# 7.03 Mean and median as measures of center

Lesson

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 score. In this lesson, we will learn about two more ways we can summarize numerical data sets.

### The mean

The mean of a data set is an average score.

#### Exploration

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$20 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$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 using more mathematical language, we are sharing the total luggage equally among three groups. As a mathematical expression, we find:

$\frac{17+18+22}{3}=\frac{57}{3}=19$17+18+223=573=19

Each person carries $19$19 kg. This amount is the mean of the data set.

Summary

If we replace every number in a numerical data set with the mean, the sum of the numbers in the data set will be the same.

To calculate the mean, use the formula:

$\text{mean}=\frac{\text{sum of scores}}{\text{number of scores}}$mean=sum of scoresnumber of scores

#### Worked example

Find the mean of this data set:

$4,7,1,2,3$4,7,1,2,3

Think: There are $5$5 scores, so we should add these numbers all together and divide by $5$5.

Do: $4+7+1+2+3=17$4+7+1+2+3=17, and $17\div5=3.4$17÷​5=3.4.

Reflect: 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$3.4, even though it isn't possible to have $0.4$0.4 siblings! The mean is a way to summarize data - it is not part of the data set itself.

### The median

The median of a data set is another kind of average.

#### Exploration

Seven people were asked about their weekly income, and their responses form this data set:

### Outcomes

#### 6.SP.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.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.5

Summarize numerical data sets in relation to their context.

#### 6.SP.5c

Find the quantitative measures of center (median and/or mean) for a numerical data set and recognize that this value summarizes the data set with a single number. Interpret mean as an equal or fair share. Find measures of variability (range and interquartile range) as well as informally describe the shape and the presence of clusters, gaps, peaks, and outliers in a distribution.