Calculate and interpret measures of centre and spread

Ontario Grade 11 Mathematics

Data Management

Calculating and interpreting measures of centre and spread

A list of numbers can be difficult to understand at a glance. Measures of centre describe a central or typical value. Measures of spread describe how much the values differ from one another. Together, they give a fuller description of one-variable data. One-variable data record one feature, such as the number of minutes students spend travelling to school. In this lesson, you will calculate measures and explain what they say about the data.

What you will learn

Calculate the mean, median, and mode of a data set.
Calculate the range, interquartile range, and sample standard deviation.
Interpret measures of centre and spread in the context of the data.
Compare data sets using both centre and spread.

1. Review and describe the centre

A data value is one recorded observation. Before finding a median or quartiles, arrange the values from least to greatest. Their positions matter, so ordering them helps prevent mistakes.

The mean is the sum of the values divided by the number of values. It uses every observation. A very high or low value can pull the mean toward it. The median is the middle value in an ordered list. If there are two middle values, find their mean. The mode is the value that occurs most often. A data set may have more than one mode, or no mode if no value repeats.

Each measure answers a different question. The mean is the average across all recorded values. The median identifies the halfway position in the ordered data. The mode identifies the most common value. For example, a median travel time of 18 minutes means at least half the recorded times are 18 minutes or less and at least half are 18 minutes or more.

\bar{x}=\frac{\sum x}{n}

Mean: add the values and divide by the number of values.
Median: order the data and find the middle value or the mean of the two middle values.
Mode: find the value or values that occur most often.

2. Describe spread with range and quartiles

Spread describes how far apart the data values are. The range is the largest value minus the smallest value. It is quick to calculate, but it depends only on those two values. One unusual value can make the range much larger.

Quartiles are values that divide ordered data into four parts. The first quartile,

Q_1

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

Q_3

, is the median of the upper half. The interquartile range, or IQR, is the difference between these quartiles. It describes the spread of the middle half of the data.

For an even number of values, divide the ordered list into equal lower and upper halves. For an odd number of values, leave out the overall median before finding the medians of the two halves. Use this method consistently when calculating quartiles.

\text{range}=\text{maximum}-\text{minimum},\quad \text{IQR}=Q_3-Q_1

Range uses the smallest and largest values.
IQR describes the spread of the middle half of the ordered data.
A larger range or IQR indicates more spread by that measure.

3. Standard deviation: typical distance from the mean

Standard deviation is another measure of spread. It describes how far values typically are from the mean. A smaller standard deviation means the values tend to be closer to the mean. A larger one means they tend to be farther away.

In this lesson, standard deviation is calculated for a sample: a set of observations used to describe a group. The sample standard deviation uses

n-1

in the calculation. Here,

n

is the number of observations,

x

represents one data value, and

\bar{x}

is the mean. For the calculation, subtract the mean from each value, square each difference, add the squares, divide by

n-1

, and take the square root.

Squaring makes differences above and below the mean contribute positively. Taking the square root returns the result to the same units as the original data. Standard deviation uses every value, so an extreme value can affect it. Interpret it alongside the mean and the context rather than assuming every value lies the same distance from the mean.

s=\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}

Sample standard deviation uses $n-1$ in the calculation.
Standard deviation is expressed in the same units as the data.
A larger standard deviation indicates more spread around the mean.

4. Choose and interpret measures in context

A useful interpretation connects a measure to the situation. A mean of 24 minutes for travel time describes the average of the recorded trips. It does not mean every trip took 24 minutes. Include units when the data have units.

If a data set has an extreme value, the median and IQR can give a useful description of the middle values. They rely on positions in the ordered data. The mean and standard deviation use every value and can show the effect of an extreme value. Neither pair is always best; consider the data and the question.

Compare both centre and spread when looking at two groups. Two groups can have the same mean but different spreads. In that case, their averages match, but one group’s values may be less consistent. Explain what the measures show rather than reporting numbers without context.

Interpret each measure in the setting of the data.
Consider whether an extreme value affects the measures.
Use centre and spread together for a fuller comparison.

Worked example

Travel times with a high value

Five students report one-way travel times of 12, 15, 15, 18, and 40 minutes. Find the mean, median, mode, range, and IQR. Interpret the results.

