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Lesson 18Using Data to Solve Problems

Let's compare data sets using visual displays.

Learning Targets:

  • I can decide whether mean and MAD or median and IQR would be more appropriate for describing the center and spread of a data set.
  • I can draw an appropriate graphical representation for a set of data.
  • I can explain what the mean and MAD or the median and IQR tell us in the context of a situation and use them to answer questions.

18.1 Wild Bears

In one study on wild bears, researchers measured the head lengths and head widths, in inches, of 143 wild bears. The ages of the bears ranged from newborns (0 years) to 15 years. The box plots summarize the data from the study.

Two sets of box plots for "head length in inches" and "head width in inches". The numbers 2 through 20, in increments of 2, are indicated and there are tick marks midway between each indicated number. For "head length in inches" the top box plot is labeled "male bears" and the five-number summary are as follows: Minimum value, 9. Maximum value, 19. Q1, 12 point 5. Q2, 13 point 5. Q3, 15 point 5. The bottom box plot is labeled "female bears" and the five-number summary are as follows: Minimum value, 10. Maximum value, 15 point 5. Q1, 12. Q2, 12 point 5. Q3, 13 point 5. For "head width in inches" the top box plot is labeled "male bears" and the five-number summary are as follows: Minimum value, 4. Maximum value, 10. Q1, 5 point 5. Q2, 6 point 5. Q3, 8. The bottom box plot is labeled "female bears" and the five-number summary are as follows: Minimum value, 4 point 5. Maximum value, 7 point 5. Q1, 5. Q2, 6. Q3, 7 point 5.
  1. 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. 
  2. Trade questions with your partner.
    1. Decide if each question is a statistical question.
    2. Use the box plots to answer each question.

18.2 Math Homework (Part 1)

Over a two-week period, Mai had the following number of math homework problems each school day.

2 15 20 0 5 25 1 0 10 12

  1. Calculate the following. Show your reasoning.

    1. The mean number of math homework problems.
    2. The mean absolute deviation (MAD).
  2. Interpret the mean and MAD. What do they tell you about the number of homework problems Mai had over these two weeks?

  3. Find or calculate the following values and show your reasoning.

    1. The median, quartile, maximum, and minimum of the same data on Mai’s math homework problems.
    2. The interquartile range (IQR).
  4. Which pair of measures of center and variability—mean and MAD, or median and IQR—do you think summarize the distribution of Mai’s math homework assignments better? Explain your reasoning.

You may use the applet below to help if you choose to. Enter the values needed to calculate the IQR and the mean when prompted.

open applet in presentation mode

18.3 Math Homework (Part 2)

Jada wanted to know whether a dot plot, a histogram, or a box plot would best summarize the center, variability, and other aspects of her homework data.

2 15 20 0 5 25 1 0 10 12

  1. Use the axis to make a dot plot to represent the data. Mark the position of the mean, which you calculated earlier, on the dot plot using a triangle ( \Delta ). From the triangle, draw a horizontal line segment to the left and right sides to represent the MAD.

    Missing image 6.8.E1.Image.03a.blank

  2. Use the five-number summary from the previous task and the grid to draw a box plot that represents Jada’s homework data.

    Missing image 6.8.E1.Image.03b.blank

  3. Work with your group to draw three histograms to represent Jada’s homework data. The width of the bars in each histogram should represent a different number of homework problems, as specified.

    1. One bar represents 10 problems.
      a blank grid with number of math problems as the x axis label
    2. One bar represents 5 problems.
      a blank grid with number of math problems as the x axis label
    3. One bar represents 2 problems.
      a blank grid with number of math problems as the x axis label
  4. Which of the five representations should Jada use to summarize her data? Should she use a dot plot, box plot, or one of the histograms? Explain your reasoning.

You can use the applet to make each type of graph if you choose to.

open applet in presentation mode

18.4 Will the Yellow Perch Survive?

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. The following 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

  1. 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.

    A blank grid. The horizontal axis is labeled “fish length in centimeters” and the numbers 0 through 30, in increments of 5, are indicated. The vertical axis has the numbers 0 through 40, in increments of 5, indicated. There are 4 evenly spaced horizontal gridlines between each indicated number.
  2. How many fish were measured? How do you know?

  3. Use the histogram to answer the following questions.

    1. How would you describe the shape of the distribution?
    2. Estimate the median length for this sample. Describe how you made this estimate.
    3. 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.
    4. Would you use the mean or the median to describe a typical length of the fish being studied? Explain your reasoning.

  4. Based on your work so far:

    1. Would you describe a typical age for the yellow perch in this sample as: “young,” “adult,” or “old”? Explain your reasoning.
    2. 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.

Lesson 18 Summary

The dot plot shows the distribution of 30 cookie weights in grams.

