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D1.5 · Compare data sets with mean, median, mode, and range

Learn to compare data sets with mean, median, mode, and range through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Data

Use mean, median, mode, and range to describe what is alike and what is different

Imagine two groups record how many pages they read in one sitting. Looking at just one number cannot tell the whole story. A data set is a group of recorded numbers. Four useful measures describe different parts of a data set: mean, median, mode, and range. Every example in this lesson uses hypothetical data made for learning. You will practise finding each measure, then comparing what the measures show.

What you will learn

Find the mean, median, mode, and range of a data set.
Compare two data sets using the same measures.
Explain what each measure tells you about the data.

Four ways to describe a data set

Before finding a measure, copy the values carefully. Put them in order from least to greatest when you need the median or range. Ordering also helps you check that no value was missed.

The mean is the total shared equally among all the values. Add the values, then divide by how many values there are. For example, the values 2, 4, and 6 have a total of 12. Sharing 12 equally among 3 values gives a mean of 4.

The median is the middle value after the data are put in order. If there is an odd number of values, one value is in the middle. If there is an even number, the two middle values share the middle place. Find the mean of those two values to get the median.

The mode is the value that appears most often. A set can have one mode or more than one mode if values tie for appearing most often. If every value appears equally often, there is no mode. The range is the largest value minus the smallest value. It tells how far apart the two end values are.

For example, in the ordered list 2, 4, 4, 6, the total is 16 and there are 4 values. The mean is 4. The two middle values are 4 and 4, so the median is 4. The mode is 4 because it appears twice. The range is 4, found by subtracting 2 from 6. In words, mean equals the total of the values divided by the number of values.

Mean: add the values, then divide by the number of values.
Median: order the values and find the middle, or the mean of the two middle values.
Mode: find the value or values that appear most often.
Range: subtract the smallest value from the largest.

Compare the same measures

To compare two data sets, find the same measures for each set. Compare mean with mean, median with median, mode with mode, and range with range. Then explain what the results tell you.

The measures describe different features. The mean uses every value. The median tells where the middle of the ordered list is. The mode points to the most repeated value. The range only uses the smallest and largest values. One measure cannot tell you everything about a data set.

A frequency count means how many times a value appears. A bar chart can show these counts. Each bar represents a value, and its height shows that value's frequency. A taller bar means that value appears more often. A bar chart can help you see the mode, but you still need the values to calculate the other measures.

A careful comparison names the measure and says what it shows. For example, if one set has a greater range, its largest and smallest values are farther apart. That does not automatically mean that its mean is greater. Matching means also do not prove that two sets contain the same values.

\text{range}=\text{largest value}-\text{smallest value}

Compare like with like: mean with mean, median with median, mode with mode, and range with range.
The four measures describe different features of the data.
Explain what a difference means instead of only saying which number is greater.

A steady routine

Use this routine to avoid mixing up the measures. First, copy the values and put them in order. Add the values and count them to find the mean. Use the ordered list to find the median. Count how often each value appears to find the mode. Finally, subtract the smallest value from the largest to find the range.

For an even number of values, use the mean of the two middle values. Do not choose just one. If all values appear the same number of times, there is no mode. If two or more values tie for appearing most often, report all of them as modes.

When you compare data sets, keep your working clear. Check that the total includes each value once for every time it appears. Check that the range uses the two end values in the ordered list. These checks help make your comparison accurate.

Order the values before finding the median.
For an even number of values, find the mean of the two middle values.
A set can have more than one mode or no mode.

Practise the comparison

As you work through the examples, notice that two sets can match on some measures and differ on others. State all four measures when asked to compare the sets. Then describe a difference in plain language.

For practice, use two short lists of numbers from a stated, hypothetical situation. Find the four measures for each list. Compare each measure and write a sentence about what the comparison shows. Check your totals, ordering, and counts before deciding.

A comparison can include both similarities and differences.
Check calculations against the original values.

Worked example

Same mean, different middle and spread

These hypothetical data sets show numbers of pages read in one sitting. Set A: 4, 6, 6, 8, 11. Set B: 3, 5, 7, 9, 11. Find and compare the four measures.

Frequency in hypothetical Set A

The value 6 appears twice; each other value appears once.

Find the means
Add each set and divide by its five values. Set A totals 35, and Set B also totals 35. Both means are 7.
$35\div5=7$
Find the medians and modes
Both lists are already in order. The third value is the middle because each list has five values. Set A's middle is 6, and its mode is 6 because it appears twice. Set B's middle is 7. Each value in Set B appears once, so it has no mode.
$\text{median}_A=6,\quad \text{median}_B=7$
Find and compare the ranges
Subtract the smallest value from the largest in each set. Set A has a range of 7, while Set B has a range of 8. The sets have the same mean, but different medians, modes, and ranges. Set B's values stretch one more page from smallest to largest.
$11-4=7,\quad 11-3=8$

Answer: Set A: mean 7, median 6, mode 6, range 7. Set B: mean 7, median 7, no mode, range 8.

