Can you make the mean the same value as one (or more) of the data points? How many ways? What do you notice?
Can you make more than one data set with a mean of 4?
Will the mean ever be outside of the data set?
Set up the points to 4,\,6 and 11. How far is each value from the mean? Use negative values for below the mean and positive values for above the mean. What is the sum of these values?

The mean is the balance point of a data set. This means that the sum of the distances from the mean of all of the points below the mean is equal to the sum of the distances from the mean of all of the points above the mean.

A number line from 0 to 10 with plotted points 4, 5, and 9. Ask your teacher for more information.

When given a data set the balance point can be found by plotting the points on a line plot.

The points to the left are a total of 3 units away from the mean, 6.

The point on the right is 3 units away from the mean, 6.

This creates a balanced distance of 3 on either side of the mean.

To find the balance point, when it is not given, points can be moved one-by-one towards the middle.

A line plot ranging from 0 to 11 in steps of 1. Showing Step 1 to Step 4 to find the balance point. Ask your teacher for more information.

A line plot ranging from 0 to 11 in steps of 1. Showing Step 5 to Step 8 to find the balance point. Ask your teacher for more information.

If the balance point is located between two values, we can find the halfway point between those values by averaging the numbers.

A line plot ranging from 0 to 10 in steps of 1. The number of dots is as follows: at 7, 3; at 8, 3. Points 7 and 8 are encircled with label balance point.

The balance point for the data set is located halfway between 7 and 8.

We can calculate the halfway point by finding the sum of the two values and dividing by two.\dfrac{7+8}{2}=\dfrac{15}{2}=7.5

Adding or removing a data point might throw off the balance of the data set resulting in a new balance point.

Examples

Example 1

A classroom recorded the number of pets for each student. The results for the class are represented in the given line plot.

A line plot titled Number of pets, ranging from 1 to 6 in steps of 1. The number of dots is as follows: at 0, 3; at 1, 3; at 2, 3; at 3, 1; at 4, 2; 6, 1.

What was the total number of pets for the entire class?

Worked Solution

Create a strategy

Multiply the total number of each number of pets by the number of students that have that many pets.

Apply the idea

\displaystyle \text{Total number of pets}	\displaystyle =	\displaystyle \begin{aligned}&(0\cdot 3)+(1\cdot 3)+(2\cdot 3) + (3\cdot 1) \\ &+ (4\cdot 2) + (5\cdot 0) + (6\cdot 1)\end{aligned}	Multiply each number of pets by its frequency
	\displaystyle =	\displaystyle 0+3+6 + 3 + 8 + 0 + 6	Evaluate the values in the parentheses
	\displaystyle =	\displaystyle 26	Find the sum

What is the mean number of pets per student?

Worked Solution

Create a strategy

Create a line plot of the data set. Alternate moving each far left and far right point towards the center until finding the balance point of the data set, which is the mean.

Apply the idea

Start on the far left and move one of the points at 0 one unit to the right.

A line plot ranging from 1 to 6 in steps of 1. Moving one dot from point 0 to point 1. Ask your teacher for more information.

Alternate to the far right and move the point at 6 one unit to the left.

A line plot ranging from 1 to 6 in steps of 1. Moving one dot from point 6 to point 5. Ask your teacher for more information.

Alternate back to the far left and move another point at 0 one unit to the right.

Alternate back to the far right and move the point at 5 one unit to the left.

A line plot ranging from 1 to 6 in steps of 1. Moving one dot from point 5 to point 4. Ask your teacher for more information.

Alternate back to the far left and move the point at 0 one unit to the right.

Alternate back to the far right and move one of the points at 4 one unit to the left.

A line plot ranging from 1 to 6 in steps of 1. Moving one dot from point 4 to point 3. Ask your teacher for more information.

Continuing this process we end up with all points on 2.

A line plot ranging from 1 to 6 in steps of 1. The number of dots is as follows: at 2, 13. Ask your teacher for more information.

This creates a balance point at 2. The mean is 2 pets per student in the class.

Reflect and check

Alternatively, we can calculate the mean by finding the sum of all the points and dividing by the total number of data points.

\displaystyle \text{Mean}	\displaystyle =	\displaystyle \dfrac{0\cdot 3+1\cdot 3+2\cdot 3 + 3\cdot 1+ 4\cdot 2 +5\cdot 0 + 6 \cdot 1}{13}	Multiply each number of pets by its frequency and divide by the total number of students
	\displaystyle =	\displaystyle \dfrac{26}{13}	Evaluate the multiplication and addition
	\displaystyle =	\displaystyle 2	Evaluate the quotient for the mean

Example 2

During a fitness challenge, Alex recorded the number of push-ups completed each day. The table shows the number of push-ups Alex did.

Day	1	2	3	4	5	6	7	8	9	10
Push-ups	15	20	15	10	15	20	10	15	20	20

Create a line plot of the data.

Worked Solution

Create a strategy

The number line should include values in between 10 and 20 based upon the given data. Then, count the number of times that Alex completed each number of push-ups, this number will tell us how many dots to put on the number line..

Apply the idea

For this problem, we do not need to know which days Alex completed the number of push-ups, only how many days he completed them.

A line plot titled Number of push-ups, ranging from 10 to 20 in steps of 1. The number of dots is as follows: at 10, 2; at 15, 4; at 20, 4.

Alex completed 10 push-ups 2 days, so we will place 2 points above 10.

Alex completed 15 push-ups 4 days, so we will place 4 points above 15.

Alex completed 20 push-ups 4 days, so we will place 4 points above 20.

What was the total number of push-ups completed?

Worked Solution

Create a strategy

Multiply the number of completed push-ups by the frequency.

