D1.8 Choose suitable measures of centre and spread

Ontario Grade 11 Mathematics

Data Management

A practical guide to describing one-variable data

A list of numbers can be summarized in different ways. A measure of centre describes a typical or central value. A measure of spread describes how much the values vary. The best choice depends on the data and the question. For example, a value far from the rest may make one summary less representative, while another may still give a useful picture. In this lesson, you will compare common measures and explain why particular measures suit particular data.

What you will learn

Describe what measures of centre and spread tell us about a data set.
Choose measures that suit the data and the question being asked.
Explain how an unusually high or low value can affect some measures more than others.

1. Start by looking at the data

A data set is a group of recorded values for one feature, such as travel time, height, or weekly sales. Before choosing a summary, put numerical values in order from least to greatest. An ordered list makes it easier to identify the middle and compare values.

Check that the values are numerical. Travel times in minutes are numerical, so it makes sense to calculate their mean or range. Favourite colours are not numerical values, so those calculations would not make sense for that list.

Next, look at the pattern. Are most values close together? Is one value far above or below the others? A value that is far from most of the data is often called an outlier. You can notice a possible outlier by comparing the values and considering what they represent. You do not need a formal test to make a sensible choice of summary.

Finally, think about the question. If it asks for the most common result, the mode may be useful. If it asks for a typical value and how much the values vary, you will usually choose one measure of centre and one measure of spread.

Sort numerical values before finding the median or quartiles.
Check whether a value is far from the rest.
Let both the data pattern and the question guide your choice.

2. Measures of centre

The mean is found by adding all the values and dividing by the number of values. It uses every value in the data set. This makes it useful when the values are reasonably balanced and no unusually high or low value pulls the result away from most of the data. If the data values are represented by

x_1, x_2, \ldots, x_n

, then

n

is the number of values.

For example, if six values have a total of 24, their mean is 4. The division works because it shares the total equally across the six values. In symbols, the mean is the sum of the values divided by the number of values.

The median is the middle value after the data are ordered. If there are two middle values, the median is the mean of those two values. The median marks the middle position: about half the values are at or below it and about half are at or above it. Because it depends on position rather than the size of every value, an unusually high or low value usually affects it less than the mean.

The mode is the value that occurs most often. A data set may have one mode, more than one mode, or no mode if no value repeats. The mode is useful when the most frequent value matters, such as the shoe size sold most often. It may not describe a typical value well if many values appear only once.

Each measure answers a slightly different question. The mean uses every value, the median identifies the middle position, and the mode identifies the most frequent value. Choose the measure that best fits the data and the purpose. The symbol

\bar{x}

is commonly used for the mean.

\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}

Mean: uses every value and can be pulled by an unusually high or low value.
Median: identifies the middle position and is less affected by extreme values.
Mode: identifies the most frequent value.

3. Measures of spread

Spread describes how far apart the data values are. The range is the greatest value minus the least value. It is quick to calculate and describes the full span from one end of the data to the other. However, it uses only the two end values, so one unusually high or low value can make the range large.

The interquartile range, shortened to IQR, describes the spread of the middle half of the data. Quartiles are values that divide an ordered data set into four parts. The first quartile, written

Q_1

, is the median of the lower half. The third quartile, written

Q_3

, is the median of the upper half. The IQR is the difference between these quartiles.

Use one consistent method to find the quartiles. If there is an odd number of values, leave out the overall median when making the lower and upper halves. If there is an even number, split the ordered list into two equal halves. Find the median of each half. The IQR is less affected by extreme values than the range because it describes the middle half rather than the full span.

Centre and spread work together to describe a data set. A median with an IQR is often suitable when a value is far from the others. A mean with a range can be suitable when values are fairly balanced and there is no unusual value. These are useful guidelines, not automatic rules. Always look at the data and the question.

\text{range}=\text{greatest value}-\text{least value},\quad \text{IQR}=Q_3-Q_1

Range measures the distance from the least value to the greatest value.
IQR describes the spread of the middle half of the ordered data.
Consider centre and spread together when choosing a useful summary.

4. Make and explain a choice

A good choice includes both the measures and a reason. Do not just say that the median is suitable. Explain that one value is much larger than the rest, so the mean could be pulled toward it. Or, when choosing the mean, explain that the values are fairly close and there is no unusual value.

The purpose of the data matters too. If someone asks for the most common result, the mode may answer the question directly. If someone wants to know how consistent a set of values is, include a measure of spread as well as a measure of centre. A centre alone does not tell how much the values vary.

Do not choose a measure only because it is familiar or easy to calculate. Look at the values, consider what the question asks, and select measures that give a fair picture of the data.

Connect your choice to the question and a feature of the data.
Mention an outlier or a close cluster when it supports your choice.
A useful summary often includes one measure of centre and one of spread.

