Topics
8. Statistics
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United States of America
Grade 6

8.03 Mean and median as measures of center

Lesson
Practice

Introduction

Sometimes we want to talk about a data set without having to refer to every single result. In other words, we want to summarize the data set to learn more about it and make comparisons. In the last lesson, we introduced the mode, the most frequently occurring data value. In this lesson, we will learn about two measures of center that we can use to summarize numerical data sets.

The mean

The mean of a data set is an average of all the data values in the set. Let's look at an example.

Three friends are planning a trip to Alice Springs. They plan to fly there, and discover that the airline imposes a weight limit on their luggage of 20\text{ kg} per person. On the night before the flight they weigh their luggage and find that their luggage weights form this data set: 17,\,18,\,22

One of them has packed too much. They decide to share their luggage around so that they all carry the same amount. How much does each person carry now? Thinking about it mathematically, we are sharing the total luggage equally among three groups. We can represent that as an expression: \dfrac{17+18+22}{3}=\dfrac{57}{3}=19

Each person carries 19\text{ kg}. This amount is the mean of the data set.

To calculate the mean, use the formula:

Mean

=\dfrac{\text{sum of all data values}}{\text{number of data values}}

Examples

Example 1

Find the mean of this data set:4,\,7,\,1,\,2,\,3

Worked Solution
Create a strategy

Add the 5 data values together and divide by number of data values.

Apply the idea
\displaystyle \text{mean}\displaystyle =\displaystyle (4+7+1+2+3) \div 5Add all the data values
\displaystyle =\displaystyle 17 \div 5Divide the sum of the data values by the number of data values in the set
\displaystyle =\displaystyle 3.4Evaluate the division
Reflect and check

Even though all the numbers in the data set are whole numbers, the mean is a decimal. If the data set was produced from a survey question "How many siblings do you have?", we would say the mean number of siblings was 3.4, even though it isn't possible to have 0.4 siblings. The mean is a way to summarize data - it is not part of the data set itself.

Idea summary

The mean of a data set is the average of all the data values in the set.

Mean

=\dfrac{\text{sum of all data values}}{\text{number of data values}}

The median

The median of a numerical data set is another measure of center. It is the "middle" value, and how to find the median changes depending on the number of values in the data set. If there are an odd number of values, the median will be the middle value. If there are an even number of values, the median will be the number in between the middle two values.

The image shows 2 sets of values ordered from smallest to largest and their medians. Ask your teacher for more information.

Because we are finding the middle value, half the values will be greater than or equal to the median, and half will be less than or equal to the median. Let's look at an example.

Seven people were asked about their weekly income, and their responses form this data set: \$300,\,\$400,\,\$430,\,\$470,\,\$490,\,\$2900The mean of this data set is \dfrac{\$5390}{7}=\$770, but this amount doesn't represent the data set very well because six out of seven people earn much less than this.

Instead we can find the median, which is the middle of the data set. To find the median we remove the biggest and the smallest values: \$400,\,\$400,\,\$430,\,\$470,\,\$490

Then the next biggest and the next smallest: \$400,\,\$430,\,\$470

Then the next biggest and the next smallest: \$430

There is only one number left, and this is the median - so for this data set the median is \$430. This weekly income is much closer to the other values in the data set, and summarizes the set better.

Examples

Example 2

Find the median of the following values: 11,\,17,\,3,\,14,\,19,\,7

Worked Solution
Create a strategy

There is an even number of values, the median will be the number in between the middle two values.

First, put the list in order from smallest to largest to more easily remove values. Then remove the smallest and largest values until we are down to two numbers.

Apply the idea

11,\,17,\,3,\,14,\,19,\,7

Order the values from smallest to largest:3,\,7,\,11,\,14,\,17,\,19

Remove the smallest and largest values: 7,\,11,\,14,\,17

Remove the smallest and largest values: 11,\,14

Find the mean of the remaining two numbers: \dfrac{11+14}{2}

Add in the numerator then divide by the denominator: \dfrac{25}{2}

The median of the data set is 12.5.

Example 3

Find the median of the following scores: 3,\,18,\,10,\,19,\,12,\,5,\,6,\,20,\,7

Worked Solution
Create a strategy

There is an odd number of values, the median will be the middle value.

We need to put the scores in order and list down the four largest, four smallest numbers, and the middle score.

Apply the idea

3,\,5,\,6,\,7,\,10,\,12,\,18,\,19,\,20

The four largest scores are: 12,\,18,\,19,\,20

The four smallest scores are: 3,\,5,\,6,\,7

The middle score is: 10

The median of the scores is 10.

Idea summary

To find the median of a numerical data set:

  • If there are an odd number of values, the median will be the middle value.

  • If there are an even number of values, the median will be the number in between the middle two values.

Outcomes

6.SP.A.2

Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

6.SP.A.3

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number

6.SP.B.5

Summarize numerical data sets in relation to their context, such as by:

6.SP.B.5.C

Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data was gathered.

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