Grade by Grade
D. Data · Data, Statistics and Mathematical Modelling

Describing One-Variable Data: Mean, Median, Mode and Range

D1.2 A1About 65 minutes
Learning goals
  • I can calculate the mean, median, mode and range of a data set, including from a frequency table.
  • I can explain how an outlier affects the mean and the median.
  • I can choose the best measure of centre for a situation and justify my choice.
  • I can work backward from a mean to find a missing value.

Words to know

one-variable data
Data about one characteristic, such as the height of each student.
measure of central tendency
A single number that describes the centre of a data set: the mean, median or mode.
mean
The sum of the values divided by the number of values.
median
The middle value when the data is in order (or the mean of the two middle values).
mode
The value that occurs most often.
range
The greatest value minus the least value.
outlier
A value much greater or much less than most of the data.
frequency table
A table showing how many times each value (or group of values) occurs.
weighted mean
A mean in which some values count more than others, for example because they come from a bigger group.

Same average, different story

Two Grade 9 classes write the same math test. Both classes have a mean of 72%. Are the classes the same?

Not necessarily. In one class, almost everyone scored between 65% and 80%. In the other, some students scored above 95% and some below 50%. One number for the centre is not enough. You also need to describe the spread.

In this lesson you will review the measures of centre you know, learn when to use each one, calculate them from a frequency table, and see how one unusual value can change the story.

Remember from Grade 8

You already know the mean, median, mode and range, and in Grade 7 you saw how adding or removing a value changes them. Grade 9 is about choosing the right measure, explaining your choice, and working with bigger data sets using technology.

Measures of central tendency

A measure of central tendency is one number that describes the middle or typical value of a data set.

  • Mean: add all the values, then divide by how many values there are.

mean = (sum of values) ÷ (number of values)

  • Median: put the values in order. The median is the middle value. If there are two middle values (an even number of values), the median is halfway between them (their mean).
  • Mode: the value that appears most often. A data set can have no mode, one mode or several modes.

A measure of spread

  • Range = greatest value − least value. It tells how far apart the extremes are.

In the next lesson you will learn a better measure of spread, the interquartile range.

Worked example

Example 1: Homework minutes

Eleven students recorded how many minutes they spent on homework one evening:

25, 40, 30, 55, 35, 30, 45, 20, 30, 60, 35

Find the mean, median, mode and range.

  1. Order the data: 20, 25, 30, 30, 30, 35, 35, 40, 45, 55, 60. There are n = 11 values.
  2. Mean: the sum is 20 + 25 + 30 + 30 + 30 + 35 + 35 + 40 + 45 + 55 + 60 = 405. Mean = 405 ÷ 11 ≈ 36.8 minutes.
  3. Median: with 11 values, the middle one is the 6th: (11 + 1) ÷ 2 = 6. The 6th value is 35 minutes.
  4. Mode: 30 appears three times, more than any other value. Mode = 30 minutes.
  5. Range: 60 − 20 = 40 minutes.

Mean ≈ 36.8 min, median = 35 min, mode = 30 min, range = 40 min. A typical student did about 35 minutes of homework, but the times varied by 40 minutes from least to most.

Homework minutes for 11 students1520253035404550556065median 35 mean ≈ 36.8
A dot plot of the homework data. Each dot is one student. The median (35) and mean (about 36.8) are close together because the data has no extreme values.

Outliers pull the mean

An outlier is a value that is much greater or much less than the rest of the data. The mean uses every value, so an outlier can pull it a long way. The median only cares about the middle position, so it barely moves.

Eight players on a hockey team scored these numbers of goals in a season: 2, 3, 3, 4, 5, 5, 6, 41.

  • Mean = 69 ÷ 8 = 8.625. Only one player scored more than 8.6 goals, so the mean does not describe a typical player.
  • Median = (4 + 5) ÷ 2 = 4.5. This is a much better description of a typical player.

Without the star player (41 goals), the mean would be 28 ÷ 7 = 4 and the median would be 4. One value moved the mean from 4 to 8.6!

Goals by 8 players on one hockey team051015202530354045median 4.5mean ≈ 8.6
The star player's 41 goals is an outlier. It pulls the mean up to about 8.6, far to the right of where most of the dots are. The median (4.5) stays with the main group.
MeasureBest used when...Watch out for...
MeanThe data is roughly symmetric with no outliers; you need a total (mean × n = total)Outliers and skewed data pull it toward the long tail
MedianThere are outliers or the data is skewed (incomes, house prices, wait times)It ignores how far the other values are from the middle
ModeThe data is categorical (favourite sport, shoe brand) or you want the most common valueThere may be no mode or several modes
RangeYou want a quick sense of spreadOne outlier can make it huge
Choosing a measure.
GoalsNumber of games
03
15
26
34
42
Data for Example 2.
Worked example

Example 2: Mean and median from a frequency table

A soccer team recorded the number of goals it scored in each of 20 games.

(The data is in the table above.)

Find the mean, median and mode.

  1. Check the total number of games: 3 + 5 + 6 + 4 + 2 = 20.
  2. Mean: multiply each number of goals by its frequency and add: 0 × 3 + 1 × 5 + 2 × 6 + 3 × 4 + 4 × 2 = 0 + 5 + 12 + 12 + 8 = 37 goals. Mean = 37 ÷ 20 = 1.85 goals per game.
  3. Median: with 20 values, the median is halfway between the 10th and 11th values. Count up: values 1 to 3 are 0 goals, values 4 to 8 are 1 goal, values 9 to 14 are 2 goals. So the 10th and 11th values are both 2. Median = 2 goals.
  4. Mode: 2 goals happened most often (6 games). Mode = 2 goals.

