Grade by Grade
D. Data · Data Literacy and Probability

Range and the Measures of Central Tendency

D1.5About 50 minutes
Learning goals
  • I can find the mean, median and mode of a data set.
  • I can find the range and explain that it measures how spread out the data is.
  • I can explain how an outlier changes the mean and the range.

Words to know

measures of central tendency
The mean, median and mode. Each one is a way to describe the centre, or a typical value, of a data set.
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's halfway between the two middle values.
mode
The value that appears most often. A data set can have no mode or more than one mode.
range
The greatest value minus the least value. It measures how spread out the data is.
outlier
A value that is much bigger or much smaller than the rest of the data.

Remember from Grade 5

You found the mean, median and mode of whole numbers and decimals.

  • Mean: add all the values, then divide by how many there are.
  • Median: put the values in order and find the middle.
  • Mode: the value that appears most often.

These three are called the measures of central tendency, because each describes the centre of the data. This year we add a measure of spread: the range.

The range: how spread out is the data?

The range is the greatest value minus the least value.

Look at the two sets of data below. Both are the number of minutes six students spent on homework. Both have a mean of 10 and a median of 10. But they are very different!

  • Set A: 8, 9, 10, 10, 11, 12. Range: 12 − 8 = 4. The values are close together.
  • Set B: 2, 5, 10, 10, 15, 18. Range: 18 − 2 = 16. The values are spread out.
Two Sets with the Same MeanSet A02468101214161820Set B02468101214161820Number of minutes
Each dot is one student. Set A's dots are bunched between 8 and 12. Set B's dots are spread from 2 to 18. Both sets have a mean and median of 10.

The centre alone doesn't tell the whole story. A small range means the values are consistent, or similar to each other. A large range means they vary a lot.

Worked example

Example 1: All four measures

Liam skated these numbers of minutes each day for a week:

30, 45, 20, 45, 60, 35, 45

Find the mean, median, mode and range.

  1. Order the data: 20, 30, 35, 45, 45, 45, 60.
  2. Mean: 20 + 30 + 35 + 45 + 45 + 45 + 60 = 280. There are 7 values. 280 ÷ 7 = 40.
  3. Median: with 7 values, the middle one is the 4th: 45.
  4. Mode: 45 appears three times, more than any other value: 45.
  5. Range: 60 − 20 = 40.

Mean 40 min, median 45 min, mode 45 min, range 40 min.

Worked example

Example 2: Decimals and an even number of values

Snowfall each day for 6 days, in centimetres: 2.5, 0.5, 4.0, 3.5, 1.5, 6.0. Find the mean, median, mode and range.

  1. Order: 0.5, 1.5, 2.5, 3.5, 4.0, 6.0.
  2. Mean: 0.5 + 1.5 + 2.5 + 3.5 + 4.0 + 6.0 = 18.0. 18.0 ÷ 6 = 3.0.
  3. Median: there are 6 values, so there are two middle ones: 2.5 and 3.5. Halfway between is (2.5 + 3.5) ÷ 2 = 3.0.
  4. Mode: no value appears more than once, so there is no mode.
  5. Range: 6.0 − 0.5 = 5.5.

Mean 3.0 cm, median 3.0 cm, no mode, range 5.5 cm.

Watch out for outliers

Suppose a seventh day had a huge storm with 27.0 cm of snow.

  • New mean: (18.0 + 27.0) ÷ 7 = 45.0 ÷ 7 ≈ 6.4 cm. It jumped from 3.0 to about 6.4!
  • New median: 0.5, 1.5, 2.5, 3.5, 4.0, 6.0, 27.0, so 3.5 cm. It barely moved.
  • New range: 27.0 − 0.5 = 26.5 cm. It jumped too.

An outlier pulls the mean and the range a lot, but the median hardly changes. When there is an outlier, the median is often the best description of a typical value.

