Chapter 3 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

M3.7.3 Chapter 3

Mean, median and mode

Five friends have 2, 3, 3, 4 and 8 pencils. Shared out equally, how many pencils does each friend get?

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

  • Find the mean, median and mode of a set of numbers
  • Find the range to see how spread out the data is

A list of numbers can be long. An average is one number that stands for the whole list. There are three common averages: the mean, the median and the mode.

Five friends have 2, 3, 3, 4 and 8 pencils. Show each friend as a tower of cubes, one cube for each pencil. Share the cubes out until all towers are level. That height is the mean.

Draw it first

Level the towers to find the mean

  1. Step 1: five towers of cubes. Amina 2, Ravi 3, Mei 3, Kofi 4, Leila 8. Step 2: the same cubes shared out so every tower is level. Each tower has 4 cubes. The cubes that moved are hatched.2Amina3Ravi3Mei4Kofi8Leila

    1 Build a tower for each friend. The tallest tower has 8 cubes and the shortest has 2.

  2. Step 1: five towers of cubes. Amina 2, Ravi 3, Mei 3, Kofi 4, Leila 8. Step 2: the same cubes shared out so every tower is level. Each tower has 4 cubes. The cubes that moved are hatched.4Amina4Ravi4Mei4Kofi4Leila

    2 Move cubes from the tall towers to the short ones until all five towers are level. Each tower has 4 cubes.

To find the mean without cubes, add all the numbers. Then divide by how many there are. Here 20 ÷ 5 = 4. The mean is the total shared equally.

The median is the middle number when the list is in order. For 2, 3, 3, 4, 8 the middle one is 3. The mode is the number that appears most often. Here the mode is also 3. If no number appears more than once, there is no mode. If two numbers tie, both are modes: 1, 1, 2, 2 has the modes 1 and 2.

A dot plot of 2, 3, 3, 4, 8 pencils. The scale runs from 1 to 9. A dashed line marks median 3 pencils. The axis label is pencils.median 3 pencils123456789pencils
The dots stand in order along the line. Two values come before the middle one and two come after it. The middle value is the second dot at 3.

The range is the largest number take away the smallest. It is not an average. It tells you how spread out the numbers are.

Worked example

Find the mean, median, mode and range of the pencils 2, 3, 3, 4 and 8.

  1. Add the five numbers: 2 + 3 + 3 + 4 + 8 = 20.
  2. Mean: share the total between 5 friends. 20 ÷ 5 = 4.
  3. Median: the numbers are already in order. The middle one is 3.
  4. Mode: 3 appears twice, more than any other number. The mode is 3.
  5. Range: the largest is 8 and the smallest is 2. 8 − 2 = 6.

Answer: Mean 4, median 3, mode 3 and range 6.

A dot plot of 2, 3, 3, 4, 8 pencils with the scale from 1 to 9. Lines mark mean 4 pencils, median 3 pencils and mode 3 pencils. A bracket shows range 6 pencils. The axis label is pencils.median 3 pencilsmode 3 pencilsmean 4 pencils123456789range 6 pencilspencils
The mean, median and mode are not always the same. Here the mean is 4 and the median and mode are 3.

What if the list has an even number of values? Then there are two middle numbers. The median is halfway between them.

Worked example

Find the median of 6, 2, 9 and 4.

  1. Put them in order: 2, 4, 6, 9.
  2. Four numbers have two middle ones, 4 and 6. Add them: 4 + 6 = 10.
  3. Halfway between them is half of the sum: 10 ÷ 2 = 5.

Answer: 5

A dot plot of 2, 4, 6 and 9 with the scale from 1 to 9. A dashed line marks median 5, halfway between 4 and 6. The axis label is number.median 5123456789number
The median 5 is between the two middle dots. No dot sits on it.

Worked example

The mean of 4 numbers is 6. Three of them are 5, 7 and 4. Find the fourth number.

  1. The total of 4 numbers with mean 6 is 6 × 4 = 24.
  2. The three known numbers add up to 5 + 7 + 4 = 16.
  3. The fourth number is 24 − 16 = 8.
  4. Check: the mean of 5, 7, 4 and 8 is 24 ÷ 4 = 6.

Answer: 8

Four towers of cubes, each 6 cubes tall after sharing. Ana, Leila, Mei and Diego. The mean is 6.6Ana6Leila6Mei6Diego
Towers of 5, 7, 4 and 8 cubes, shared out. Each ends with 6 cubes, which is the mean.

See it three ways

With things

Ana, Mei and Kofi have 2, 4 and 6 beans. Put all the beans in one pile. Deal them out one at a time until the pile is gone.

As a picture

Three towers of 4 cubes each, after sharing. Ana, Mei and Kofi each have 4.4Ana4Mei4Kofi

In symbols

2 + 4 + 6 = 12. 12 ÷ 3 = 4. The mean is 4.

Try it

The mean length of names

You need: paper and a pencil, five names, such as your family or friends.

  1. Write five names. Count the letters in each name.
  2. Draw a tower of squares for each name, one square for each letter.
  3. Add up all the squares. Divide by 5.
  4. Say what number of letters a name has on average.

What you will see: The names Ana, Mei, Ravi, Leila and Diego have 3, 3, 4, 5 and 5 letters. That is 20 letters in all. The mean is 20 ÷ 5 = 4 letters.

If you cannot try it, this is what happens.
Five towers of cubes, one cube for each letter. Ana 3, Mei 3, Ravi 4, Leila 5, Diego 5.3Ana3Mei4Ravi5Leila5Diego

A wrong idea many people have

The median is the middle number of the list as it is written.

Why it feels right: The word "middle" makes you look at the middle of the page.

