Unit 8, Lesson 18
Using Data to Solve Problems
Grade 6
18.1
Warm-up
In one study on wild bears, researchers measured the head lengths and head widths, in inches, of 143 wild bears. The box plots summarize the data from the study.
A set of two box plots, head length in inches, from 2 to 20 by ones. Top box plot labeled male bears. Whisker from 9 to 12 point 5. Box from 12 point 5 to 15 point 5 with vertical line at 13 point 5. Whisker from 15 point 5 to 18 point 5. Bottom box plot labeled female bears. Whisker from 10 to 12. Box from 12 to 13 point 5 with vertical line at 12 point 5. Whisker from 13 point 5 to 15 point 5.
A set of two box plots, head width in inches, from 2 to 20 by ones. Top box plot labeled male bears. Whisker from 4 to 5 point 5. Box from 5 point 5 to 8 with vertical line 6 point 5. Whisker from 8 to 10. Bottom box plot whisker from 4 point 5 to 5. Box from 5 to 6 point 5 with vertical line at 6. Whisker from 6 point 5 to 7 point 5.
- Write four statistical questions that could be answered using the box plots: two questions about the head length and two questions about the head width.
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Trade questions with your partner.
- Decide if each question is a statistical question.
- Use the box plots to answer each question.
18.4
Activity
Scientists studying the yellow perch, a species of fish, believe that the length of a fish is related to its age. This means that the longer the fish, the older it is. Adult yellow perch vary in size, but they are usually between 10 and 25 centimeters.
Scientists at the Great Lakes Water Institute caught, measured, and released yellow perch at several locations in Lake Michigan. This summary is based on a sample of yellow perch from one of these locations.
| length of fish in centimeters | number of fish |
|---|---|
| 0 to less than 5 | 5 |
| 5 to less than 10 | 7 |
| 10 to less than 15 | 14 |
| 15 to less than 20 | 20 |
| 20 to less than 25 | 24 |
| 25 to less than 30 | 30 |
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Use the data to make a histogram that shows the lengths of the captured yellow perch. Each bar should contain the lengths shown in each row in the table.
- How many fish were measured? How do you know?
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Use the histogram to answer these questions.
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How would you describe the shape of the distribution?
- Estimate the median length for this sample. Describe how you made this estimate.
- Predict whether the mean length of this sample is greater than, less than, or nearly equal to the median length for this sample of fish? Explain your prediction.
- Would you use the mean or the median to describe a typical length of the fish being studied? Explain your reasoning.
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Based on your work so far:
- Would you describe a typical age for the yellow perch in this sample as: young, adult, or old? Explain your reasoning.
- Some researchers are concerned about the survival of the yellow perch. Do you think the lengths (or the ages) of the fish in this sample are something to worry about? Explain your reasoning.
Student Lesson Summary
The data displays show the distribution of stickers on 30 pages.
A box plot and dot plot for "stickers on a page". The numbers 8 through 34, in increments of two, are indicated. The box plot is above the dot plot. The five-number summary for the box plot is as follows: Minimum value, 9. Maximum value, 34. Q1, 16. Q2, 20.5. Q3, 26. For the dot plot a triangle is indicated at 21 stickers. A horizontal line is drawn below the triangle and begins at 15.4 and ends at 26.6. The data for the dot plot are as follows: 9 stickers, 1 dot. 10 stickers, 1 dot. 11 stickers, 2 dots. 12 stickers, 1 dot. 14 stickers, 1 dot. 16 stickers, 2 dots. 17 stickers, 1 dot. 18 stickers, 2 dots. 19 stickers, 1 dot. 20 stickers, 3 dots. 21 stickers, 1 dot. 22 stickers, 3 dots. 23 stickers, 1 dot. 24 stickers, 2 dots. 26 stickers, 2 dots. 28 stickers, 1 dot. 30 stickers, 1 dot. 32 stickers, 2 dots. 33 stickers, 1 dot. 34 stickers, 1 dot.
The mean number of stickers, marked by the triangle, is 21. This tells us that if the total number of stickers were redistributed so that each page had the same number of stickers, then each page would have 21. The MAD is 5.6 stickers, which suggests that a page typically has between 15.4 stickers and 26.6 stickers.
The box plot for the same data set is shown above the dot plot. The median shows that half of the pages have greater than or equal to 20.5 stickers, and half have less than or equal to 20.5 stickers. The box shows that the IQR is 10 and that the middle half of the pages have between 16 and 26 stickers.
In this case, the median number of stickers is very close to the mean number of stickers, and the IQR is about twice the MAD. This tells us that the two pairs of measures of center and spread are very similar.
Now let’s look at another example of 30 different pages.
A box plot and a dot plot for "stickers on a page". The numbers 8 through 34, in increments of two, are indicated. For the box plot: The minimum value is 9 and the maximum value is 26. Q1 has a value of 20, Q2 has a value of 23, and Q3 has a vlaue of 24. For the dot plot: A triangle is indicated at 21 stickers. A line indicates the distance from 17.6 to 24.4 stickers. The data is as follows: 9 stickers, 1 dot. 10 stickers, 1 dot. 13 stickers, 1 dot. 14 stickers, 1 dot. 16 stickers, 1 dot. 17 stickers, 1 dot. 19 stickers, 1 dot. 20 stickers, 2 dots. 21 stickers, 2 dots. 22 stickers, 3 dots. 23 stickers, 6 dots. 24 stickers, 5 dots. 25 stickers, 4 dots. 26 stickers, 1 dot.
Here the mean is 21 stickers, and the MAD is 3.4 stickers. This suggests that a page typically has between 17.6 and 24.4 stickers. The median number of stickers is 23, and the box plot shows that the middle half of the data are between 20 and 24 stickers. These two pairs of measures paint very different pictures of the variability of the number of stickers on a page.
From the dot plot, we can see that the median (23 stickers) is closer to the middle of the big cluster of values. If we were to ignore the pages with very few stickers, the median and IQR would give a more accurate picture of how many stickers are typically on a page.
When a distribution is not symmetrical, the median and IQR are often better measures of center and spread than the mean and MAD are. However the decision about which pair of measures to use depends on what we want to know about the group we are investigating.
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