Unit 8, Lesson 4
The Mean
Accelerated 6
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The preschool teacher wants the kittens distributed equally among the boxes. How might that be done? How many kittens will end up in each box?
Another preschool room has 6 boxes. No 2 boxes have the same number of kittens, and there is an average of 3 kittens per box. Draw or describe at least 2 different arrangements of kittens that match this description.
Five servers are scheduled to work the number of hours shown. They decide to share the workload, so each one would work equal hours.
Server A: 3
Server B: 6
Server C: 11
Server D: 7
Server E: 4
On the first grid, draw 5 bars whose heights represent the hours worked by Servers A, B, C, D, and E.
Then, think about how you would rearrange the hours so that each server gets a fair share. On the second grid, draw a new graph to represent the rearranged hours. Be prepared to explain your reasoning.
Explain why we can also find the mean by finding the value of the expression .
Here are dot plots showing how long Diego’s trips to school took in minutes and how long Andre’s trips to school took in minutes. The dot plots include the means for each data set, and those means are marked by triangles.
A dot plot, Diego's travel time in minutes, 6 to 23 by ones. It has one dot each at 7, 9, 12, 13, 14 and a triangle at 11.
A dot plot, Andre's travel time in minutes, 6 to 23 by ones. It has 2 dots at 12, one dot each at 13, 16, 17, and a triangle at 14.
Here is a dot plot showing lengths of Lin’s trips to school.
| time in minutes | distance from the mean | left or right of the mean? |
|---|---|---|
| 22 | ||
| 18 | ||
| 11 | ||
| 8 | ||
| 11 |
Sometimes a general description of a distribution does not give enough information, and a more precise way to talk about center or spread would be more useful. The mean, or average, is a number we can use for the center to summarize a distribution.
We can think about the mean in terms of “fair share” or “leveling out.” That is, a mean can be thought of as a number that each member of a group would have if all the data values were combined and distributed equally among the members.
For example, suppose there are 5 containers, each of which has a different amount of water: 1 liter, 4 liters, 2 liters, 3 liters, and 0 liters.
There are 5 identical tape diagrams that are each partitioned into 4 equal parts. The first diagram has 1 part shaded. The second diagram has 4 parts shaded. The third diagram has 2 parts shaded. The fourth diagram has 3 parts shaded. The fifth diagram has no parts shaded.
To find the mean, first we add up all of the values. We can think of this as putting all of the water together: .
To find the “fair share,” we divide the 10 liters equally into the 5 containers: .
In general, to find the mean of a data set with values, we add all of the values and divide the sum by .
The mean is often used as a measure of center of a distribution. One way to see this is that the mean of a distribution can be seen as the “balance point” for the distribution. Why is this a good way to think about the mean? Let’s look at a very simple set of data on the number of stickers that are on 8 pages:
Here is a dot plot showing the data set.
Dot plot for “number of stickers”. The numbers 18 through 24 are indicated. There is a triangle indicated at 21 stickers. The data are as follows: 18 stickers, 0 dots. 19 stickers, 1 dot. 20 stickers, 2 dots. 21 stickers, 2 dots. 22 stickers, 2 dots. 23 stickers, 1 dot. 24 stickers, 0 dots.
The distribution shown is completely symmetrical. The mean number of stickers is 21, because . If we mark the location of the mean on the dot plot, we can see that the data points could balance at 21.
A dot plot for "number of stickers". The numbers 18 through 24 are indicated. There is a triangle indicated at 21 stickers. There are lines indicating the distance between 18 and 21, the distance between 19 and 21, the distance between 21 and 22, and the distance between 21 and 23. The data are as follows: 18 stickers, 0 dots. 19 stickers, 1 dot. 20 stickers, 2 dots. 21 stickers, 2 dots. 22 stickers, 2 dots. 23 stickers, 1 dot. 24 stickers, 0 dots.
Even when a distribution is not completely symmetrical, the distances of values below the mean, on the whole, balance the distances of values above the mean.
The average is another name for the mean of a data set. To find the average, add all the numbers in the data set. Then divide by how many numbers there are.
The average is 7.5.
The mean is one way to measure the center of a data set. It can be thought of as a balance point. To find the mean, add all the numbers in the data set. Then divide by how many numbers there are.
The mean is 11. So, the typical travel time is 11 minutes.
A measure of center is a value that seems typical for a data distribution.
Mean and median are both measures of center.