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Without calculating, tell whether each pair of data sets have the same mean and whether they have the same mean absolute deviation.
Set A
1
3
3
5
6
8
10
14
Set B
21
23
23
25
26
28
30
34
Set X
1
2
3
4
5
Set Y
1
2
3
4
5
6
Set P
47
53
58
62
Set Q
37
43
68
72
For students who have a difficult time starting without calculating, help them to compare the values in the ones place for the first and third pairs of data.
The purpose of the discussion is to bring out methods students used to notice whether the pairs of data sets have the same mean or the same MAD or both.
Poll the class for each pair of data sets as to whether they had the same mean, the same MAD, both, or neither.
After students have had a chance to register their vote, ask some students to explain their reasoning for their answer.
Earlier, students compared two groups when data from both populations are known. In this activity, students compare two groups using only samples from each group. Students construct informal arguments to explain why the different samples come from populations that are meaningfully different or not (MP3).
Arrange students in groups of 2. Allow students 3–5 minutes quiet work time. Follow with partner and whole-class discussions.
Consider the question: Do tenth-grade students' backpacks generally weigh more than seventh-grade students' backpacks?
Here are dot plots showing the weights of backpacks for a random sample of students from these two grades:
<p>A dot plot for “backpack weight in pounds,” labeled "grade 7," and the numbers 1 through 22 are indicated. The data are as follows: 2 pounds, 1 dot. 3 pounds, 2 dots. 4 pounds, 3 dots. 5 pounds, 2 dots. 6 pounds, 1 dot. 7 pounds, 2 dots. 9 pounds, 1 dot. 11 pounds, 1 dot. 12 pounds, 1 dot. 13 pounds, 1 dot. </p>
<p>A dot plot for “backpack weight in pounds.” labeled "grade 10," and the numbers 1 through 22 are indicated. The data are as follows: 10 pounds, 1 dot. 11 pounds, 2 dots. 12 pounds, 1 dot. 13 pounds, 2 dots. 14 pounds, 2 dots. 15 pounds, 2 dots. 16 pounds, 1 dot. 18 pounds, 2 dots. 20 pounds, 1 dot. 22 pounds, 1 dot.</p>
The mean weight of this sample of seventh-grade backpacks is 6.3 pounds. Do you think the mean weight of backpacks for all seventh-grade students is exactly 6.3 pounds?
The purpose of the discussion is for students to think about how comparing groups by using data from samples differs from comparing groups when the population is known.
Ask partners to share their decision about whether the groups had a meaningful difference with the class.
Consider asking these questions for discussion:
In this activity, students look at different samples from the same population to see that their means are relatively close based on the MADs of the samples to motivate a general rule for determining whether two groups are meaningfully different. This concept can be reversed to say that if two samples have means that are not very close, then the samples likely came from populations that are quite different. A general rule is given to determine whether two populations are meaningfully different based on the mean and MAD from a sample of each (MP6).
Keep students in groups of 2. Allow students 5 minutes of partner work time, and then pause the class to assign samples and explain the general rule.
Ask students to pause after the third question in order to explain the general rule and assign a sample to each group. After all students have paused, assign each group 1 of the 10 samples to work with for the last two questions. Further, explain to students:
Give students 5 more minutes of partner work time, and follow with a whole-class discussion.
Here are 10 random samples from the same population of seventh-grade students' backpack weights.
sample 1, mean: 5.8 poundsA dot plot titled "sample one," with the numbers 0 through 18 indicated. The data are as follows: 1 pound, 3 dots. 2 pounds, 1 dot. 4 pounds, 3 dots. 5 pounds, 1 dot. 6 pounds, 1 dot. 7 pounds, 1 dot. 10 pounds, 4 dots. 12 pounds, 1 dot.
sample 2, mean: 9.2 poundsA dot plot titled "sample two," with the numbers 0 through 18 indicated. The data are as follows: 4 pounds, 1 dot. 5 pounds, 1 dot. 6 pounds, 1 dot. 7 pounds, 1 dot. 8 pounds, 2 dots. 9 pounds, 3 dots. 10 pounds, 1 dot. 12 pounds, 4 dots. 15 pounds, 1 dot.
sample 3, mean: 5.5 poundsA dot plot titled "sample three," with the numbers 0 through 18 indicated. The data are as follows: 1 pound, 1 dot. 2 pounds, 1 dot. 3 pounds, 3 dots. 4 pounds, 1 dot. 5 pounds, 1 dot. 6 pounds, 3 dots. 7 pounds, 1 dot. 8 pounds, 1 dot. 9 pounds, 1 dot. 10 pounds, 2 dots.
