Compare data sets using measures of centre and spread

Ontario Grade 11 Mathematics

Data Management

Ontario Grade 11 MBF3C | Study-guide label D1.9

Suppose two school clubs track how many kilometres their members walk in a week. The clubs could have the same average distance but very different patterns: one group’s distances might be close together, while the other group’s distances vary widely. A comparison based only on the average would miss that difference. In this lesson, you will use measures of centre and spread to describe and compare data sets. A data set is a group of recorded values for one feature, such as weekly walking distance.

What you will learn

Identify the centre and spread of a one-variable data set.
Calculate and interpret the mean, median, range, and interquartile range.
Compare data sets by considering both a typical value and how much the values vary.

1. Review: organize the data first

A data value is one recorded measurement or count. In a one-variable data set, every value describes the same feature, using the same unit. For example, a set of weekly walking distances should be measured in kilometres throughout.

To find the median or quartiles, put the values in order from least to greatest. The median is the middle value. If there are two middle values, average those two. The mean is the sum of all the values divided by how many values there are. These are different ways to describe the centre, or a typical location, of a data set.

Spread describes how far apart the values are. Range is the difference between the greatest and least values. The interquartile range, often shortened to IQR, describes the spread of the middle half of the ordered data. To find it, find the first quartile,

Q_1

, and third quartile,

Q_3

, then subtract

Q_1

from

Q_3

. Quartiles split ordered data into four parts. For this lesson, when there is an odd number of values, leave the overall median out when finding the lower and upper halves.

\mathrm{IQR}=Q_3-Q_1

Keep the same variable and units when comparing data sets.
Order the values before finding the median, quartiles, or range.
Centre describes a typical location; spread describes variation.

2. What the measures tell you

The mean uses every value. This makes it useful when you want the balance point of the data, but an unusually high or low value can pull it away from most of the values. Such a value is sometimes called an outlier: a value that sits far from the rest. The median depends on the middle position, so one unusually large value usually affects it less than it affects the mean.

The range uses only the two extreme values. It is quick to calculate, but it does not tell you whether the other values are clustered or spread out. The IQR focuses on the middle half and is less affected by an extreme value than the range.

A fair comparison uses measures that answer the same question in both sets. If the sets have similar centres, compare their spread to see which is more consistent. If their spreads are similar, compare their centres to see which has the higher typical value. If one set has an extreme value, report both the mean and median when useful, and explain why they differ.

The mean can shift when an extreme value is added.
The median gives the middle position in ordered data.
A larger range or IQR indicates greater spread for the measure used.

3. Compare carefully and explain your choice

A numerical comparison should include an interpretation. Instead of saying only that one range is 8 and another is 3, say that the first set covers a wider interval from its minimum to maximum. Likewise, a larger IQR means the middle half of that set occupies a wider interval.

Consider what the data represent before deciding which measure is most helpful. For scores with no unusually distant values, the mean and range may give a useful comparison. If a set contains a very high value while most values are close together, the median and IQR may better describe the typical middle of the data. There is not one measure that is best for every situation.

A strong comparison names the measure, gives the result for each set, and states what the difference means in context. It also avoids claiming more than the data show. For example, a larger median walking distance describes the recorded group’s middle value; it does not prove why the group walked farther.

Compare the same measures in both sets.
State what a numerical difference means in the setting.
Use centre and spread together for a fuller comparison.

Measures used in a comparison

Measure	What it describes	How to calculate or find it
Mean	Balance point using every value	Add the values and divide by the number of values
Median	Middle position in ordered data	Find the middle value, or average the two middle values
Range	Distance from least to greatest	Greatest value minus least value
IQR	Spread of the middle half	Third quartile minus first quartile

Worked example

Example 1: Compare two walking groups

Weekly walking distances, in kilometres, are recorded for five members of each of two clubs. Club A: 2, 3, 4, 5, 6. Club B: 1, 3, 4, 5, 7. Compare the centres and spreads using the mean, median, and range.

Find the mean for each club
Add each club’s five distances and divide by five, because the mean shares the total evenly across the recorded members.
$\bar{x}_A=\frac{2+3+4+5+6}{5}=4,\quad \bar{x}_B=\frac{1+3+4+5+7}{5}=4$
Find each median
The values are already in order. With five values, the third value is in the middle, so it is the median.
$\operatorname{median}(A)=4,\quad \operatorname{median}(B)=4$
Find each range
Subtract the least value from the greatest value. This measures the full distance between the endpoints of each set.
$\operatorname{range}(A)=6-2=4,\quad \operatorname{range}(B)=7-1=6$

Answer: Both clubs have the same mean of 4 km and median of 4 km. Club B has the larger range, 6 km compared with Club A’s 4 km, so its recorded distances vary more from lowest to highest.

