Grade by Grade
D. Data · Data and Probability

Mean, Median and Mode

D1.5About 50 minutes
Learning goals
  • I can find the median of a data set, with an odd or an even number of values.
  • I can identify the mode or modes, or tell that there is no mode.
  • I can find the mean by adding the values and dividing by how many there are.
  • I can explain what the mean, the median and the mode each tell about the data.

Words to know

median
The middle value when the data is put in order. Half the values are at or below it and half are at or above it.
mode
The value that appears most often. A data set can have one mode, more than one mode, or no mode.
mean
The fair share: add all the values, then divide the total equally by the number of values.
in order
Arranged from least to greatest.
unusual value
A value that is much bigger or much smaller than the rest of the data.

Remember from Grade 3

  • The mode is the value that appears most often. 2, 5, 5, 8 has mode 5. 1, 3, 6 has no mode.
  • The mean is the fair share. You found it by levelling cube towers, or by adding the values and sharing the total equally.

In Grade 4 you will learn a new measure, the median, and find the mean using division.

What is typical?

Five friends recorded how many minutes they read on Saturday: 25, 40, 15, 30, 20.

If someone asked, "About how long do these friends usually read?", what would you say? There are three different ways to answer with one number: the median, the mode and the mean. Each one tells us something a little different.

The median: the middle value

  1. Put the values in order from least to greatest.
  2. Find the value in the middle.

If there is an even number of values, there are two middle values. The median is halfway between them.

15202530405 values: the middle one, 25, is the median101214184 values: halfway between 12 and 14 is 13
Top: 15, 20, 25, 30, 40. The middle value, 25, is the median. Bottom: 10, 12, 14, 18. The two middle values are 12 and 14, so the median is 13.
Worked example

Example 1: Median with an odd number of values

Find the median of the reading minutes: 25, 40, 15, 30, 20.

  1. Put them in order: 15, 20, 25, 30, 40.
  2. There are 5 values. Cross off one from each end at the same time: 15 and 40, then 20 and 30.
  3. One value is left in the middle: 25.

The median is 25 minutes. Two friends read less than 25 minutes and two read more.

Worked example

Example 2: Median with an even number of values

Find the median of 12, 18, 10, 14.

  1. Put them in order: 10, 12, 14, 18.
  2. Cross off from each end: 10 and 18. Two values are left in the middle: 12 and 14.
  3. Find the number halfway between 12 and 14. Count: 12, 13, 14. Or add and halve: 12 + 14 = 26, half of 26 is 13.

The median is 13. It doesn't have to be one of the values in the data.

The mean: add, then divide

  1. Add all the values to find the total.
  2. Divide the total by how many values there are.

This is the same fair share you found in Grade 3. Dividing is just a quick way to share the total equally.

Worked example

Example 3: Find all three

A basketball player scored these points in 6 games: 14, 6, 22, 10, 16, 10. Find the mode, the median and the mean.

  1. Mode: 10 appears twice. Every other value appears once. The mode is 10.
  2. Median: in order: 6, 10, 10, 14, 16, 22. There are 6 values, so the two middle values are 10 and 14. Halfway between 10 and 14 is 12.
  3. Mean: add: 14 + 6 + 22 + 10 + 16 + 10 = 78. There are 6 games. 78 ÷ 6 = 13. Check: 6 × 13 = 78. ✓

Mode 10 points, median 12 points, mean 13 points.

What does each one tell us?

  • Mode: the most common value. What happened most often?
  • Median: the middle. Half the values are at or below it, and half are at or above it.
  • Mean: the fair share. If all the values were evened out, this is what each one would be.

Often the three are close together. But one unusual value can pull the mean far away from the others.

Worked example

Example 4: An unusual value

Five students sold cookies at the bake sale: 3, 4, 4, 5, 34. Find the mean, median and mode. Which one best describes how many cookies a typical student sold?

  1. Mean: 3 + 4 + 4 + 5 + 34 = 50. 50 ÷ 5 = 10.
  2. Median: the values are already in order. The middle value is 4.
  3. Mode: 4 (it appears twice).
  4. Four of the five students sold 3, 4 or 5 cookies. Nobody sold close to 10! The one big value, 34, pulled the mean up.

Mean 10, median 4, mode 4. The median (or the mode) describes a typical student best.

