot plots on the same scale from 0 to 9. The axis label is letters. Names has dots at 3, 4, 4, 4, 4, 4, 5: mean 4 letters, range 2 letters. Words has dots at 2, 2, 3, 4, 4, 6, 7: mean 4 letters, range 5 letters.Namesmean 4 letters0123456789range 2 letterslettersWordsmean 4 letters0123456789range 5 lettersletters

A wrong idea many people have

If two groups have the same mean, the groups are the same.

Why it feels right: The mean is one number that feels like the whole story.

The test that breaks it: Red and Blue both have a mean of 4 goals. Red never scored more than 5. Blue once scored 0 and once scored 9.

What is true: The mean says what is typical. The range says how spread out the data is. Use both.

Key words

spread
how far apart the values are, shown by the range.
outlier
a value far from the rest.

Back to your guess

The answer is A: No. The means are equal, but one team may be much less steady than the other.

The mean says what is typical. It does not say how spread out the scores are. Red scored 3 to 5 goals every match. Blue scored from 0 to 9. To compare groups, use an average and the range.

Check yourself

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

  1. Group A and group B both have a mean of 10. The range of A is 3. The range of B is 12. Which is true?

    Tap your answer.

    1. Not this one. A small range means the values are close together. Group A has the smaller range, so it is the steadier group. Go to M3.7.4. Try another.

    2. Not this one. The same mean does not make the same group. The ranges differ, so the groups differ. Go to M3.7.4. Try another.

    3. Yes, that is right. A bigger range means the values are more spread out. B has the bigger range.

    See the answer

    Group B is more spread out.

    A bigger range means the values are more spread out. B has the bigger range.

    • If you chose A: A small range means the values are close together. Group A has the smaller range, so it is the steadier group. Go to M3.7.4.
    • If you chose B: The same mean does not make the same group. The ranges differ, so the groups differ. Go to M3.7.4.

  2. Six children read for 5, 6, 6, 6, 7 and 8 minutes. A seventh reads for 46 minutes. Which average describes a typical reading time best?

    Tap your answer.

    1. Not this one. The mean of 12 is bigger than six of the seven times. The 46 minutes is an outlier and pulls the mean up. Go to M3.7.4. Try another.

    2. Yes, that is right. The median is the middle value, 6. The outlier 46 does not move it. The mean is 84 ÷ 7 = 12.

    3. Not this one. The range is not an average. It tells you the spread, not what is typical. Go to M3.7.3. Try another.

    See the answer

    The median, 6 minutes

    The median is the middle value, 6. The outlier 46 does not move it. The mean is 84 ÷ 7 = 12.

    • If you chose A: The mean of 12 is bigger than six of the seven times. The 46 minutes is an outlier and pulls the mean up. Go to M3.7.4.
    • If you chose C: The range is not an average. It tells you the spread, not what is typical. Go to M3.7.3.

  3. Group P and group Q both have a mean of 5. Which group is more spread out, and what is its range?

    Two dot plots on the same scale from 0 to 9. The axis label is number. Group P has dots at 4, 4, 5, 5, 5, 6, 6. Group Q has dots at 1, 3, 4, 5, 6, 7, 9.Group P0123456789numberGroup Q0123456789number
    Tap your answer.

    1. Yes, that is right. Group Q has dots from 1 to 9, so its range is 9 − 1 = 8. Group P has dots from 4 to 6, so its range is 2.

    2. Not this one. Group P has all its dots between 4 and 6. It is the steadier group. Look at the group with the dots furthest apart. Go to M3.7.4. Try another.

    3. Not this one. 9 is the largest value, not the range. The range is the largest take away the smallest: 9 − 1. Go to M3.7.3. Try another.

    See the answer

    Group Q, range 8

    Group Q has dots from 1 to 9, so its range is 9 − 1 = 8. Group P has dots from 4 to 6, so its range is 2.

    • If you chose B: Group P has all its dots between 4 and 6. It is the steadier group. Look at the group with the dots furthest apart. Go to M3.7.4.
    • If you chose C: 9 is the largest value, not the range. The range is the largest take away the smallest: 9 − 1. Go to M3.7.3.

  4. A school wants to know which shoe size most children wear. Which average should it use?

    Tap your answer.

    1. Not this one. The mean could be a size like 5.3, which no shoe has. It also does not tell you which size sold most. Go to M3.7.4. Try another.

    2. Not this one. The median is the middle size. The most popular size can be somewhere else. Go to M3.7.4. Try another.

    3. Yes, that is right. The mode is the value that appears most often. That is the size most children wear.

    See the answer

    The mode

    The mode is the value that appears most often. That is the size most children wear.

    • If you chose A: The mean could be a size like 5.3, which no shoe has. It also does not tell you which size sold most. Go to M3.7.4.
    • If you chose B: The median is the middle size. The most popular size can be somewhere else. Go to M3.7.4.

  5. Team X has a mean of 6 goals a match and a range of 2. Team Y has a mean of 6 goals a match and a range of 7. Which team scores more steadily?

    Tap your answer.

    1. Not this one. Team Y has the bigger range, so its scores jump about more. Steady means a small range. Go to M3.7.4. Try another.

    2. Yes, that is right. Both teams score about the same on average. Team X has the smaller range, so its scores stay closer together.

    3. Not this one. The means are equal, but the ranges are not. The range is the number that shows steadiness. Go to M3.7.4. Try another.

    See the answer

    Team X

    Both teams score about the same on average. Team X has the smaller range, so its scores stay closer together.

    • If you chose A: Team Y has the bigger range, so its scores jump about more. Steady means a small range. Go to M3.7.4.
    • If you chose C: The means are equal, but the ranges are not. The range is the number that shows steadiness. Go to M3.7.4.

  6. Both groups have a mean of 5. Group A is 4, 5 and 6. Group B is 1, 5 and 9. What is the range of the group that is more spread out?

    See the answer

    8

    Group B is more spread out. Its range is 9 − 1 = 8. Group A has a range of 6 − 4 = 2.

  7. Team A scores a mean of 7 goals a match. Team B scores a mean of 5. How many more goals a match does Team A score on average?

    See the answer

    2 goals

    Take the means away: 7 − 5 = 2.

  8. From an earlier chapter

    Review: find the median of 12, 4, 8 and 6.

    See the answer

    7

    In order: 4, 6, 8, 12. The two middle numbers are 6 and 8. Halfway between them is (6 + 8) ÷ 2 = 7.

Keep it

Copy this into your exercise book.

Compare groups with an average and the range. The same mean can hide a different spread. An outlier pulls the mean, so use the median. Use the mode for the most popular.

Draw: Draw two dot plots on one scale. Give them the same mean but different ranges. Write both ranges.

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 (968 KB). It opens on any phone.