Check: For both clubs, the five values add to 20 km. Their means are therefore 20 divided by 5, or 4 km. The range calculations use the stated smallest and largest values.

Worked example

Example 2: An unusually high value

Two small teams record the number of items packed in one shift. Team A records 8, 9, 10, 11, 12 items. Team B records 8, 9, 10, 11, 32 items. Compare their centres and their range and IQR. Use the median-of-halves method described in this lesson.

Calculate the means
Find each total and divide by five. Team B’s value of 32 is much larger than its other values, so it has a stronger effect on that team’s mean.
$\bar{x}_A=\frac{50}{5}=10,\quad \bar{x}_B=\frac{70}{5}=14$
Compare the medians
Each ordered set has five values, so its third value is the median. The medians match even though the means do not.
$\operatorname{median}(A)=10,\quad \operatorname{median}(B)=10$
Calculate range and IQR
For each range, subtract the minimum from the maximum. For the IQR, leave out the middle value, find the median of each remaining half, and subtract the lower-half median from the upper-half median.
$\operatorname{range}(A)=12-8=4,\quad \operatorname{range}(B)=32-8=24$
Interpret the quartiles
For Team A, the lower half is 8 and 9, and the upper half is 11 and 12. For Team B, the lower half is 8 and 9, and the upper half is 11 and 32. Averaging each pair gives the quartiles; the IQRs are equal.
$Q_{1,A}=8.5,\ Q_{3,A}=11.5,\ \operatorname{IQR}_A=3;\quad Q_{1,B}=8.5,\ Q_{3,B}=21.5,\ \operatorname{IQR}_B=13$

Answer: The means suggest a higher average for Team B, but its median is the same as Team A’s. Team B’s much higher mean and range reflect its recorded value of 32. Its IQR is also larger under the stated method, so its middle half spans a wider interval in this small data set.

Check: Team A’s total is 50 and Team B’s is 70. In Team B, the upper-half median is the average of 11 and 32, which is 21.5; subtracting 8.5 gives an IQR of 13.

Common mistakes and how to avoid them

Calculating the median before ordering the values.

Correction: Order the values from least to greatest first, then locate the middle position.

Treating the range as a typical value.

Correction: The range measures the distance between the two endpoints; use a centre measure to describe a typical location.

Comparing the mean of one set with the median of another as if they were the same measure.

Correction: Compare like with like first, such as mean with mean and median with median.

Claiming that a larger range means every value is farther from the centre.

Correction: Range uses only the minimum and maximum. It does not describe every value’s distance from the centre.

Lesson summary

Centre measures describe where values tend to sit; the mean and median describe centre in different ways.
Spread measures describe variation; range uses endpoints, while IQR describes the middle half.
A useful comparison interprets both centre and spread in context and uses the same measures for each data set.

Check your understanding

Question 1

Set P is 3, 4, 5, 6, 7. Set Q is 3, 4, 5, 6, 12. Which statement is correct?

Both sets have median 5, and Q has a larger range.
Q has median 6, and both sets have range 4.
Both sets have mean 5, and P has a larger range.
P has median 4, and Q has a larger IQR.

Show answer and explanation

Both sets have median 5, and Q has a larger range.

Both ordered sets have 5 in the middle. P has range 7 minus 3, or 4; Q has range 12 minus 3, or 9. So Q has the larger range.

Question 2

A data set has values 2, 5, 8, and 9. What is its median?

5
6.5
8
24

Show answer and explanation

6.5

There are four values, so average the two middle values, 5 and 8. Their average is 6.5.

Question 3

Why might the median be more helpful than the mean when one value is far above the rest?

The median uses the greatest value twice.
The median depends on the middle position and is usually less affected by that extreme value.
The median is always larger than the mean.
The median measures the full distance between the endpoints.

Show answer and explanation

The median depends on the middle position and is usually less affected by that extreme value.

The median is based on the middle position in ordered data. An unusually high value can pull the mean upward more strongly.

Key terms

Data set: A group of recorded values for a feature.
Centre: A measure describing the typical location of values.
Spread: A description of how much the values vary.
Quartile: A value marking a division of ordered data into four parts.
Outlier: A value that sits far from the other values in a data set.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation D1.9. 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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