Unit 8, Lesson 10
Finding and Interpreting the Mean as the Balance Point
Grade 6
10.3
Activity
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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.- Which of the two data sets has a larger mean? In this context, what does a larger mean tell us?
- Which of the two data sets has larger sums of distances to the left and right of the mean? What do these sums tell us about the variability in Diego’s and Andre’s travel times?
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Here is a dot plot showing lengths of Lin’s trips to school.
- Calculate the mean of Lin’s travel times.
- Complete the table with the distance between each point and the mean as well whether the point is to the left or right of the mean.
time in minutes distance from the mean left or right of the mean? 22 18 11 8 11 - Find the sum of distances to the left of the mean and the sum of distances to the right of the mean.
- Use your work to compare Lin’s travel times to Andre’s. What can you say about their average travel times? What about the variability in their travel times?
Student Lesson Summary
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:
- 19
- 20
- 20
- 21
- 21
- 22
- 22
- 23
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.
In this plot, each point on either side of the mean has a mirror image. For example, the two points at 20 and the two at 22 are the same distance from 21, but each pair is located on either side of 21. We can think of them as balancing each other around 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 arrows pointing to the dots at 20 and 22. There are lines indicating the distance between 20 and 21 and the distance between 21 and 22. 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.
Similarly, the points at 19 and 23 are the same distance from 21 but are on either side of it. They, too, can be seen as balancing each other around 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 arrows pointing to the dots at 19 and 23. There are lines indicating the distance between 19 and 21 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.
We can say that the distribution of the stickers has a center at 21 because that is its balance point, and that the eight pages, on average, have 21 stickers.
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.
A measure of center is a value that seems typical for a data distribution.
Mean and median are both measures of center.