sample 4, mean: 7.3 poundsA dot plot titled "sample four," with the numbers 0 through 18 indicated. The data are as follows: 1 pound, 1 dot. 4 pounds, 2 dots. 5 pounds, 2 dots. 6 pounds, 1 dot. 7 pounds, 2 dots. 8 pounds, 2 dots. 10 pounds, 3 dots. 12 pounds, 1 dot. 13 pounds, 1 dot.
sample 5, mean: 7.2 poundsA dot plot titled "sample five," with the numbers 0 through 18 indicated. the data are as follows: 3 pounds, 2 dots. 4 pounds, 3 dots. 5 pounds, 1 dot. 8 pounds, 3 dots. 9 pounds, 2 dots. 10 pounds, 2 dots. 11 pounds, 1 dot. 12 pounds, 1 dot.
sample 6, mean: 6.6 poundsA dot plot titled "sample six," with the numbers 0 through 18 indicated. The data are as follows: 1 pound, 1 dot. 2 pounds, 2 dots. 4 pounds, 1 dot. 5 pounds, 3 dots. 8 pounds, 2 dots. 9 pounds, 3 dots. 10 pounds, 1 dot. 11 pounds, 1 dot. 12 pounds, 1 dot.
sample 7, mean: 5.2 poundsA dot plot titled "sample seven," with the numbers 0 through 18 indicated. The data are as follows: 1 pound, 1 dot. 2 pounds, 3 dots. 3 pounds, 1 dot. 4 pounds, 2 dots. 5 pounds, 1 dot. 6 pounds, 1 dot. 7 pounds, 1 dot. 8 pounds, 3 dots. 9 pounds, 2 dots.
sample 8, mean: 5.3 poundsA dot plot titled "sample eight," with the numbers 0 through 18 indicated. The data are as follows: 0 pounds, 1 dot. 1 pound, 1 dot. 2 pounds, 1 dot. 4 pounds, 4 dots. 5 pounds, 2 dots. 6 pounds, 1 dot. 8 pounds, 4 dots. 9 pounds, 2 dots.
sample 9, mean: 6.3 poundsA dot plot titled "sample nine," with the numbers 0 through 18 indicated. The data are as follows: 0 pounds, 1 dot. 1 pound, 1 dot. 5 pounds, 3 dots. 6 pounds, 1 dot. 7 pounds, 4 dots. 8 pounds, 3 dots. 9 pounds, 1 dot. 12 pounds, 1 dot.
sample 10, mean: 6.4 poundsA dot plot titled "sample ten," with the numbers 0 through 18 indicated. The data are as follows: 0 pounds, 3 dots. 1 pound, 1 dot. 4 pounds, 2 dots. 6 pounds, 2 dots. 7 pounds, 1 dot. 9 pounds, 2 dots. 10 pounds, 1 dot. 11 pounds, 2 dots. 18 pounds, 1 dot.
A sample of tenth-grade students’ backpacks has a mean weight of 14.8 pounds. The MAD for this sample is 2.7 pounds. Your teacher will assign you one of the samples of seventh-grade students’ backpacks to use.
The purpose of the discussion is for students to understand the general rule for determining if two samples suggest a meaningful difference between their populations.
Select at least one group assigned to each of the samples to share their responses to the last two questions and record for all to see. Note that all 10 samples from the seventh-grade students have means that are within 2 MADs of one another, but the mean from the tenth-grade-student sample is at least 2 MADs away from the mean of each of the seventh-grade-student samples.
Note that the general rule only has two possible outcomes: “There is a meaningful difference.” or “There is not enough information to say there is a meaningful difference.” If the means are less than 2 MADs apart, the general rule cannot say whether two samples were drawn from populations that contain identical data.
Ask students, “Based only on the dot plots for the 10 samples, would you have guessed that they all might have come from the same population? Explain your reasoning.” (Maybe. There is a lot of overlap among all of the samples.)
In this activity, students practice using the general rule developed in an earlier activity by estimating the measure of center for a population and comparing populations based on those estimates as well as the associated measure of variability. Students must construct viable arguments for their conclusions about whether there is a meaningful difference between the groups (MP3).
Keep students in groups of 2.
Explain to students that different regions had different raw materials and techniques for constructing metal. One way of testing ancient metal is by looking at the carbon content in the steel. In some cases, this content could determine the region where the metal was made.
Ask students how the general rule from the previous activity might be adapted to use median and interquartile range (IQR) rather than mean and MAD.
Allow students 10 minutes of partner work time, and follow with a whole-class discussion.
