Unit 1, Lesson 15
Comparing Data Sets
Algebra 1
15.1
Warm-up
Bowling Partners
Instructional Routines
Materials
None
Activity Narrative
The mathematical purpose of this activity is for students to compare different distributions using shape, measures of center, and measures of variability. This Warm-up prompts students to compare four distributions representing recent bowling scores for potential teammates. It gives students a reason to use language precisely (MP6) and gives you the opportunity to hear how they use terminology and talk about characteristics of the images in comparison to one another.
Launch
Arrange students in groups of 2–4. Tell students that bowling is a game in which a higher score is better and the maximum score is 300. It is typical for non-professional bowlers to score a little over 100.
Activity
NoneEach histogram shows the bowling scores for the last 25 games played by each person. Choose 2 of these people to join your bowling team. Explain your reasoning.
Person A
- mean: 118.96
- median: 111
- standard deviation: 32.96
- interquartile range: 44
Histogram from 60 to 300 by 10's. Bowling scores for person A. Beginning at 60 up to but not including 70, height of bar at each interval is 0, 0, 5, 4, 3, 2, 3, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.
Person B
- mean: 131.08
- median: 129
- standard deviation: 8.64
- interquartile range: 8
Histogram from 60 to 300 by 10's. Bowling scores for person B. Beginning at 60 up to but not including 70, height of bar at each interval is 0, 0, 0, 0, 0, 0, 13, 8, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.
Person C
- mean: 133.92
- median: 145
- standard deviation: 45.04
- interquartile range: 74
Histogram from 60 to 300 by 10's. Bowling scores for person C. Beginning at 60 up to but not including 70, height of bar at each interval is 3, 2, 1, 0, 1, 2, 2, 1, 1, 5, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0.
Person D
- mean: 116.56
- median: 103
- standard deviation: 56.22
- interquartile range: 31.5
Histogram from 60 to 300 by 10's. Bowling scores for person D. Beginning at 60 up to but not including 70, height of bar at each interval is 0, 3, 3, 4, 5, 4, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2.
Activity Synthesis
Ask each group to share one bowler they would choose and their reasoning. If none of the groups select a certain player, ask why this player was not chosen, or give reasons why another team may want this player on their team. If time allows, ask if there is any additional information that might make that player more desirable. For example, knowing the conditions behind each score might be helpful. Player C might be a new bowler, and the lower scores might have been when Player C was learning, but the newer scores may all be closer to 200. Player D might not take practice seriously, but can bowl a perfect game in competition.
Because there is no single correct answer, attend to students’ explanations, and ensure that the reasons given are correct. During the discussion, ask students to explain how they used the statistics given as well as the histograms.
Student Lesson Summary
To compare data sets, it is helpful to look at the measures of center and measures of variability. The shape of the distribution can help choose the most useful measure of center and measure of variability.
When distributions are symmetric or approximately symmetric, the mean is the preferred measure of center and should be paired with the standard deviation as the preferred measure of variability. When distributions are skewed or when outliers are present, the median is usually a better measure of center and should be paired with the interquartile range (IQR) as the preferred measure of variability.
Once the appropriate measure of center and measure of variability are selected, these measures can be compared for data sets with similar shapes.
For example, let’s compare the number of seconds it takes football players to complete a 40-yard dash at two different positions. First, we can look at a dot plot of the data to see that the tight-end times do not seem distributed symmetrically, so we should probably find the median and IQR for both sets of data to compare information.
The median and IQR could be computed from the values, but can also be determined from a box plot.
Box plot from 4 point 25 to 5 point 75 by point 25’s. Wide receiver time in seconds. Whisker from 4 point 31 to 4 point 405. Box from 4 point 405 to 4 point 665 with vertical line at 4 point 5. Whisker from 4 point 665 to 4 point 8.
Box plot from 4 point 25 to 5 point 75 by point 25’s. Tight end times in seconds. Whisker from 4 point 56 to 4 point 685. Box from 4 point 685 to 5 point 225 with vertical line at 4 point 87. Whisker from 5 point 255 to 5 point 7.
This shows that the tight-end times have a greater median (about 4.9 seconds) compared to the median of wide-receiver times (about 4.5 seconds). The IQR is also greater for the tight-end times (about 0.5 seconds) compared to the IQR for the wide-receiver times (about 0.25 seconds).
This means that the tight ends tend to be slower in the 40-yard dash when compared to the wide receivers. The tight ends also have greater variability in their times. Together, this can be taken to mean that, in general, a typical wide receiver is faster than a typical tight end is, and the wide receivers tend to have more similar times to one another than the tight ends do to one another.
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