Grade by Grade
D. Data · Data and Probability

Will Another Group Have the Same Mean, Median and Mode?

D2.2 D1.5 A1About 45 minutes
Learning goals
  • I can explain what a population is.
  • I can predict whether the mean, median and mode will be about the same for a different population, and give a reason.
  • I can test my prediction by collecting data.
  • I can reflect on my prediction, even when it doesn't come true.

Words to know

population
The whole group you want to learn about, like all the students in Grade 1 or all the grown-ups in your family.
prediction
A thoughtful guess about what will happen, based on what you know.
test
Collect data to check whether a prediction is right.
reflect
Think back about what happened and what you learned.

Remember from Grade 3

In Grade 3 you predicted whether a different group would have the same mean and mode, and you tested your predictions. In Grade 4 you will add the median, and think carefully about why two groups might be the same or different.

Would another class get the same answers?

Room 9 counted the letters in each student's first name. Their mean, median and mode were all 5 letters.

They wondered: Would the Grade 1 class get the same results? What about a group of grown-ups?

Each group we study is called a population. Different populations can give different data, but sometimes they give very similar data.

How to make a good prediction

Ask yourself: Is there a reason these two populations would be different?

  • If the groups are different in a way that matters (age, size, season, a special event), predict that the measures will be different, and say which will be higher.
  • If there's no reason for a difference, predict that the measures will be about the same.

Even when you predict "about the same", the numbers are rarely exactly the same. Data always has some chance in it.

Worked example

Example 1: Letters in first names

Prediction: Room 9 (Grade 4) and Room 3 (Grade 1) will have about the same mean, median and mode for the number of letters in first names. Test it with this data.

Room 9: 4, 5, 5, 6, 3, 5, 7, 4, 6

Room 3: 6, 4, 6, 5, 3, 8, 6, 4, 3

  1. Reason for the prediction: names don't get longer as you get older, so there is no reason for the classes to be different.
  2. Room 9: in order 3, 4, 4, 5, 5, 5, 6, 6, 7. Total 45, and 45 ÷ 9 = 5. Mean 5, median 5, mode 5.
  3. Room 3: in order 3, 3, 4, 4, 5, 6, 6, 6, 8. Total 45, and 45 ÷ 9 = 5. Mean 5, median 5, mode 6.
  4. Compare: the means and medians are the same. The modes are different (5 and 6), but close.

The prediction was mostly right: the mean and median match. The mode was a little different, which can happen just by chance.

Worked example

Example 2: Hand spans

Prediction: Grade 1 students will have a smaller mean, median and mode hand span than Grade 4 students. Test it. Hand spans in centimetres:

Grade 4: 13, 14, 15, 15, 16, 16, 16

Grade 1: 11, 12, 12, 12, 13, 14, 17

  1. Reason: younger children usually have smaller hands.
  2. Grade 4: total 105, and 105 ÷ 7 = 15. Mean 15 cm. Median (4th value) 15 cm. Mode 16 cm.
  3. Grade 1: total 91, and 91 ÷ 7 = 13. Mean 13 cm. Median 12 cm. Mode 12 cm.
  4. All three Grade 1 measures are smaller. Notice that one Grade 1 student (17 cm) has a bigger hand span than every Grade 4 student in the data! The measures describe the groups, not every person.

The prediction came true: the Grade 1 mean, median and mode are all smaller.

MeasureGrade 4Grade 1
Mean15 cm13 cm
Median15 cm12 cm
Mode16 cm12 cm
Hand spans: comparing two populations.
04812161513Mean1512Median1612ModeGrade 4Grade 1Hand span (cm)Hand spans: two populations
In every pair, the Grade 1 bar (right) is shorter. All three Grade 1 measures are smaller, just as predicted.
Worked example

Example 3: When a prediction doesn't come true

Room 9 predicted that Room 14 (also Grade 4) would read about the same number of books over the winter break. Test it.

Room 9: 2, 3, 3, 4, 4, 4, 8

Room 14: 4, 6, 7, 8, 8, 9, 14

  1. Room 9: total 28, and 28 ÷ 7 = 4. Mean 4, median 4, mode 4.
  2. Room 14: total 56, and 56 ÷ 7 = 8. Mean 8, median 8, mode 8.
  3. Room 14's measures are double Room 9's. The prediction was not right.
  4. Reflect: Room 9 asked Room 14 about it. Room 14 had a winter reading challenge! That was a difference Room 9 didn't know about.

The prediction didn't come true, and that's okay. Room 9 learned something new: a reading challenge can make a big difference.

Worked example

Example 4: Same mean, different median and mode

Two hockey teams scored these goals in 5 games. Team X: 2, 2, 5, 7, 9. Team Y: 1, 6, 6, 6, 6. Find the mean, median and mode for each.

