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Mean as the balancing point

AP.STATS:
UNC‑1 (EU)
,
UNC‑1.I (LO)
,
UNC‑1.I.2 (EK)
CCSS.Math: ,
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 left brace, 5, comma, 7, right brace.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Find the mean of left brace, 5, comma, 6, comma, 7, right brace.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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 left brace, 1, comma, 3, comma, 5, right brace.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Notice how the 1 and 5 were "balanced" on either side of the 3:
Find the mean of left brace, 4, comma, 7, comma, 10, right brace.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Can you see how the data points are always balanced around the mean? Let's try one more!
Find the mean of left brace, 2, comma, 3, comma, 5, comma, 6, right brace.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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

What is the total distance start color #e84d39, start text, b, e, l, o, w, end text, end color #e84d39 the mean in this example?
Choose 1 answer:
Choose 1 answer:

What is the total distance start color #1fab54, start text, a, b, o, v, e, end text, end color #1fab54 the mean in this example?
Choose 1 answer:
Choose 1 answer:

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:
start fraction, 2, plus, 3, plus, 6, plus, 9, divided by, 4, end fraction, equals, start color #e07d10, 5, end color #e07d10
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

Problem 1

Which of the lines represents the mean of the data points shown below?
Choose 1 answer:
Choose 1 answer:

Problem 2

Which of the lines represents the mean of the data points shown below?
Choose 1 answer:
Choose 1 answer:

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.
Choose the fourth data point.
Choose 1 answer:
Choose 1 answer:

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