Choosing a useful summary

What you notice or need	Possible measure of centre	Possible measure of spread
A value is far from most of the data	Median	IQR
Values are fairly close, with no unusual value	Mean	Range
The most frequent value matters	Mode	Choose a spread measure if variation also matters

Worked example

A travel time far from the rest

Seven students report one-way travel times of 12, 14, 15, 16, 18, 19, and 80 minutes. Choose a suitable measure of centre and spread, and explain your choice.

Inspect the ordered values
The values are already in order. The travel time of 80 minutes is much greater than the others, which range from 12 to 19 minutes. It is far from most of the data, so a summary that is less affected by it may better describe a typical trip.
Find the median
There are seven values, so the fourth value is in the middle position. The median uses that position rather than the size of every value.
$\text{median}=16$
Find the quartiles and IQR
Leave out the overall median when forming the halves. The lower half is 12, 14, and 15, so its middle value is 14. The upper half is 18, 19, and 80, so its middle value is 19. Subtract the first quartile from the third to find the IQR.
$Q_1=14,\quad Q_3=19,\quad \text{IQR}=19-14=5$
Choose suitable summaries
The median and IQR are suitable because the 80-minute trip is far from the other values. The mean and range would be strongly affected by that value.
$\text{median}=16,\quad \text{IQR}=5$

Answer: A suitable summary is a median travel time of 16 minutes and an IQR of 5 minutes.

Check: The fourth of seven ordered values is 16. The lower- and upper-half medians are 14 and 19, and their difference is 5.

Worked example

Weekly sales without an extreme value

A small shop records the number of reusable bottles sold on eight days: 14, 15, 16, 16, 17, 18, 19, and 21. Choose a suitable measure of centre and spread, and explain your choice.

Inspect the pattern
The values are ordered and lie between 14 and 21. No value is far from the others, so an unusually high or low value is not an obvious concern.
Find the mean
Add the eight daily totals and divide by eight. Since the values are fairly close, the mean gives a reasonable description of the centre.
$\text{mean}=\frac{14+15+16+16+17+18+19+21}{8}=\frac{136}{8}=17$
Find the range
Subtract the smallest daily total from the largest. This gives the full span of the recorded sales.
$\text{range}=21-14=7$
Explain the choice
The mean and range are suitable because the values are fairly close and there is no unusual value that would distort either measure. The mean is a summary; it does not claim that exactly 17 bottles were sold on every day.
$\text{mean}=17,\quad \text{range}=7$

Answer: A suitable summary is a mean of 17 bottles per day and a range of 7 bottles.

Check: The eight values total 136, and 136 divided by 8 is 17. The largest value minus the smallest is 21 minus 14, or 7.

Common mistakes and how to avoid them

Using the mean automatically because it uses every value.

Correction: Check for an unusually high or low value first. If it pulls the mean away from most of the data, the median may better represent the centre.

Calling the range the spread of most of the data.

Correction: The range uses only the least and greatest values. Consider the IQR when an unusual value makes the range unrepresentative.

Finding quartiles before ordering the data.

Correction: Order the values first so the lower half, middle, and upper half are clear.

Giving a measure without explaining why it fits.

Correction: Link your choice to the question and a feature of the values, such as a close cluster or a value far from the rest.

Lesson summary

Measures of centre describe a typical or common value; measures of spread describe variation.
The mean can be affected by unusually high or low values, while the median is less affected.
The range uses the least and greatest values; the IQR describes the middle half.
Choose measures by examining the data and considering what the question asks.

Check your understanding

Question 1

A data set has one value much larger than all the others. Which pair is usually more suitable for describing its centre and spread?

Mean and range
Median and IQR
Mode and mean
Range and IQR

Show answer and explanation

Median and IQR

The median and IQR are less affected by an unusually large value than the mean and range.

Question 2

The ordered data are 3, 5, 6, 8, 10. What is the median?

5
6
6.4
10

Show answer and explanation

6

There are five values, so the third value is the middle value. The median is 6.

Question 3

A set of daily counts has a greatest value of 24 and a least value of 9. What is its range?

15
24
33
9

Show answer and explanation

15

Subtract the least value from the greatest: 24 minus 9 is 15.

Key terms

Data set: A group of recorded values for a feature or question.
Measure of centre: A number used to describe a typical or central value in a data set.
Measure of spread: A number used to describe how much the values vary.
Outlier: A value that is far from most of the other values in a data set.
Quartile: A value that helps divide ordered data into four parts.
Interquartile range: The difference between the third and first quartiles; it describes the spread of the middle half of the data.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation D1.8. It is a study resource, not an official curriculum publication.

Official curriculum reference

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D1.8 · Choose suitable measures of centre and spread

Ontario Grade 11 Mathematics

A practical guide to describing one-variable data

What you will learn

1. Start by looking at the data

2. Measures of centre

3. Measures of spread

4. Make and explain a choice

Choosing a useful summary

A travel time far from the rest

Weekly sales without an extreme value

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MBF3C

About this lesson