Grade by Grade
D. Data · Data and Probability

Will Another Group Have the Same Mean and Mode?

D2.2 D1.4 A1About 40 minutes
Learning goals
  • I can explain what a population is.
  • I can predict whether another population is likely to have the same mean or mode as my data.
  • I can test my prediction by collecting data and finding the mean and mode.
  • I can explain why a prediction might not come true.

Words to know

population
The whole group of people or things you collect data about, like the children in Room 9.
prediction
A thoughtful guess about what will happen, based on what you know.
test
Collect data to check whether a prediction was right.
mean
The fair share when the data is evened out.
mode
The value or category that appears most often.

Remember from Grade 2

In Grade 2 you predicted whether another group would have the same mode as your class. In Grade 3 you'll predict and test the mean too, and use likelihood words like likely and unlikely in your predictions.

What is a population?

A population is the group you collect data from. It could be:

  • the children in your class
  • the grown-ups in your family
  • the trees in your yard

If you collect data from a different population, will you get the same mean and mode? Sometimes yes, sometimes no. Good data detectives predict, then test.

Think before you predict

Ask yourself: How are the two populations alike? How are they different?

  • If the groups are very alike (two Grade 3 classes in the same town), the mean and mode are more likely to be the same or close.
  • If the groups are very different (Grade 3 children and Kindergarten children), the mean and mode are less likely to be the same.
Worked example

Example 1: Teeth lost: Grade 3 and Kindergarten

Eight children in Room 9 (Grade 3) told how many baby teeth they have lost: 4, 6, 5, 4, 3, 8, 4, 6. Find the mean and mode. Then predict: will a Kindergarten class have the same mean and mode?

  1. Mode: 4 appears three times. The mode is 4.
  2. Mean: 4 + 6 + 5 + 4 + 3 + 8 + 4 + 6 = 40. Shared by 8 children: 5 each (5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 40). The mean is 5.
  3. Think: Kindergarten children are younger. Most children start losing baby teeth at around age 6, so many Kindergarten children have not lost any yet.
  4. Prediction: it is unlikely that the Kindergarten class will have the same mean and mode. Their mean and mode will probably be smaller.

Room 9: mean 5, mode 4. Prediction: the Kindergarten class will probably have a smaller mean and mode.

Worked example

Example 2: Test the prediction

Eight children in a Kindergarten class answered: 0, 0, 1, 0, 2, 0, 3, 2. Find the mean and mode. Was the prediction right?

  1. Mode: 0 appears four times. The mode is 0.
  2. Mean: 0 + 0 + 1 + 0 + 2 + 0 + 3 + 2 = 8. Shared by 8 children: 1 each.
  3. Compare: Grade 3 mean 5, mode 4. Kindergarten mean 1, mode 0.

The prediction was right: the Kindergarten mean and mode are both smaller.

PopulationData (teeth lost)MeanMode
Room 9 (Grade 3)4, 6, 5, 4, 3, 8, 4, 654
Kindergarten class0, 0, 1, 0, 2, 0, 3, 210
Comparing two populations. Source: interviews by Room 9.
Room 9 (Grade 3): mode 4012345678mean 5Kindergarten class: mode 0012345678mean 1number of baby teeth lost (each dot = 1 child)
Each dot is one child. The Room 9 dots sit far to the right of the Kindergarten dots, so their mean (5) and mode (4) are bigger than the Kindergarten mean (1) and mode (0).
Worked example

Example 3: When a prediction doesn't come true

In Room 9, the mode for favourite winter activity was skating. Rania predicts that Room 11, another Grade 3 class at the same school, will likely have the same mode. Room 11's data: skating 6, tobogganing 9, snow forts 4, skiing 5. Was she right?

  1. The two classes are alike: same grade, same school, same winter. So the same mode was a sensible prediction.
  2. Room 11's biggest frequency is 9, for tobogganing.
  3. Room 11's mode is tobogganing, not skating.

The prediction didn't come true, and that's okay. Likely is not certain. Small groups can be different just by chance. Rania made a good prediction and learned something by testing it.

Room 11: Favourite Winter Activity02468106Skating9Tobogganing4Snow forts5SkiingVotes
Room 11's data from Example 3. The tallest bar is tobogganing (9), so that is the mode, not skating. The scale counts by 2s; the dotted lines mark the odd numbers.
Worked example

Example 4: Same mean, different mode

Six children in Room 9 counted the letters in their first names: 4, 5, 3, 6, 5, 7. Six children in Room 11: 5, 4, 6, 4, 8, 3. Compare the means and modes.