Apply the idea

\displaystyle \text{Total number of push-ups}	\displaystyle =	\displaystyle (10 \cdot 2) + (15 \cdot 4) + (20 \cdot 4)	Multiply each number of push-ups by its frequency
	\displaystyle =	\displaystyle 20 + 60 + 80	Evaluate the values in the parentheses
	\displaystyle =	\displaystyle 160	Find the sum

Alex completed a total of 160 push-ups over the 10 days.

What is the balance point of push-ups completed each day?

Worked Solution

Create a strategy

Use the line plot created in part (a) to alternate moving each far left and far right point towards the center until find the balance point of the data set.

Apply the idea

Start on the far left and move one of the points at 10 one unit to the right.

A line plot ranging from 10 to 20 in steps of 1. Moving one dot from point 10 to point 11. Ask your teacher for more information.

Alternate to the far right and move one of the points at 20 one unit to the left.

A line plot ranging from 10 to 20 in steps of 1. Moving one dot from point 20 to point 19. Ask your teacher for more information.

Alternate back to the far left and move the point at 10 one unit to the right.

Alternate back to the far right and move one of the points at 20 one unit to the left.

Alternate back to the far left and move one of the points at 11 one unit to the right.

A line plot ranging from 10 to 20 in steps of 1. Moving one dot from point 11 to point 12. Ask your teacher for more information.

Alternate back to the far right and move one of the points at 20 one unit to the left.

Continue this process until all the points have been balanced at 16.

A line plot ranging from 10 to 20 in steps of 1. The number of dots is as follows: at 16, 10. Point 16 is encircled and labeled as balance point.

The mean number of push-ups completed per day is 16 each day.

Reflect and check

A line plot titled Number of push-ups, ranging from 10 to 20 in steps of 1. The number of dots is as follows: at 10, 2; at 15, 4; at 20, 4. Ask your teacher for more information.

Each point on the far left and right sides alternate and get moved one place closer towards the center until all the points are at the balance point.

The 2 points at 10 move 6 units to the right.

The 4 points at 20 move 4 units to the left.

The 4 points at 15 move 1 unit to the right.

The points on the left side of the mean moved a total of 16 units.

The points on the right side of the mean moved a total of 16 units.

Example 3

Use the given line plot to identify the mean of a group of friends shoe sizes.

A line plot titled Shoe sizes, ranging from 4 to 9 in steps of 1. The number of dots is as follows: at 4, 1; at 5, 1; at 6, 3; at 7, 3; at 8, 1; at 9, 1.

Worked Solution

Create a strategy

Use the line plot to alternate moving each far left and far right point towards the center until you find the balance point of the data set.

Apply the idea

Start on the far left of the line plot and move the point at 4 one unit to the right.

A line plot titled Shoe sizes, ranging from 4 to 9 in steps of 1. Moving one dot from point 4 to point 5. Ask your teacher for more information.

Alternate to the far right side of the line plot and move the point at 9 one unit to the left.

A line plot titled Shoe sizes, ranging from 4 to 9 in steps of 1. Moving one dot from point 9 to point 8. Ask your teacher for more information.

Alternate back to far left side of the line plot and move one of the points at 5 one unit to the right.

A line plot titled Shoe sizes, ranging from 4 to 9 in steps of 1. Moving one dot from point 5 to point 6. Ask your teacher for more information.

Alternate back to the far right side of the line plot and move one of the points at 8 one unit to the left.

A line plot titled Shoe sizes, ranging from 4 to 9 in steps of 1. Moving one dot from point 8 to point 7. Ask your teacher for more information.

Alternate back to the far left side of the line plot and move the point at 5 one unit to the right.

Alternate back to the far right side of the line plot and move the point at 8 one unit to the left.

The balance point for this data is the value halfway between 6 and 7.

A line plot ranging from 4 to 9 in steps of 1. The number of dots is as follows: at 6, 5; at 7, 5. Point 6 and 7 is encircled and labeled as balance point.

To find the value halfway between 6 and 7 we can find the sum of these values and divide by 2.

\displaystyle \text{Halfway}	\displaystyle =	\displaystyle \dfrac{6+7}{2}	Find the halfway value between 6 and 7
	\displaystyle =	\displaystyle \dfrac{13}{2}	Evaluate the sum
	\displaystyle =	\displaystyle 6.5	Evaluate the quotient

The mean of the shoe sizes for this group of friends is 6.5.

Reflect and check

Alternatively, we can find the mean by finding the sum of all the shoe sizes and divide by the number of friends.

\displaystyle \text{Mean}	\displaystyle =	\displaystyle \dfrac{4\cdot 1+5\cdot 1+6\cdot 3+7\cdot 3+8\cdot 1+9\cdot 1}{10}	Multiply each shoe size by its frequency and divide by the total number of friends
	\displaystyle =	\displaystyle \dfrac{65}{10}	Evaluate the multiplication and addition
	\displaystyle =	\displaystyle 6.5	Evaluate the quotient for the mean

Idea summary

The mean is the balance point of a data set and is best when there are not any values which are far away from the rest.

To find the balance point from a line plot we alternate moving each point on the far left side and far right side one unit closer to the center of the line plot. Eventually, all the points should be at balance point, which is the mean of the data set.

The points located to the left of the balance point are the same distance away from the balance point as the points located on the right side.

Outcomes

6.PS.2

The student will represent the mean as a balance point and determine the effect on statistical measures when a data point is added, removed, or changed.

6.PS.2a

Represent the mean of a set of data graphically as the balance point represented in a line plot (dot plot).

9.05 Mean as a balance point

Ideas

Mean as a balance point

Exploration

Examples

Example 1

Example 2

Example 3

Outcomes

6.PS.2

6.PS.2a

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