Snowfall (cm): what one stormy day does6 days: mean 3.0, median 3.0, range 5.504812162024287 days with the storm: mean ≈ 6.4, median 3.5, range 26.50481216202428mean = median = 3.0median 3.5mean ≈ 6.4 (pulled right!)outlier
The storm day (27.0 cm) sits far from the other dots. It drags the mean from 3.0 to about 6.4, but the median only moves from 3.0 to 3.5.

Check your answers

  • The mean and median must be between the least and greatest values.
  • The range is a difference, a single number, not "20 to 60".
  • Always order the data before finding the median or range.
Just for fun

Why did the mean get dragged off the couch?

An outlier showed up and pulled it! The median just stayed comfy in the middle.

Hands-on activity

Paper airplane flights

You need: a sheet of paper, a measuring tape (or a metre stick), and space indoors or outside where nothing can break.

  1. Fold a paper airplane.
  2. Throw it 7 times from the same line. Measure each flight to the nearest tenth of a metre (10 cm), for example 3.4 m.
  3. Record the 7 distances in a list.
  4. Find the mean, median, mode (if there is one) and range.
  5. Was any throw an outlier? What happens to the mean and range if you leave it out?

Keep your data! You will compare it with a second design in the next lesson.

Your turn

1

A hockey team scored these numbers of goals in 9 games: 3, 1, 4, 2, 4, 0, 6, 4, 3. Put the data in order.

2

Find the mean number of goals.

3

Find the median.

4

Find the mode.

5

Find the range.

6

Find the range of: 14.5, 9.0, 12.5, 18.0, 10.5.

7

What does a range of 0 tell you about a data set?

More practice: warm-up

1

Find the range: 12, 7, 19, 3.

2

Find the median: 5, 8, 2, 9, 6.

3

Find the mean: 4, 6, 8.

4

Find the mode: 7, 3, 7, 1, 9, 7.

5

Which measure tells how spread out data is: mean, median, mode or range?

More practice: on your own

1

Sunflower heights (cm): 152.5, 160.0, 148.5, 171.0, 160.0. Find the mean, median, mode and range.

2

Find the median of 11, 4, 9, 15, 6, 13.

3

Two classes collected cans for a food drive. Class A: range 8 cans. Class B: range 30 cans. Which class's students collected more similar amounts?

4

Data: 5, 6, 6, 7, 41. Which value is an outlier? Which describes a typical value better, the mean or the median?

5

Mira says the range of 3, 8, 12, 20 is "3 to 20". What is the correct way to write it?

Challenge

1

The mean of 4 test scores is 80. Three of the scores are 72, 85 and 90. What is the fourth score?

2

Make a set of 5 whole numbers with a mean of 10, a median of 9 and a range of 12.

3

Data: 12, 15, 15, 18, 20. A new value is added, and the range becomes 14 while the median stays 15. What could the new value be? Find all the possible whole numbers.

Check your understanding

Quiz

1.

What is the range of 15, 22, 9, 30, 18?

2.

What is the median of 3, 8, 1, 6?

3.

Which is not a measure of central tendency?

4.

An outlier is added to a data set. Which measure usually changes the least?

5.

Set A has a range of 3. Set B has a range of 25. Which is true?

6.

What is the mean of 2.5, 3.0, 4.5, 6.0?

For parents and teachers

What this lesson teaches: finding the mean, median and mode (the measures of central tendency) and the range (a measure of spread, new in Grade 6) for data sets with whole numbers and decimals (D1.5).

How to help at home:

  • Use real data: family members' heights, daily temperatures, times to walk to school.
  • Ask "Is that answer between the smallest and the largest value?" to catch mean and median errors.
  • Show how one outlier (such as one very long trip) changes the mean and range but not the median.

Common difficulties:

  • Forgetting to order the data before finding the median.
  • Writing the range as "9 to 30" instead of the difference, 21.
  • With an even number of values, picking one of the two middle values instead of finding halfway between them.
  • Dividing by the wrong number when finding the mean.