Mean = 1.85, median = 2, mode = 2 goals per game.

Worked example

Example 3: Working backward from the mean

Amira's first four test marks are 68, 74, 81 and 77. What mark does she need on the fifth test to have a mean of 76?

  1. For a mean of 76 over 5 tests, the total must be 76 × 5 = 380.
  2. Her total so far is 68 + 74 + 81 + 77 = 300.
  3. She needs 380 − 300 = 80.
  4. Check: (68 + 74 + 81 + 77 + 80) ÷ 5 = 380 ÷ 5 = 76. ✓

She needs 80% on the fifth test.

Using technology

With large data sets, nobody adds by hand. In a spreadsheet (Google Sheets or Excel), if the data is in cells A1 to A200:

  • =AVERAGE(A1:A200) gives the mean
  • =MEDIAN(A1:A200) gives the median
  • =MODE(A1:A200) gives a mode
  • =MAX(A1:A200)-MIN(A1:A200) gives the range

You still need to choose which measure fits the situation and explain what it means.

Just for fun

Why did the mean get kicked out of the quiet study room?

Because every time an outlier walked in, it got pulled into the conversation.

Hands-on activity

Sentence length detective

You need: a novel or chapter book, paper and a pencil (or a spreadsheet).

  1. Open the book to any page. Count the number of words in each of the first 21 sentences. Write the counts in a list.
  2. Find the mean, median, mode and range of your data. Show your work for the median (write the values in order).
  3. Make a dot plot of your data.
  4. Are there any outliers (very long or very short sentences)? Which measure of centre best describes a typical sentence in this book? Explain.
  5. Bonus: repeat with a page from a picture book or a newspaper and compare.

What to look for: values written in order, the median as the 11th value, a correct mean, and a reason for the choice of measure that mentions outliers or symmetry.

Your turn

1

Find the mean, median, mode and range: 12, 15, 15, 18, 20, 22, 24.

2

Find the median of 3.2, 4.1, 2.8, 3.9, 4.4, 3.6 (litres of sap collected from six taps).

3

The heights (cm) of 8 Grade 9 students are 150, 165, 158, 172, 160, 155, 168, 190. Find the mean and median.

4

The daily low temperatures (°C) for a week in January were −12, −8, −15, −5, −10, −9, −14. Find the mean, median and range.

5

A data set is 8, 9, 9, 10, 11, 12, 13, 14, 15, 40. Which is a better measure of centre, the mean or the median? Calculate both.

6

Which measure of centre makes sense for favourite winter sports (skating, skiing, hockey, snowboarding)?

7

A basketball player's points in 4 games were 12, 18, 9 and 15. How many points does she need in game 5 for a mean of 15?

More practice: warm-up

1

Find the mean of 4, 7, 9, 10.

2

Find the median of 5, 1, 8, 3, 6.

3

Find the mode of 2, 4, 4, 5, 7, 7, 7, 9.

4

Find the range of 31, 18, 44, 27.

5

True or false: the median is always one of the data values.

More practice: on your own

1

The number of pets owned by 25 students: 0 pets: 6 students; 1 pet: 9; 2 pets: 5; 3 pets: 3; 4 pets: 2. Find the mean and median.

2

Seven house prices on a street (in thousands of dollars) are 520, 545, 560, 575, 590, 610, 2 400. Which measure would a real-estate report use to describe a typical price? Why?

3

The mean of 6 numbers is 14. Five of the numbers are 10, 12, 15, 16 and 18. What is the sixth number?

4

Create a set of 5 whole numbers with a mean of 10, a median of 8 and a mode of 7.

5

A data set of 9 values has a mean of 20. A tenth value of 50 is added. What is the new mean?

Challenge

1

Class A (20 students) has a mean test mark of 70%. Class B (30 students) has a mean of 80%. What is the mean for all 50 students? Why is it not 75%?

2

Is it possible for a set of 4 numbers to have a mean of 10, a median of 10, a mode of 10 and a range of 0? A range of 8? Give an example of each or explain why not.

3

A teacher drops each student's lowest of 5 quiz marks. Jordan's marks are 55, 78, 82, 85, 90. By how much does dropping the lowest mark raise the mean? Does it change the median?

Check your understanding

Quiz

1.

What is the median of 14, 9, 21, 6, 17, 12?

2.

Which measure of centre is most affected by an outlier?

3.

For which data would the mode be the only sensible measure of centre?

4.

Five numbers have a mean of 12. Four of them are 10, 11, 13 and 15. What is the fifth?

5.

Incomes in a town are skewed: most people earn a moderate amount and a few earn very large amounts. Which statement is true?

6.

The range of a data set is 25 and the least value is −8. What is the greatest value?

7.

In a frequency table, 4 students read 1 book, 6 read 2 books and 10 read 3 books. What is the mean number of books?

For parents and teachers

Big idea: D1.2 asks students to represent and statistically analyse one-variable data in various ways. This lesson reviews centre and spread and focuses on choosing and justifying a measure.

How to help: Have your teen explain the median of an even-sized set out loud, including why they ordered the data first. Look for averages in the news ("average home price", "median income") and ask why that measure was chosen.

Common difficulties:

  • Forgetting to order the data before finding the median.
  • For a frequency table, dividing by the number of rows instead of the total frequency.
  • Thinking the mean is always the "best" average.
  • Subtracting negatives when finding a range of temperatures: −5 − (−15) = 10.