# Mean as the balancing point

CCSS Math: 6.SP.A.3
Explore how we can think of the mean as the balancing point of a data distribution.
You know how to find the mean by adding up and dividing. In this article, we'll think about the mean as the balancing point. Let's get started!

## Part 1: Find the mean

Find the mean of $\{5, 7\}$.
The mean is $6$.
Find the mean of $\{5, 6, 7\}$.
The mean is $6$.
Interesting! In the first two problems, the data was "balanced" around the number six. Try the next one without finding the total or dividing. Instead, think about how the numbers are balanced around the mean.
Find the mean of $\{1, 3, 5\}$.
The mean is $3$.
Notice how the $1$ and $5$ were "balanced" on either side of the $3$:
Find the mean of $\{4, 7, 10\}$.
The mean is $7$.
Can you see how the data points are always balanced around the mean? Let's try one more!
Find the mean of $\{2,3, 5, 6\}$.
The mean is $4$.

## Part 2: A new way of thinking about the mean

You might have noticed in Part 1 that it's possible to find the mean without finding the total or dividing for some simple data sets.
Key idea: We can think of the mean as the balancing point , which is a fancy way of saying that the total distance from the mean to the data points below the mean is equal to the total distance from the mean to the data points above the mean.

### Example

In Part 1, you found the mean of $\{2,3, 5, 6\}$ to be $\goldD4$. We can see that the total distance from the mean to the data points below the mean is equal to the total distance from the mean to the data points above the mean because $\redD{1} + \redD{2} = \greenD{1} + \greenD{2}$:

#### Reflection questions

What is the total distance $\redD{\text{below}}$ the mean in this example?
What is the total distance $\greenD{\text{above}}$ the mean in this example?

## Part 3: Is the mean always the balancing point?

Yes! It is always true that the total distance below the mean is equal to the total distance above the mean. It just happens to be easier to see in some data sets than others.
For example let's consider the data set $\{2, 3, 6, 9\}$.
Here's how we can calculate the mean:
$\dfrac{2 + 3 + 6 + 9}{4} = \goldD 5$
And we can see that the total distance below the mean is equal to the total distance above the mean because $\redD{2} + \redD{3} = \greenD{1} + \greenD{4}$:

## Part 4: Practice

### Problem 1

Which of the lines represents the mean of the data points shown below?
Line $\maroonD C$ represents the mean because it is the line where the distance below the line is equal to the distance above the line:

### Problem 2

Which of the lines represents the mean of the data points shown below?
Line $\purpleC B$ represents the mean because it is the line where the distance below the line is equal to the distance above the line:
The mean of four data points is $5$. Three of the four data points and the mean are shown in the diagram below.
Step 1: So far, the distance below the mean is $\redD1$, and the distance above the mean is $\greenD4$:
Step 2: To make the distance below the mean equal the distance above the mean, the fourth data point must be $\redD3$ below the mean of $\goldD5$:
The fourth data point is $2$.