Grade by Grade
D. Data · Data and Probability

Mean and Mode: What Is Typical?

D1.4About 45 minutes
Learning goals
  • I can find the mode (or modes) of a set of data, or tell that there is no mode.
  • I can find the mean by levelling out (evening out) the data.
  • I can find the mean by adding all the numbers and sharing the total equally.
  • I can explain what the mean and the mode tell us about the data.

Words to know

mode
The value or category that appears most often. Data can have one mode, more than one mode, or no mode.
mean
The fair share: what each would get if all the data were evened out so every value is the same.
level out
Move amounts from bigger values to smaller values until they are all the same.
data set
A group of data, like 2, 6, 4.
value
One piece of data in a data set, like the 6 in 2, 6, 4.

Remember from Grade 2

The mode is the answer or number that appears most often.

  • 3, 5, 5, 7: the mode is 5.
  • 2, 2, 4, 4, 6: there are two modes, 2 and 4.
  • 1, 3, 6, 8: no mode, because nothing appears more than once.

A fair share of apples

Three friends went apple picking at an orchard in the fall. Amir picked 2 apples, Bella picked 6, and Cole picked 4.

They decide to share fairly, so each friend has the same number. How many apples does each friend get?

The answer to this question is called the mean. The mean is the fair share.

Three wicker baskets holding different numbers of red apples in an apple orchard in the fall.
Each basket holds a different number of apples. Sharing them out so every basket has the same amount gives the mean, the fair share. AI-generated image

Way 1: Level it out

Make a cube tower for each friend. Then move cubes from tall towers to short towers until all the towers are the same height.

Bella's tower has 6. Move 2 cubes from Bella's tower to Amir's tower. Now Amir has 4, Bella has 4 and Cole has 4. All the towers are level!

264444BeforeAfter levelling
Before: towers of 2, 6 and 4 cubes. After levelling: 4, 4 and 4. The mean is 4.

Way 2: Add, then share

  1. Add all the values: 2 + 6 + 4 = 12 apples in all.
  2. Share the total equally among the 3 friends: 12 shared by 3 is 4 each (4 + 4 + 4 = 12).

Both ways give the same answer: the mean is 4.

Worked example

Example 1: Mean of four numbers

Four children made snowballs at recess: 5, 3, 8 and 4. What is the mean? What is the mode?

  1. Add: 5 + 3 = 8, 8 + 8 = 16, 16 + 4 = 20 snowballs.
  2. There are 4 children. Share 20 equally by 4: 5 + 5 + 5 + 5 = 20, so each gets 5.
  3. Check by levelling: the 8 gives 1 to the 4 and 2 to the 3. Now 5, 5, 5, 5. ✓
  4. Mode: every value appears once, so there is no mode.

The mean is 5 snowballs. There is no mode.

Worked example

Example 2: Mean and two modes

Our hockey team's goals in 5 games: 2, 3, 2, 5, 3. Find the mean and the mode.

  1. Mode: 2 appears twice and 3 appears twice. 5 appears once. So there are two modes: 2 and 3.
  2. Mean: add 2 + 3 + 2 + 5 + 3 = 15 goals.
  3. Share 15 equally among 5 games: 3 + 3 + 3 + 3 + 3 = 15, so 3 each.

Mean: 3 goals per game. Modes: 2 and 3.

What do they tell us?

  • The mode tells us the most common value: what happened most often.
  • The mean tells us the fair share: what each would have if everything were evened out.

Sometimes they are the same, and sometimes they are very different.

Worked example

Example 3: When the mean and mode are different

Four friends read books over the winter break: 1, 1, 1 and 9 books. Find the mean and the mode. What does each tell us?

  1. Mode: 1 appears three times. The mode is 1.
  2. Mean: 1 + 1 + 1 + 9 = 12. Share by 4: 3 + 3 + 3 + 3 = 12. The mean is 3.
  3. The mode tells us most friends read just 1 book.
  4. The mean tells us that if the books were shared out evenly, each friend would have 3. The one friend who read 9 books pulls the mean up.

Mode: 1. Mean: 3. The mode describes most of the friends. The mean shows the fair share.