When anthropologists find steel artifacts, they can test the amount of carbon in the steel to learn about the people that made the artifacts. Here are the box plots showing the percentage of carbon in samples of steel that were found in two different regions:
<p>Two box plots from 0 point 32 to 0 point 76 by 0 point 4's. Top box plot labeled "region 1". Whisker from about 0 point 41 to 48. Box from about 0 point 62 to about 0 point 67 with vertical line at 0 point 64. Whisker from about 0 point 67 to about 0 point 7. Bottom box plot labeled "region 2". Whisker from about 0 point 37 to 45. Box from about 0 point 45 to 0 point 48 with vertical line at about 0 point 46. Whisker from 0 point 48 to about 0 point 57. </p>
Is there any steel found in region 1 that has:
more carbon than some of the steel found in region 2?
less carbon than some of the steel found in region 2?
Based only on the box plots, do you think there is a meaningful difference between all the steel artifacts found in regions 1 and 2?
A sample of artifacts known to come from region 1 has a median of 0.64% carbon in the steel and an interquartile range of 0.05%.
A sample from region 2 has a median of 0.47% carbon in the steel and an IQR of 0.03%.
What is the difference between the sample medians for these two regions?
The anthropologists who conducted the study conclude that there is a meaningful difference between the steel from these regions. Do you agree? Explain or show your reasoning.
The purpose of the discussion is for students to understand how to adapt the general rule for determining a meaningful difference between populations to median and IQR.
Consider asking these questions for discussion:
“Why did this problem use median and IQR instead of mean and MAD?” (Since the distribution for region 1 is not symmetric, it makes more sense to use the median. Also the box plots will show the median and IQR, but there is not a good way to know the mean and MAD.)
“Is there any overlap in the data from the two regions?” (Yes. The smallest percentage of carbon from the region 1 is well below the median from region 2, while the typical percentage of carbon from region 1 is much greater than from region 2.)
“On the box plot in the activity, draw a dot two IQRs above the median for region 2. Then draw a star two IQRs below the median for region 1. How do these help you see that there is a meaningful difference in the medians?” (The dot is at 0.53 and the star is at 0.54. Since the median for region 1 is not below the dot nor is the median for region 1 above the star, there must be a meaningful difference.)
“A piece of steel is found in a place between the two regions sampled. Would testing the percentage of carbon from this metal be useful in determining the region from which it came?” (Yes. Since there is a meaningful difference in the percentage of carbon in the steel from the two regions, it should give a good indication which region created the metal.)
Sometimes we want to compare two different populations. For example, is there a meaningful difference between the weights of pugs and beagles? Here are histograms showing the weights for a sample of dogs from each of these breeds:
A histogram for two different populations: On the horizontal axis, the numbers 6 through 11, in increments of zero point 5, are indicated. The label “pug weights in kilograms” is indicated for the numbers 6 through 8 and “beagle weights in kilograms” is indicated for the numbers 9 through 11. On the vertical axis, the numbers 0 through 8 are indicated. The data represented by the bars are as follows: Pug weights in kilograms: Weight from 6 up to 6 point 5, 5. Weight from 6 point 5 up to 7, 5. Weight from 7 up to 7 point 5, 7. Weight from 7 point 5 up to 8, 3. A triangle is located at 6 point 9 kilograms. Beagle weights in kilograms: Weight from 9 up to 9 point 5, 3. Weight from 9 point 5 up to 10, 3. Weight from 10 up to 10 point 5, 8. Weight from 10 point 5 up to 11, 6. A triangle is located at 10 point 1.
The red triangles show the mean weight of each sample, 6.9 kg for the pugs and 10.1 kg for the beagles. The red lines show the weights that are within 1 MAD of the mean. We can think of these as typical weights for the breed. These typical weights do not overlap. In fact, the distance between the means is , or 3.2 kg, over 6 times the larger MAD! So we can say there is a meaningful difference between the weights of pugs and beagles.
Is there a meaningful difference between the weights of male pugs and female pugs? Here are box plots showing the weights for a sample of male pugs and a sample of female pugs:
Two box plots labeled “male pug weights in kilograms” and “female pug weights in kilograms” are indicated. The numbers 4 through 8 point 5, in increments of zero point 5, are indicated. The five-number summary for the box plots are as follows: Male pug weights in kilograms: Minimum value, 6 point 4. Maximum value, 8 point 3. Q1, 7 point 2. Q2, 7 point 6. Q3, 7 point 9. Female pug weights in kilograms: Minimum value, 6 point 2. Maximum value, 8. Q1, 6 point 4. Q2, 6 point 9. Q3, 7 point 3.
We can see that the medians are different, but the weights between the first and third quartiles overlap. Based on these samples, we would say there is not a meaningful difference between the weights of male pugs and female pugs.
In general, if the measures of center for two samples are at least two measures of variability apart, we say the difference in the measures of center is meaningful. Visually, this means the ranges of typical values do not overlap. If they are closer, then we don't consider the difference to be meaningful.