Order and count
The values are already ordered. There are five observations, so the third value is the median.
$12,\ 15,\ 15,\ 18,\ 40$
Find the centre
Add the five times and divide by five to find the mean. The third value is the median. Since 15 occurs twice and the other values occur once, 15 is the mode.
$\text{mean}=\frac{100}{5}=20,\quad \text{median}=15,\quad \text{mode}=15$
Find the spread
Subtract the smallest time from the largest to find the range. Leave out the overall median when finding the two quartiles. The lower half is 12 and 15; the upper half is 18 and 40.
$\text{range}=40-12=28,\quad Q_1=13.5,\quad Q_3=29,\quad \text{IQR}=29-13.5=15.5$
Interpret the measures
The 40-minute trip pulls the mean above the median. The median is 15 minutes, while the range is 28 minutes. The IQR of 15.5 minutes describes the spread between the quartiles of the middle half of the data.

Answer: Mean: 20 minutes; median: 15 minutes; mode: 15 minutes; range: 28 minutes; IQR: 15.5 minutes.

Check: The five values total 100 minutes, and 100 divided by 5 is 20. The quartiles are 13.5 and 29, so their difference is 15.5.

Worked example

Compare two sets of measurements

Two teams record the number of items they pack in five minutes. Team A records 8, 9, 10, 11, 12. Team B records 6, 8, 10, 12, 14. Treat each set as a sample. Find each mean and sample standard deviation, then compare the results.

Calculate the means
Add the five observations for each team and divide by five. Both totals are 50, so each team has a mean of 10 items.
$\bar{x}_A=\frac{50}{5}=10,\quad \bar{x}_B=\frac{50}{5}=10$
Find Team A's sample standard deviation
Team A's differences from the mean are $-2$ , $-1$ , 0, 1, and 2. Square these differences and add them. Divide by $n-1$ , which is 4, then take the square root.
$s_A=\sqrt{\frac{4+1+0+1+4}{4}}=\sqrt{2.5}\approx1.58$
Find Team B's sample standard deviation
Team B's differences from the mean are $-4$ , $-2$ , 0, 2, and 4. Their squares total 40. Divide by 4 and take the square root to find the sample standard deviation.
$s_B=\sqrt{\frac{16+4+0+4+16}{4}}=\sqrt{10}\approx3.16$
Compare the teams
The means are equal, but Team B's sample standard deviation is larger. Its recorded counts are more spread out around 10 items and are less consistent by this measure.

Answer: Both means are 10 items. Team A's sample standard deviation is about 1.58 items; Team B's is about 3.16 items.

Check: For Team A, the squared differences total 10; for Team B, they total 40. Dividing each by 4 gives 2.5 and 10. Their square roots are approximately 1.58 and 3.16.

Common mistakes and how to avoid them

Finding the median or quartiles before ordering the values.

Correction: Arrange the data from least to greatest before using positions to find the median or quartiles.

Calling the largest value the range.

Correction: Subtract the smallest value from the largest value to find the range.

Dividing by the number of observations when calculating sample standard deviation.

Correction: For the sample standard deviation used here, divide the sum of squared differences by one less than the number of observations.

Reporting a measure without units or context.

Correction: State what the measure describes and include the original data units.

Lesson summary

Mean, median, and mode describe centre in different ways.
Range, IQR, and sample standard deviation describe spread in different ways.
Sample standard deviation uses the differences from the mean and divides the sum of their squares by one less than the number of observations before taking the square root.
Interpret centre and spread together in the context of the data.

Check your understanding

Question 1

For the ordered data 3, 5, 5, 7, 10, what are the median and range?

Median 5; range 7
Median 5; range 10
Median 5; range 13
Median 6; range 7

Show answer and explanation

Median 5; range 7

The middle value is 5. The range is 10 minus 3, which is 7.

Question 2

Two groups have the same mean. Group P has a standard deviation of 1, and Group Q has a standard deviation of 4. Which statement is supported?

Group P must have a larger mean.
Group Q's values tend to be farther from its mean.
Every value in Group Q is four times a value in Group P.
The two groups must contain identical values.

Show answer and explanation

Group Q's values tend to be farther from its mean.

A larger standard deviation indicates more spread around the mean. It does not tell you the exact values in either group.

Key terms

Data value: One recorded observation in a data set.
Measure of centre: A number that describes a central or typical value in a data set.
Measure of spread: A number that describes how separated or varied the data values are.
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.
Sample: A set of observations used to describe a group.
Standard deviation: A measure of how far data values typically lie from the mean.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation D1.7. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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