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## AP®︎/College Statistics

### Course: AP®︎/College Statistics > Unit 3

Lesson 2: More on mean and median- Mean as the balancing point
- Missing value given the mean
- Missing value given the mean
- Impact on median & mean: increasing an outlier
- Impact on median & mean: removing an outlier
- Effects of shifting, adding, & removing a data point
- Estimating mean and median in data displays
- Estimating mean and median in data displays

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# Mean as the balancing point

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

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.**Notice how the 1 and 5 were "balanced" on either side of the 3:

Can you see how the data points are always balanced around the mean? Let's try one more!

## 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 left brace, 2, comma, 3, comma, 5, comma, 6, right brace to be start color #e07d10, 4, end color #e07d10. 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 start color #e84d39, 1, end color #e84d39, plus, start color #e84d39, 2, end color #e84d39, equals, start color #1fab54, 1, end color #1fab54, plus, start color #1fab54, 2, end color #1fab54:

#### Reflection questions

## 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 left brace, 2, comma, 3, comma, 6, comma, 9, right brace.

Here's how we can calculate the mean:

And we can see that the total distance below the mean is equal to the total distance above the mean because start color #e84d39, 2, end color #e84d39, plus, start color #e84d39, 3, end color #e84d39, equals, start color #1fab54, 1, end color #1fab54, plus, start color #1fab54, 4, end color #1fab54:

## Part 4: Practice

## Challenge problem

The mean of four data points is 5. Three of the four data points and the mean are shown in the diagram below.

## Want to join the conversation?

- I do not get where the mean is located on a number line. Can someone help me i'm stuck(33 votes)
- the mean is a measure of central data so it will always be in the middle of the data plot. this is why your data plot should be balanced on both sides. but don't do the literal middle of the data plot because that won't always be correct. you have to consider all of the data you have.(26 votes)

- I need a badge if anyone can upvote this I will upvote you back.(34 votes)
- please vote me up for a badge(27 votes)
- we have same 1st name(1 vote)

- can someone please explain the last question to me(20 votes)
- You can take "mean" as a balance point between left and right, making the both side has exactly the same strength.

In this case, we can see that the energy above the mean number is (7-5)*2=4, so you need 4 energy below the mean to balance the strength. Then, below the mean, we already have (5-4) = 1 energy, therefore we still need 3 energy to make it balanced.

As a result, if you plot a dot at "2", you will have (5-2) = 3 energy.(14 votes)

- Vote me a upvote I need a badge and ill upvote you 😛(23 votes)
- i get it but the bar is messing me up(11 votes)
- The bar is just another way of finding the mean(11 votes)

- I dont really understand this(9 votes)
- Probably too late but here's an example:

1 2. 3. 4. 5 6. 7. 8. 9. 10. 11.

the numbers with dots are points.

5 is the mean

the distance from 5 to 4 is one. the distance from 5 to 3 is two. the distance from 5 to 2 is three. 1+2+3=6 so the distance to the left of the mean is 6. the same on the other side.

Does that make sense?(13 votes)

- 42 is the answer to life, the universe, and everything.(11 votes)
- Hi! How can I prove that in all cases the total distance below the mean is equal to the total distance above the mean?(6 votes)
- The statement that the total distance below the mean equals the total distance above the mean is equivalent to the statement that the sum of the
**directed**distances from the mean is 0.

Let n be the number of data values, and let x_1, x_2, ... , x_n be the data values. Let m be the mean of these values.

We need to prove that sum j from 1 to n of (x_j - m) = 0.

By definition of mean, m = (sum j from 1 to n of x_j)/n.

So mn = sum j from 1 to n of x_j.

Therefore,

sum j from 1 to n of (x_j - m)

= (sum j from 1 to n of x_j) - (sum j from 1 to n of m)

= (sum j from 1 to n of x_j) - mn

= (sum j from 1 to n of x_j) - (sum j from 1 to n of x_j)

= 0.

Have a blessed, wonderful day!(8 votes)

- This was a little hard for me so I don't get it(8 votes)
- you know what else is hard??(4 votes)