  1. Team X: total 25, and 25 ÷ 5 = 5. Mean 5. Median 5. Mode 2.
  2. Team Y: total 25, and 25 ÷ 5 = 5. Mean 5. Median 6. Mode 6.
  3. The means are the same, but the medians and modes are different.

Two populations can share one measure and differ on another. That's why we check all three.

012345678910Team Xmean 5012345678910Team Ymean 5Goals in a gamemedian 5, mode 2median 6, mode 6
Both teams have a mean of 5 goals (the triangle is the balance point). But their dots are spread differently, so the median and mode are different.

Making good predictions

  • Give a reason for every prediction.
  • Use careful words: "about the same", "probably higher".
  • Collect enough data to test it. A few values can surprise you.
  • Reflect afterwards, especially when you were wrong. A wrong prediction often teaches you the most! Mathematicians and scientists are wrong all the time, and they keep going.
Just for fun

A careful prediction: "It will probably snow this week."

A not-so-careful prediction: "It will snow exactly 47 cm on Tuesday at 2:07 p.m., and a moose will build a snowman."

Words like probably and about the same are a data detective's best friends!

Hands-on activity

Predict and test: kids' and grown-ups' hand spans

You need: a ruler, paper and a pencil.

  1. Predict: Will grown-ups have the same mean, median and mode hand span as kids? Write your prediction and a reason.
  2. Measure: spread your fingers wide on the table. Measure from the tip of your thumb to the tip of your little finger, to the nearest centimetre. Measure at least 5 kids (you, siblings, cousins, friends) and 5 grown-ups.
  3. Calculate: find the mean, median and mode for each group. If the mean doesn't divide evenly, say which two numbers it is between.
  4. Compare and reflect: Did your prediction come true? Was anyone surprising, like a kid with big hands?

Math talk: "If we measured 50 grown-ups instead of 5, do you think the results would change much?"

Your turn

1

What is a population?

2

Predict: Would the mean height of Grade 4 students be the same as the mean height of Grade 8 students? Give a reason.

3

Predict: Would two Grade 4 classes in the same school have about the same median number of pets? Give a reason.

4

Example 1: which measures were the same for Room 9 and Room 3?

5

Example 2: how much greater is the Grade 4 mean hand span than the Grade 1 mean?

6

What is the median hand span for Grade 1?

7

Give one reason a good prediction might still not come true.

More practice: warm-up

1

Find the mean of 2, 4, 6.

2

Find the median of 3, 8, 5.

3

Find the mode of 1, 1, 2.

4

Would you predict that Grade 4 students and grown-ups have the same mean shoe length? Why?

5

After you test a prediction, what should you do?

More practice: on your own

1

Predict: Would the mean age of a Grade 4 class be the same as the mean age of the school's teachers? Give a reason.

2

Example 3: Room 14 read 4, 6, 7, 8, 8, 9, 14 books. Check the mean, median and mode.

3

Team X: 2, 2, 5, 7, 9. Which measure is the same as Team Y's, and which are different?

4

Kit predicted her class and the class next door would have the same mode for number of siblings. The modes were 1 and 2. Was her prediction a bad one?

5

Write a reason why grown-ups' mean hand span would probably be greater than kids'.

Challenge

1

Make two data sets of 5 numbers with the same median but different means.

2

Two new students, Natalie (7 letters) and Ava (3 letters), join Room 9. Room 9 had 9 students with a mean of 5 letters. What is the new mean?

3

Plan a test for this question: "Do students who play hockey hold a plank for the same median time as students who don't?" Write a prediction with a reason, and say what data you would collect.

Check your understanding

Quiz

1.

What is a population in data?

2.

For which data would you predict two populations to have about the same mean?

3.

You predict Grade 1 students will have a smaller mean hand span than Grade 4 students. What is the best reason?

4.

Your prediction did not come true. What should you do?

5.

Team X: 2, 2, 5, 7, 9. Team Y: 1, 6, 6, 6, 6. Which measure is the same for both teams?

6.

Room 14 had a winter reading challenge. How does this help explain why its mean was higher than Room 9's?

For parents and teachers

What this lesson teaches: making and testing predictions about whether the mean, median and mode of a data set will be the same for data from different populations (D2.2), using the measures from D1.5. Persevering and reflecting when a prediction doesn't come true (A1) is a key part.

How to help at home:

  • Do the hand span activity, or compare other data between kids and grown-ups (shoe length, how many minutes to tie shoes).
  • Always ask for a reason: "Why do you think the grown-ups will be different?"
  • Celebrate wrong predictions as chances to learn.

Common difficulties:

  • Predicting with no reason, or saying the measures will be exactly the same.
  • Thinking that if a group's mean is bigger, every person in it is bigger.
  • Arithmetic slips when finding three measures for two groups. Encourage organized work: one column per group.