Grade by Grade
D. Data · Data Literacy and Probability

How Adding or Removing Data Changes the Mean, Median and Mode

D1.5About 50 minutes
Learning goals
  • I can find the mean, median and mode of a data set.
  • I can predict and explain how adding or removing a value changes the mean, median and mode.
  • I can explain why an outlier changes the mean much more than the median.
  • I can find the value that must be added to reach a target mean.

Words to know

measure of central tendency
One number that describes the centre of a data set: the mean, median or mode.
mean
The total of all the values divided by the number of values (the "fair share").
median
The middle value when the data is in order. With an even number of values, it is halfway between the two middle values.
mode
The value that appears most often. A data set can have no mode, one mode or more than one mode.
outlier
A value much greater or much less than the rest of the data.

Remember from Grade 6

You found the mean, median, mode and range and used them to compare data sets. In Grade 5 you saw that an outlier can pull the mean. Now you will predict exactly how each measure changes when data is added or removed.

One big game

Jordan plays basketball. In 5 games he scored 12, 15, 15, 18 and 20 points.

  • Mean: 12 + 15 + 15 + 18 + 20 = 80, and 80 ÷ 5 = 16 points.
  • Median: the middle of 12, 15, 15, 18, 20 is 15 points.
  • Mode: 15 appears twice, so the mode is 15 points.

Then, in game 6, Jordan has an amazing game and scores 40 points! What happens to his mean, median and mode? Make a prediction before you read on.

A basketball resting on the floor of an empty school gym, with a hoop in the background.
One game with a very high score can pull the mean up a lot, while the median and mode barely move. AI-generated image
Worked example

Example 1: Adding an outlier

Add 40 to Jordan's data: 12, 15, 15, 18, 20, 40. Find the new mean, median and mode.

  1. Mean: the total grows to 80 + 40 = 120. 120 ÷ 6 = 20. The mean went up by 4.
  2. Median: 6 values, so the median is halfway between the 3rd and 4th values: (15 + 18) ÷ 2 = 16.5. It went up by only 1.5.
  3. Mode: still 15. Adding a 40 didn't change which value appears most.

Mean 20, median 16.5, mode 15. The outlier pulled the mean a lot, but the median only a little, and the mode not at all.

Jordan's Points per Game10152025303540Points scored
Dot plot: the five original games (12, 15, 15, 18, 20) and the added game of 40 points, which sits far to the right. It is an outlier.

Why does the mean move so much?

The mean is a fair share: it uses every value. A big value adds a lot to the total, so the fair share grows. The median only cares about position (which value is in the middle), so one huge value just shifts the middle over by half a step. The mode only changes if the new value changes which value is most common.

Rules you can use to predict

101520253035405 games10152025303540after adding 40mean 16median 15mean 20 (up 4)median 16.5 (up 1.5)outlier
Jordan's points before and after the 40-point game. The mean (filled triangle) is pulled 4 points toward the outlier, but the median (open triangle) moves only 1.5.
If you add a value that is...the mean will...
greater than the meango up
less than the meango down
equal to the meanstay the same
an outlier (far from the rest)change a lot
Removing a value works the opposite way: removing a value greater than the mean makes the mean go down, and removing a value less than the mean makes it go up.

The median depends on the middle position. Adding a value greater than the median makes the median stay the same or go up. Adding a value less than the median makes it stay the same or go down. Even an outlier moves the median only a little.

The mode changes only if the added or removed value changes which value is most common.

Worked example

Example 2: Removing a value

Go back to the first 5 games: 12, 15, 15, 18, 20. The coach decides not to count game 1 (12 points), because Jordan left early with a sore ankle. What happens?

  1. New data: 15, 15, 18, 20.
  2. Mean: 68 ÷ 4 = 17. Removing a value less than the mean (12 < 16) made the mean go up.
  3. Median: (15 + 18) ÷ 2 = 16.5.
  4. Mode: still 15.

Mean 17, median 16.5, mode 15.

Worked example

Example 3: Changing the mode

Start again with 12, 15, 15, 18, 20. Jordan scores 18 in the next game. What happens to the mode?

  1. New data: 12, 15, 15, 18, 18, 20.
  2. Now 15 and 18 each appear twice.
  3. There are two modes: 15 and 18.
  4. Mean: 98 ÷ 6 ≈ 16.3 (went up a little, since 18 > 16). Median: (15 + 18) ÷ 2 = 16.5.

The modes are 15 and 18. Adding one value created a second mode.