A box plot and a histogram for "cookie weights in grams". 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 grams. 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 grams, 1 dot. 10 grams, 1 dot. 11 grams, 2 dots. 12 grams, 1 dot. 14 grams, 1 dot. 16 grams, 2 dots. 17 grams, 1 dot. 18 grams, 2 dots. 19 grams, 1 dot. 20 grams, 3 dots. 21 grams, 1 dot. 22 grams, 3 dots. 23 grams, 1 dot. 24 grams, 2 dots. 26 grams, 2 dots. 28 grams, 1 dot. 30 grams, 1 dot. 32 grams, 2 dots. 33 grams, 1 dot. 34 grams, 1 dot.

The mean cookie weight, marked by the triangle, is 21 grams. This tells us that if the weights of all of the cookies were redistributed so they all had the same weight, each cookie would weigh 21 grams. The MAD is 5.6 grams, which suggests that a cookie typically weighs between 15.4 grams and 26.6 grams.

The box plot for the same data set is shown above the dot plot. The median shows that half of the weights are greater than or equal to 20.5 grams, and half are less than or equal to 20.5 grams. The box shows that the IQR is 10 and that the middle half of the cookies weigh between 16 and 26 grams.

In this case, the median weight is very close to the mean weight, 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 cookies.

A box plot and a dot plot for "cookie weights in grams". 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 grams. A line indicates the distance from 17.6 to 24.4 grams. The data is as follows: 9 grams, 1 dot. 10 grams, 1 dot. 13 grams, 1 dot. 14 grams, 1 dot. 16 grams, 1 dot. 17 grams, 1 dot. 19 grams, 1 dot. 20 grams, 2 dots. 21 grams, 2 dots. 22 grams, 3 dots. 23 grams, 6 dots. 24 grams, 5 dots. 25 grams, 4 dots. 26 grams, 1 dot.

Here the mean is 21 grams, and the MAD is 3.4 grams. This suggests that a cookie typically weighs between 17.6 and 24.4 grams. The median cookie weight is 23 grams, and the box plot shows that the middle half of the data are between 20 and 24 grams. These two pairs of measures paint very different pictures of the variability of the cookie weights.

The median (23 grams) is closer to the middle of the big cluster of values. If we were to ignore the smaller cookies, the median and IQR would give a more accurate picture of how much a cookie typically weighs.

When a distribution is not symmetrical, the median and IQR are often better measures of center and spread than the mean and MAD. However the decision on which pair of measures to use depends on what we want to know about the group we are investigating.

Lesson 18 Practice Problems

  1. Twenty students timed how long each of them took to solve a puzzle. Here is a dot plot of their solution times.

    1. Predict which measure of center —the mean or the median—will be larger. Explain your reasoning.
    Dot plot for “puzzle-solving time in minutes.” The numbers 0 through 14, in increments of 2, are indicated. The data are as follows: 2 minutes, 1 dot; 6 mintues, 1 dot; 7 minutes, 2 dots; 8 minutes, 3 dots; 9 minutes, 7 dots; 10 minutes; 2 dots; 11 minutes; 3 dots; 12 minutes; 1 dot.
    1. Calculate the mean and the median. Was your prediction correct?
    2. Which pair of measures of center and variability—mean and MAD, or median and IQR—would you choose to describe the distribution of the solution times? Explain your reasoning.

  2. A local library showed a movie during a children’s festival. Jada recorded the ages of 100 children who watched the movie. Here is a histogram for her data.

    Answer each question. Explain your reasoning for each answer.

    1. Can the histogram be used to estimate the mean age of the movie watchers?
    A histogram: The horizontal axis is labeled "age in years" and the numbers 1 through 10 are indicated. On the vertical axis the numbers 0 through 50, in increments of 10, are indicated. The data represented by the bars are as follows: Age from 1 up to 3, 10. Age from 3 up to 5, 25. Age from 5 up to 7, 50. Age from 7 up to 9, 15.
    1. Can the histogram be used to estimate the median age of the movie watchers?  
    2. Can the histogram be used to find the exact median age of the movie watchers?  
    3. Is it possible to tell from the histogram if there was at least one 3-year-old who watched the movie?  

  3. In a game, players are shown a picture that has 14 butterfly shapes hidden in it. They are given 3 minutes to find as many hidden butterfly shapes as they can.

    Twenty fourth-grade students and twenty sixth-grade students played this game. The dot plots represent the numbers of butterfly shapes the two groups found.

    dot plots showing number of butterflies found byt fourth and sixth grade

    Write a few sentences comparing the numbers of hidden butterflies found for these two groups of students.

  4. In a round of mini-golf, Elena records the number of strokes it takes to hit the ball in the hole for each green. Find the range, median, and interquartile range for the number of strokes needed on each green.

    2 3 1 4 5 2 3 4 3