The test that breaks it: Take 8, 2, 9, 4, 6. The middle one as written is 9. Now put them in order: 2, 4, 6, 8, 9. The middle one is 6.

What is true: Put the numbers in order first. The median is the middle of the ordered list.

Key words

average
one number that stands for a whole list.
mean
the total shared equally: add them up, then divide by how many.
median
the middle number when the list is in order.
mode
the number that appears most often. A list can have no mode, or two.
range
the largest number take away the smallest.

Back to your guess

The answer is A: 4 pencils

The five friends have 2 + 3 + 3 + 4 + 8 = 20 pencils. Shared equally between 5 friends, each gets 20 ÷ 5 = 4 pencils. That number is called the mean.

Check yourself

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

  1. Find the mean of 2, 3 and 10.

    Tap your answer.

    1. Not this one. That is the middle number, the median. The mean is the total shared equally: add them, then divide. Go to M3.7.3. Try another.

    2. Not this one. That is the total. You must still share it between the 3 numbers. Go to M3.7.3. Try another.

    3. Yes, that is right. 2 + 3 + 10 = 15. Shared between 3 numbers, 15 ÷ 3 = 5.

    See the answer

    5

    2 + 3 + 10 = 15. Shared between 3 numbers, 15 ÷ 3 = 5.

    • If you chose A: That is the middle number, the median. The mean is the total shared equally: add them, then divide. Go to M3.7.3.
    • If you chose B: That is the total. You must still share it between the 3 numbers. Go to M3.7.3.

  2. Find the median of 8, 2, 9, 4 and 6.

    Tap your answer.

    1. Not this one. That is the middle number as written, but the list is not in order. Order it first. Go to M3.7.3. Try another.

    2. Yes, that is right. In order: 2, 4, 6, 8, 9. The middle number is 6.

    3. Not this one. That is the mean. The median is the middle of the ordered list. Go to M3.7.3. Try another.

    See the answer

    6

    In order: 2, 4, 6, 8, 9. The middle number is 6.

    • If you chose A: That is the middle number as written, but the list is not in order. Order it first. Go to M3.7.3.
    • If you chose C: That is the mean. The median is the middle of the ordered list. Go to M3.7.3.

  3. The dot plot shows the goals scored by a team in 8 matches. What is the mode?

    A dot plot of goals scored in 8 matches. The scale runs from 1 to 8. There is one dot at 2, three dots at 3, one at 4, two at 5 and one at 7. The axis label is goals.12345678goals
    Tap your answer.

    1. Yes, that is right. The tallest stack has 3 dots, at 3 goals. So the mode is 3.

    2. Not this one. That is the biggest number, not the most common. The mode is the tallest stack of dots. Go to M3.7.3. Try another.

    3. Not this one. There are 2 dots at 5, but 3 dots at another number. The mode has the most dots. Go to M3.7.3. Try another.

    See the answer

    3 goals

    The tallest stack has 3 dots, at 3 goals. So the mode is 3.

    • If you chose B: That is the biggest number, not the most common. The mode is the tallest stack of dots. Go to M3.7.3.
    • If you chose C: There are 2 dots at 5, but 3 dots at another number. The mode has the most dots. Go to M3.7.3.

  4. Find the range of 12, 5, 9, 20 and 7.

    Tap your answer.

    1. Not this one. That is the largest number. The range is the largest take away the smallest. Go to M3.7.3. Try another.

    2. Not this one. You added the largest and the smallest. The range is a difference, so take away. Go to M3.7.3. Try another.

    3. Yes, that is right. The largest is 20 and the smallest is 5. 20 − 5 = 15.

    See the answer

    15

    The largest is 20 and the smallest is 5. 20 − 5 = 15.

    • If you chose A: That is the largest number. The range is the largest take away the smallest. Go to M3.7.3.
    • If you chose B: You added the largest and the smallest. The range is a difference, so take away. Go to M3.7.3.

  5. The mean of 3 numbers is 6. Two of the numbers are 4 and 5. What is the third number?

    Tap your answer.

    1. Not this one. That is the mean, not the missing number. The total of the 3 numbers is 6 × 3 = 18. Go to M3.7.3. Try another.

    2. Yes, that is right. The total is 6 × 3 = 18. The two known numbers make 4 + 5 = 9. The third number is 18 − 9 = 9.

    3. Not this one. You added 4, 5 and 6 as if the mean were a third number. Find the total, 18, then take away 4 and 5. Go to M3.7.3. Try another.

    See the answer

    9

    The total is 6 × 3 = 18. The two known numbers make 4 + 5 = 9. The third number is 18 − 9 = 9.

    • If you chose A: That is the mean, not the missing number. The total of the 3 numbers is 6 × 3 = 18. Go to M3.7.3.
    • If you chose C: You added 4, 5 and 6 as if the mean were a third number. Find the total, 18, then take away 4 and 5. Go to M3.7.3.

  6. Find the mean of 3, 7, 4 and 6.

    See the answer

    5

    3 + 7 + 4 + 6 = 20. Shared between 4 numbers, 20 ÷ 4 = 5.

  7. Find the median of 9, 3, 7, 12 and 5.

    See the answer

    7

    In order: 3, 5, 7, 9, 12. The middle number is 7.

  8. From an earlier chapter

    Review: Hana asked 40 children which game they like best. 8 chose football. What is the angle of the football sector in a pie chart?

    See the answer

    72°

    One child is 360 ÷ 40 = 9°. Eight children is 8 × 9 = 72°.

Keep it

Copy this into your exercise book.

Mean: add up, then divide by how many. Median: the middle of the ordered list. Mode: the most common, if any. Range: largest take away smallest. The range shows the spread.

Draw: Draw towers of 2, 4 and 6 cubes. Next to them draw three towers of 4 cubes. Write mean = 4.

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.