Check: Both totals are 35. Each set has five values, so both means are 7.

Worked example

Even number of values and two modes

These hypothetical data sets show points earned in two rounds of a game. Set C: 2, 4, 6, 8. Set D: 3, 3, 5, 7, 7. Find and compare the four measures.

Frequency in hypothetical Set D

Both 3 and 7 occur twice, so both are modes.

Find the means
Set C totals 20 across four values, so its mean is 5. Set D totals 25 across five values, so its mean is also 5.
$20\div4=5,\quad 25\div5=5$
Find the medians
Set C has four values, so its two middle values are 4 and 6. Their mean is 5, so the median is 5. Set D has five values, so its middle value is 5.
$\frac{4+6}{2}=5$
Find the modes and ranges
Every value in Set C appears once, so there is no mode. In Set D, both 3 and 7 appear twice, so both are modes. Set C's range is 6, while Set D's range is 4. The sets share a mean and median, but differ in mode and range.
$8-2=6,\quad 7-3=4$

Answer: Set C: mean 5, median 5, no mode, range 6. Set D: mean 5, median 5, modes 3 and 7, range 4.

Check: For Set C, the two middle values are 4 and 6. Their mean is 5, not 4 or 6.

Worked example

Same first three measures, different range

These hypothetical data sets show numbers of minutes spent on a class activity. Set E: 5, 6, 6, 7, 8. Set F: 2, 6, 6, 7, 11. Find and compare the four measures.

Frequency in hypothetical Set F

The value 6 occurs twice; the other values occur once.

Calculate the means
Both sets total 32 and each has five values. Each mean is 6.4 minutes.
$32\div5=6.4$
Compare the median and mode
The third value in each ordered list is 6, so both medians are 6. The value 6 appears twice in each set, more often than any other value, so both modes are 6.
$\text{median}=6,\quad \text{mode}=6$
Compare the ranges
Set E's range is 3. Set F's range is 9. The sets have the same mean, median, and mode, but Set F has a larger range. Its smallest and largest values are farther apart.
$8-5=3,\quad 11-2=9$

Answer: Both sets have mean 6.4, median 6, and mode 6. Set E's range is 3; Set F's range is 9.

Check: Set F's total is 32: 2 + 6 + 6 + 7 + 11. Dividing by five gives 6.4.

Common mistakes and how to avoid them

Choosing one of the two middle values as the median when a set has an even number of values.

Correction: Find the mean of the two middle values. For 4 and 6, the median is 5.

Calling a value the mode just because it is large.

Correction: The mode is about how often a value appears, not how large it is.

Finding the range by adding the largest and smallest values.

Correction: Subtract the smallest value from the largest.

Saying two data sets are the same because their means match.

Correction: Compare the median, mode, and range too. Matching means do not show that every value matches.

Lesson summary

The mean is the total shared equally among the values.
The median is the middle value, or the mean of the two middle values.
The mode is the value or values that appear most often; a set can have no mode.
The range is the largest value minus the smallest value.
Compare the same measure in each set, then explain what the difference means.

Check your understanding

Question 1

A hypothetical set is 2, 4, 4, 6. What is its mean?

Show answer and explanation

The total is 16, and there are four values. The mean is 16 divided by 4, which is 4.

Question 2

A hypothetical set is 1, 3, 5, 7, 9. Which statement is correct?

The median is 3.
The mode is 5.
There is no mode.
The range is 10.

Show answer and explanation

There is no mode.

Each value appears once, so none appears more often. The median is 5 and the range is 8.

Question 3

Set A has range 4 and Set B has range 9. What can you say?

Set B's values span a greater distance from smallest to largest.
Set B must have a greater mean.
Set A must have no mode.
The sets have the same median.

Show answer and explanation

Set B's values span a greater distance from smallest to largest.

Range compares the largest and smallest values. A range of 9 is greater than a range of 4, but this alone does not tell us the means, modes, or medians.

Key terms

Data set: A group of recorded numbers.
Mean: The total of the values divided by how many values there are.
Median: The middle value after the numbers are put in order. With an even number of values, it is the mean of the two middle values.
Mode: The value or values that appear most often.
Range: The largest value minus the smallest value.
Frequency: How many times a value appears.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation D1.5. It is a study resource, not an official curriculum publication.

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