Our Backpack Data: Making Conclusions from Graphs Objectives:

• To choose appropriate visual displays to answer questions from a study. • To discover multiple ways of reading a visual display, and to answer multiple

questions from a study from a single display. • To communicate clear, responsive conclusions to a statistical question. • To use and communicate clear evidence for their conclusions. • To generate new questions, make appropriate visual displays, and use them to

make well-justified and clearly communicated evidence Statistics Prerequisites:

• Familiarity with a variety of visual displays: dot plot, histogram, box plot, scatter plot, two-way tables and plots, percentiles.

• Familiarity with the idea of describing a distribution by its shape, center, and spread.

• Familiarity with comparing groups, making conclusions about groups with evidence.

Level of students.

• Strong pre-AP statistics students • AP Statistics students • Teachers who want to learn more about how statistical thinking differs from

mathematical thinking.

Our Backpack Data: Making Conclusions from Graphs We can ask lots of questions about our individuals, their backpack weights, and their guesses. We can also generate lots of data displays about our graphs. But which graphs help us answer which questions? Below are some questions about our data, and some graphs. Part I: Analysis (15-25 min): Divide into groups of three students. Determine which graph(s) do a good job answering the question at hand. After you select a graph, make a conclusion about the question. Finally, justify your conclusion with specific evidence from the graph. Part II: Comparison/Discussion (20-30 minutes): Each group should present their choice(s) for each question, the conclusions they made, and the evidence for their conclusions. For each question, discuss the following: a) Which graphs can be used to answer the question? Which ones cannot? Explain. b) Which graphs do the best job of conveying a clear answer to the question? Explain. Part III: (Individual activity)

a) Creating your own question. Enter or download the data (Fathom data file: backpack_data_2006.ftm). Create a new question that can be answered using the data. Then create two different graphs that can be used to answer your question. For each graph, make a conclusion, and justify your conclusion with specific evidence form your graph.

b) Incorporating Class data. Enter or download the data (Fathom data file:

backpack_data_2006.ftm). In addition, create a new attribute called “group.” In this attribute, enter “adult” for the cases already in the data file. Then enter “student” for your class data. Enter the guessed and actual weights for your class. Prepare 1 or 2 questions that compare your class with adults. Use Fathom to create effective visual displays for your questions. Make well - justified conclusions, citing specific evidence from your visual displays.

Here are some of the variables that were studied: Gender: Gender of each individual (m or f) Guess: The guessed weight of an individual’s backpack (in lbs.) Actual: The actual weight of an individual’s backpack (lbs.) Difference: Guess – Actual (lbs.) Under_3_lbs. Whether the individual’s guess was less than 3 lbs. away or more than 3 lbs. away

Questions to Analyze:

1) Overall, how heavy are individual’s backpacks?

2) Which group, males or females, do a better job at guessing the weight of their backpack?

3) What fraction of people missed the correct weight of their backpack weight by less than 3 pounds?

4) Are people good at guessing their backpack weights?

5) When people guess the weight of their backpack incorrectly, do they tend to overestimate or underestimate?

Response Template (Use one for each response) Question

Choice of Graph

Your Conclusion

Evidence for your conclusions

( )countGender

f m

51015202530

under_3_lbs3 or mor under 3

Backpack Data Bar Chart

Graph A

Gen

der

0

4

8

12

f

4

8

12

m

Actual0 5 10 15 20 25 30

Collection 1 Histogram

Graph B

Genderf m

differences-15 -10 -5 0 5 10 15 20

Backpack Data Dot Plot

Graph C

Guess = actual

0

5

10

15

20

25

30

Actual0 5 10 15 20 25 30

Backpack Data Scatter Plot

Graph D

fm

differences-15 -10 -5 0 5 10 15 20

Backpack Data Box Plot

Graph E

Guess0 5 10 15 20 25 30

Backpack Data Box Plot

Actual0 5 10 15 20 25 30

Backpack Data Box Plot

Graph F

3 or

mor

unde

r 3

Genderf m

Backpack Data Breakdow n Plot

Graph G

0

20

40

60

80

100

120

differences-15 -10 -5 0 5 10 15 20

Backpack Data Percentile Plot

Graph H

Teacher Notes, Our Backpack Data: Making Conclusions from Graphs

Objectives:

• To introduce students to a variety of one-variable and two-variable displays of data.