1119mean = 36 extracubes: 2 foreach friendmode = 1 (three towers are 1 tall)books read by 4 friends
Levelling 1, 1, 1 and 9 books. The tall tower has 6 cubes above the dashed line, enough to give 2 to each short tower. Then every tower is 3 tall, so the mean is 3, but the mode is 1.

Good to know

  • The mean is not always one of the numbers in the data. In 1, 1, 1, 9, the mean is 3.
  • Sometimes sharing does not come out even. In Grade 3, our data sets will share out evenly.
  • The mean is always between the smallest and the biggest value (or equal to them if all the values are the same).
Just for fun

Why is the mean never mean?

Because it always shares fairly!

Hands-on activity

Level the towers

You need: about 30 LEGO bricks, linking cubes, coins or dry pasta pieces.

  1. Make 3 or 4 towers or piles of different sizes, like 3, 7 and 5. Write the numbers down.
  2. Find the mode, if there is one.
  3. Predict the mean.
  4. Level them out: move pieces from big piles to small piles until all are the same. Count one pile. That is the mean.
  5. Check by adding all the pieces and sharing them equally.
  6. Try again with new piles. Can you make piles whose mean is 5?

Math talk: "How do I know my piles are fair now?"

Grown-up note: small bricks, coins and pasta are a choking hazard for babies and toddlers. Keep them out of reach of younger children and put them away after playing.

Your turn

1

Find the mean of 3, 5, 7.

2

Find the mode of 4, 6, 4, 9, 4, 2.

3

Find the mean of 6 and 2.

4

Find the mean and the mode of 1, 4, 4, 7.

5

Favourite mitten colours: red, blue, red, green, blue. What is the mode?

6

Three children picked pumpkins: 2, 2 and 8. Find the mean and the mode. Which one tells how many most children picked?

7

Find the mean of 10, 10, 10.

More practice: warm-up

1

Find the mean of 2 and 4.

2

Find the mode of 5, 3, 5, 1.

3

Find the mean of 4, 4, 4, 4.

4

Find the mode of 1, 2, 3, 4.

5

Two towers have 1 cube and 5 cubes. After levelling, how many cubes are in each tower?

More practice: on your own

1

Find the mean of 3, 9, 6.

2

Find the mean of 2, 4, 5, 9.

3

Find the mean and the mode of 7, 3, 7, 3, 5.

4

Snow fell on four days in January: 6 cm, 0 cm, 10 cm and 4 cm. What is the mean snowfall? Is there a mode?

5

In Example 1, the mean was 5 snowballs. Explain in your own words what that means.

Challenge

1

Three children have a mean of 6 stickers. Two of them have 4 and 8. How many does the third child have?

2

Make a data set of 4 numbers that has a mean of 5 and a mode of 4.

3

Ben scored 0, 0, 0 and 12 goals in 4 soccer games. He says, "I usually score 3 goals." Is that a fair thing to say? Use the mean and the mode.

Check your understanding

Quiz

1.

What is the mean of 2, 4, 9?

2.

What is the mode of 3, 8, 3, 6?

3.

Which is a way to find the mean?

4.

Two towers have 3 and 7 cubes. You level them out. How many cubes are in each tower?

5.

Data: 1, 1, 1, 9. Which is true?

6.

What does the mode tell us?

For parents and teachers

What your child is learning: to find the mean and the mode(s) of data sets with whole numbers, and to explain what each tells us (Ontario expectation D1.4). The mode was taught in Grade 2. The mean is new in Grade 3, and it is taught as a fair share: levelling out towers of cubes, or adding the values and sharing the total equally. (The median is not part of the Grade 3 expectations.)

Common difficulties:

  • Confusing mean and mode. Mode = most (both start with mo). Mean = fair share.
  • Dividing by the wrong number. Share by how many values there are, not by the biggest value.
  • Expecting the mean to be one of the data values.
  • Saying "no mode" is "0". No mode means no value repeats; 0 can be a real mode (as in 0, 0, 0, 12).

At home: level out real things: crackers on plates, coins in piles, LEGO towers. Ask, "What is the fair share?" Sharing fits nicely with the Grade 3 work on division.