Worked example

Example 4: Hitting a target mean

Ella's mean mark on 4 quizzes is 75. What mark does she need on the 5th quiz to raise her mean to 78?

  1. Total for 4 quizzes: 4 × 75 = 300.
  2. Total needed for 5 quizzes: 5 × 78 = 390.
  3. Mark needed: 390 − 300 = 90.

Ella needs 90 on the 5th quiz. Check: (300 + 90) ÷ 5 = 78. ✓

Predict first, then calculate

Before calculating, say: "The new value is bigger than the mean, so the mean will go up." Then calculate and check your prediction. If it doesn't match, look for a mistake. Making a prediction and checking it is how mathematicians build confidence, even when the first guess is wrong.

Just for fun

If a billionaire walks into a café with 10 customers, the mean wealth in the room is suddenly over 90 million dollars. The median customer is still just buying a muffin!

Hands-on activity

Paper-clip chain experiment

You need: a box of paper clips (or a handful of small objects you can link or stack, like LEGO bricks), paper, a pencil and a calculator.

  1. Make 5 chains (or stacks) of different lengths: for example 4, 6, 6, 7 and 12 clips. Record the counts.
  2. Find the mean, median and mode.
  3. Predict, then test: Add a chain of 20. What happens to each measure?
  4. Remove the chain of 20 and add a chain equal to the mean. What happens?
  5. Remove your shortest chain. What happens?
  6. A quick way to see the mean: join all the chains into one long chain and share the clips out equally among the chains.

Math talk: "Which measure is the hardest to move? Which is the easiest?"

Your turn

1

Ali read for 20, 25, 30, 30 and 45 minutes on 5 days. Find the mean, median and mode.

2

Ali reads for 60 minutes on day 6. Find the new mean, median and mode.

3

Go back to the first 5 days. Remove the 45. Find the new mean, median and mode.

4

Go back to the first 5 days. Add another 30. What happens to the mean?

5

Without calculating: if Ali adds a day of 10 minutes to the first 5 days, will the mean go up or down? Why?

6

Which measure usually changes the most when you add an outlier?

7

How many minutes must Ali read on day 6 so that the mean of all 6 days is 32?

More practice: warm-up

1

Find the mean of 4, 6, 8.

2

Find the median of 3, 9, 5, 7, 1.

3

Find the mode of 2, 4, 4, 6, 7.

4

You add 100 to the data set 4, 6, 8. Will the mean go up or down?

5

Find the median of 10, 12, 14, 20.

More practice: on your own

1

Data: 8, 9, 9, 10, 14. Add 2. Find the new mean and median, and describe the change.

2

The mean of 6 numbers is 12. One number, 12, is removed. What is the new mean?

3

Snowfall (cm) on 5 days: 2, 3, 3, 4, 28. Which value is an outlier? Find the mean and median with and without it.

4

Data: 5, 7, 7, 9. What single value could you add to give two modes?

5

A team's mean score over 4 games is 3 goals. After game 5, the mean is 3.4. How many goals did they score in game 5?

Challenge

1

Data: 4, 6, 8, 10, 12 (mean 8). Add two values so that the mean stays 8 but the range becomes 16.

2

Jordan's mean is 14 points over 6 games. After game 7, his mean is 15. How many points did he score in game 7? Was it more or less than 15? Why does that make sense?

3

A data set has 7 values. Its median is 20. You remove the largest value. Can the median go up? Explain.

Check your understanding

Quiz

1.

Data: 10, 12, 14. You add 50. Which measure changes the most?

2.

You add a value equal to the mean. What happens to the mean?

3.

You remove a value that is greater than the mean. What happens to the mean?

4.

The mean of 3 tests is 80. What mark on the 4th test makes the mean 85?

5.

Data: 3, 4, 4, 5, 6. You add 5. What is the mode now?

6.

Why does an outlier change the median only a little?

For parents and teachers

Big idea: The Ontario curriculum (D1.5) asks Grade 7 students to determine the impact of adding or removing data on a measure of central tendency. Students already know how to find the mean, median and mode (Grade 6); now they predict and explain changes. The next lesson looks at how these changes alter the shape of the data.

Common difficulties:

  • Forgetting to reorder the data before finding the median after a change.
  • Using the old number of values when calculating the new mean.
  • Target-mean problems: students try to average the means. Use totals: total needed − total so far.
  • Thinking a data set can have only one mode.

How to help: Use real numbers your child cares about: game scores, minutes of practice, allowance. Ask "What if one more game had a huge score? Which number would change most?" Encourage predicting before calculating.