• To choose appropriate visual displays to answer questions from a study • To discover multiple ways of reading a visual display, and to answer multiple

questions form a single display • To communicate clear, responsive conclusions to a statistical question. • To use and communicate clear evidence for their conclusions. • To generate new questions, make appropriate visual displays, and use them to

make well-justified and clearly communicated evidence Activity Time: 40-50 minutes Setting: Part I can be done in groups of two or three. Part II can be done as a whole-class discussion. Part III is an extension activity that can be assigned individually. Statistics Prerequisites:

• Familiarity with a variety of visual displays: dot plot, histogram, box plot, scatter plot, two-way tables and plots, percentiles.

• Familiarity with the idea of describing a distribution by its shape, center, and spread.

• Familiarity with comparing groups, making conclusions about groups with evidence.

Statistics Skills:

• Understanding the meaning of a percentile, relative frequency, • Analyzing the information in a visual display:

Fathom Prerequisites: Students do not need access to Fathom to complete Parts I and II of this activity. Part III requires students to make graphs, create attributes, and use the formula editor. Fathom Skills: Students can define attributes using conditional functions, and experiment with a variety of graphs to explore answer to questions they generate.

Procedure: In this group activity, students are asked to match up different statistical questions from the PCMI backpack data to an appropriate visual display. Students should

be familiarized with the data collection process for the backpack data: 51 adults participating in the 2006 Park City Mathematics Institute were asked to estimate the weight of their book bags in pounds. Each subject book bag was then weighed with a typical bathroom scale. The guessed and actual weights were recorded in pounds. The data file backpack_data_2006.ftm contains guessed and actual weights, as well as the gender of each subject, and the difference between their guess and the actual weight (guess-actual). The instructor may choose to have students collect data in a similar fashion, and replace the data with their own. Students should be divided into groups of two or three. Each student should be given the handout above, with 8-10 copies of the response template.

Part I: In small groups, students should choose at least one graph that can answer each question. However, some questions can be answered by more than one graph. Students should be prepared not only to choose graphs, but also to prepare an answer for each question and justify their conclusion with specific evidence from the graph(s) they chose. Each small group should complete one response template for each answer they provide. Some small groups may elect to answer a question in more than one way, but should be encouraged to provide at least one complete response for each question.

Part II: After small group discussion, each group should be invited to present their results by attaching each response with its corresponding graph. The instructor should go through each statistical question one at a time, and invite 2-3 different groups to describe and explain their choices. The whole class can then compare the different conclusions found from each group, and discuss their explanations. Because some of the questions are broad enough to invite interpretation, the instructor should look for opportunities to highlight different interpretations of the same question. Students should be encouraged to ask questions that require their classmates to communicate their reasoning with detail and clarity.

Part III: If students have access to Fathom or another data analysis program, part III can be an open-ended take-home assignment that should take about 30-45 minutes. Students should be provided with the raw data, and create a new question, choose (or create) two graphs to answer their question, and make conclusions based on those graphs.

Analyzing Data: The table below can be used to determine how to use each graph to answer the question. If the graph is not usable, the box is blank.

How Heavy Better Guessers, M or F?

What fraction missed by less than 3 lbs?

Are people good at guessing? Note: “Good” is a term that will need to be defined by students. Answers will vary.

Overestimate/ Underestimate?

Graph A

Women. A majority of women and under 3 lbs, but a majority of men are 3 lbs or more.

26/52 = 50% missed by 3 lbs or under.

If we accept “under 3 lbs” as “good,” then about half are “good.”

Graph B

Combine frequencies for M and F, to notice a center weight near 11 lbs. Most individuals within 4-5 pounds. Some right skew ness. Student may also notice that men carry heavier backpacks.

Graph C

Women. 13/16 of women within 5 lbs, but a much smaller proportion of male dots are within this range.