05101520253035344534mean = 10(nobody sold 10!)median = 4Each bar is one studentCookies sold
One very tall bar (34) pulls the mean up to 10, far above what most students sold. The median, 4, stays with the typical students.
Worked example

Example 5: Median and mode from a stem-and-leaf plot

Plot: 1 | 2 5 8 / 2 | 0 3 3 7 / 3 | 1 6 (key: 2 | 3 means 23). Find the median and the mode.

  1. The leaves are already in order, so the data is in order: 12, 15, 18, 20, 23, 23, 27, 31, 36.
  2. There are 9 values. The middle one is the 5th: count 12, 15, 18, 20, 23.
  3. 23 appears twice, more than any other value.

Median 23, mode 23.

Good to know

  • Always put the data in order before finding the median.
  • Count the values carefully. If two values are the same, count both.
  • When you find the mean, divide by the number of values, not by the biggest value.
  • If the total doesn't divide evenly, the mean is between two whole numbers. You can say, for example, "the mean is between 17 and 18."
Just for fun

The mean, the median and the mode walk into a room.

The mean says, "Let's share everything fairly." The median says, "I'll just stand in the middle." The mode says, "I'm here the most!"

Hands-on activity

Grab a handful

You need: a bowl of dried pasta pieces or dried beans, paper and a pencil. (Keep dried beans and pasta away from babies and toddlers. They are a choking hazard.)

  1. Each person grabs one handful and counts the pieces. Collect at least 5 handfuls (family members, or you can grab a few times).
  2. Write the counts. Put them in order.
  3. Find the median, the mode (if there is one) and the mean (add, then divide by the number of handfuls). If it doesn't divide evenly, say which two numbers the mean is between.
  4. Talk about it: "Whose hand holds about a typical amount? Did anyone grab an unusual amount? Which measure describes our family best?"

Your turn

1

Find the median: 7, 3, 9, 5, 4.

2

Find the mode: 6, 8, 6, 2, 9.

3

Find the mean: 4, 7, 10.

4

Find the median: 8, 2, 6, 4.

5

Find the mode or modes: 3, 5, 3, 9, 5, 1.

6

Find the mean: 12, 16, 9, 15.

7

The ages of 5 cousins are 9, 9, 10, 9, 8. Find the mean, median and mode.

More practice: warm-up

1

Find the median: 11, 14, 12.

2

Find the mode: 2, 2, 3, 4, 4, 4.

3

Is there a mode for 1, 5, 7, 9?

4

Find the mean of 5 and 9.

5

Which measure is the middle value?

More practice: on your own

1

Find the mean, median and mode: 20, 25, 25, 30, 50.

2

Find the median: 31, 18, 24, 40, 28, 22.

3

Plot: 2 | 4 6 6 / 3 | 0 4 (key: 2 | 4 means 24). Find the median, mode and mean.

4

A team's mean was 3 goals per game. Does that mean they scored exactly 3 goals in every game? Explain.

5

Five friends have a mean of 8 stickers each. How many stickers do they have altogether?

Challenge

1

The heights of 5 seedlings are 6, 7, 7, 8 and 32 cm. Find the mean, median and mode. Which one best describes a typical seedling? Why?

2

Make a data set of 5 whole numbers with a mean of 6, a median of 5 and a mode of 4.

3

Four friends have a mean of 10 marbles each. A fifth friend joins with 15 marbles. What is the new mean?

Check your understanding

Quiz

1.

What is the median of 2, 9, 4, 7, 5?

2.

What is the mean of 6, 8, 10?

3.

What is the mode of 3, 7, 7, 2, 7, 3?

4.

What is the median of 10, 20, 30, 40?

5.

Which measure tells the fair share if all the values were evened out?

6.

Data: 2, 3, 3, 4, 48. Which measure is pulled up the most by the unusual value 48?

For parents and teachers

What this lesson teaches: finding the median (new in Grade 4), the mode(s) and the mean (add, then divide) for whole-number data, and explaining what each tells: most common, middle, and fair share (D1.5).

How to help at home:

  • Use real numbers: the minutes your child read each night this week, or the number of goals in their last few games. Find all three measures together.
  • Ask "Which number best describes a typical night?" and accept any well-reasoned answer.

Common difficulties:

  • Forgetting to put the data in order before finding the median.
  • With an even number of values, picking one of the middle values instead of the number halfway between them.
  • Dividing by the wrong number when finding the mean (it is the number of values).
  • Thinking the mean must be one of the data values. It often isn't.