  1. Room 9 total: 4 + 5 + 3 + 6 + 5 + 7 = 30. Shared by 6: 5. Mode: 5 (it appears twice).
  2. Room 11 total: 5 + 4 + 6 + 4 + 8 + 3 = 30. Shared by 6: 5. Mode: 4 (it appears twice).
  3. The means are the same, but the modes are different.

Both means are 5. Modes: Room 9 is 5, Room 11 is 4. A mode can change when just one or two answers change, so it often changes more than the mean does.

Making a good prediction

  • Say why: "I predict... because the two groups are alike / different in this way."
  • Use likelihood words: "It is likely that...", "It is unlikely that..."
  • Then test it with real data. A prediction that doesn't come true is not a failure. It teaches you something!
Just for fun

A prediction walks into a classroom. The teacher asks, "Are you sure you're right?"

The prediction says, "Probably!"

Hands-on activity

Predict and test at home

You need: paper and a pencil.

  1. Choose a question with a number answer, like "How many letters are in your first name?" or "How many pockets are on the clothes you are wearing?" Or choose a question with word answers, like "What is your favourite season?"
  2. Ask 5 children you know (friends, cousins, classmates). Find the mean (for number answers) and the mode.
  3. Predict: will 5 grown-ups give the same mean or mode? Write your prediction and your reason.
  4. Ask 5 grown-ups. Find their mean and mode.
  5. Compare. Was your prediction right? Why or why not?

Math talk: "How are the two groups alike? How are they different?"

Your turn

1

What is a population in data?

2

Room 9's mode for favourite season is winter. Is it certain that another Grade 3 class will also have winter as the mode? Explain.

3

Predict: whose mean number of lost baby teeth is probably bigger, a Grade 3 class or a Kindergarten class? Why?

4

Room 11 lost these numbers of teeth: 5, 3, 6, 4, 7, 5. Find the mean and the mode.

5

Compare Room 11's teeth data (mean 5, mode 5) with Room 9's (mean 5, mode 4). Which measure is the same? Which is different?

6

Which population is least likely to have the same favourite toy as a Grade 3 class: a Grade 4 class, another Grade 3 class, or a group of grandparents?

More practice: warm-up

1

Find the mode: 2, 3, 3, 1.

2

Find the mean: 2, 4, 6.

3

Is the children in your class an example of a population?

4

Two groups are very alike. Are their modes more likely or less likely to be the same?

5

After you make a prediction, what should you do to find out if it was right?

More practice: on your own

1

Pets in Room 9 (5 children): 0, 1, 2, 1, 1. Pets in Room 11 (5 children): 3, 0, 1, 0, 1. Find the mean and mode of each.

2

In the pet data, which measure was the same for both rooms? Which was different?

3

Kids in your class choose favourite snacks, and the mode is popcorn. Predict the mode for a group of parents. Give a reason.

4

Jun predicted his class and his cousin's class would have the same mode for favourite season. He tested it and the modes were different. Did Jun do something wrong?

5

Use Example 1 and 2. How much bigger is the Grade 3 mean than the Kindergarten mean?

Challenge

1

Why can a mode change more easily than a mean when you ask a different group? Use Example 4 to explain.

2

Lucas surveyed 4 children about their favourite sport and the mode was basketball. He predicts the whole school will have the same mode. Do you agree? How could he make a better prediction?

3

A Grade 3 class read for 10 minutes at quiet reading time. The mean was 4 pages. Predict the mean for a Grade 6 class reading the same kind of books at the same quiet time. Give a reason.

Check your understanding

Quiz

1.

What is a population in data?

2.

Grade 3 mean for teeth lost is 5. What is most likely for a Kindergarten class?

3.

Kindergarten data: 0, 0, 1, 0, 2, 0, 3, 2. What is the mode?

4.

Rania predicted Room 11's mode would be skating, but it was tobogganing. What does this show?

5.

Which two populations are most likely to have the same mode for favourite winter activity?

6.

Room 9 letters in names: 4, 5, 3, 6, 5, 7. What is the mean?

For parents and teachers

What your child is learning: to make and test predictions about whether the mean and mode of a data set will be the same for data collected from different populations (Ontario expectation D2.2), using the mean and mode skills from D1.4. Predicting, testing and reflecting are core mathematical processes (Strand A).

Common difficulties:

  • Thinking a prediction that doesn't come true was "wrong" or a failure. Praise the reasoning and the testing.
  • Predicting without a reason. Ask, "How are the two groups alike or different?"
  • Saying another group will definitely have the same mode. Encourage likelihood words: likely, unlikely.
  • Trusting a very small group. Data from 4 people can change a lot with one answer.

At home: compare family members and friends: "Will the grown-ups have the same favourite season as the kids?" Predict first, then ask and compare.