By making off -3 and +3 on the “differences” axes, about half of the points within these bounds.

This graph is most useful. Students can define good in many was and use this graph

The number of individuals with difference > 0 is slightly higher than the number < 0. Perhaps a slight tendency to overestimate, but unclear if true for population.

Graph D

The x-coordinates of most of the points on the scatter plot show “actual” near 10, give/take 4-5 pounds. A few are scatter considerably higher – suggesting actual weights are skewed right slightly.

Draw the lines Y = x +3, Y = x – 3 , observe about half the points within these two boundaries.

This graph is also useful. Students can define good in many was and use this graph to determine % points sufficiently close to “actual = guessed.”

The number of points above “guessed = actual” is slightly higher than the number below. Perhaps a slight tendency to overestimate, but unclear if true for population.

Graph E

The IQR for women is much smaller than for men.

The medians for each box plot are both near zero, with the men slightly skewed right. Perhaps a slight tendency to overestimate for men, hard to tell for entire population.

Graph F

“actual” boxplot shows median near 10 lbs. slightyly skewed right in the top quartile, but 1/4 of all between 10 qand 13 lbs.

Graph G

For women 11/16 under 3 lbs, a much higher proportion than 15/36 for men In the study.

26/50 dots are in the under 3 lbs category.

If we accept “under 3 lbs” as “good,” then about half are “good.”

Part II: More Issues for discussion:

a) Sureness of Conclusions: when students make conclusions, how confident are they about their answers? What facts would allow them to be more confident about their conclusions? More data? More information about the data collection process? More clear definitions?

b) Percentile Plot: Although students may be familiar with percentiles, they may have never seen a percentile plot. These are also called cumulative frequency plots, or ogives. It will be valuable to monitor group discussions so that all students understand how to read these graphs. It’s also useful to have students construct a box plot from the percentile plot as a way to assess their understanding.

c) Data Collection The data from the adults was collected between 7:30 am and 8:00 am before morning sessions at a mathematics conference. Do these facts influence the conclusions?

d) Measuring percentage errors: In defining a “good guess,” Some students may want to define “error” in terms of a percentage of the actual weights. This idea is indeed an effective way of measuring errors. Only the scatter plot can allow us to measure individual’s percentage error. For example, if a student wants to know which points fall 20% above or 20% below the actual weight, they should graph the lines “guess” = 1.2 “actual” and “guess” = 0.8 actual,” respectively.

e) Creating quantitative definitions. The question about “good guessing” is

deliberately vague. As a result, students need to create a consistent definition of “good;” perhaps this means guessing your backpack within 2 pounds, or within 10% of the actual weight. Students should be expected to clearly describe their rule, justify why it means the guess is “good,” and make correct conclusions based of their definition of “good.”

Graph H

Suppose “Good” means within 2 lbs. Then find percentile for difference = 2, and difference = -2. , subtract. That proportion are “good.”

50th %ile at 0, but 20th at -2, and 80th at +4. Slight skew right in differences. Suggest when incorrect, overestimates are larger than underestimates.

Part III: Extensions a) The first extension can be used as an end-of-topic assessment. A possible rubric for assessment follows: Criterion Grade

0 1 2 Comments

Questions: the 2 or 3 questions asked are clearly defined.

Graph chosen is/are appropriate for the question asked.

Overall Conclusion: makes appropriate conclusions without over-reaching.

Use of evidence: Student points out specific evidence in justifying their conclusion.

Communication is clear and concise. Students use of statistical terms is generally correct, and appropriate for what they have learned.

2 pt strong mastery, free of errors. 1 pt partial mastery, slight errors. 0 pt non-mastery: major errors. b) Students may collect their own backpack data, and make comparisons about guessing against adults in the PCMI data. Other variables to consider that may be related to backpack weights and guesses are:

• Whether the students lifts weights • Grade Level • Whether the backpack is light or heavy (may make it easier to estimate

weight) • Time of day (maybe backpacks are heaviest in the morning, but fewer books

are carried during lunchtime)