Statistics Project
Name: __________________________________
Section: ____
Date: ___________
Page 0
Integrated Algebra – Graphs and Statistics
This is a project based approach to teaching the above topic.
Please send comments/corrections/additions to the author: George Ludovici at gwludovici@gmail.com , thank you.
The author credits fellow participants at PCMI, especially Bill Thill, for help with the basic idea of this project.
Objective – To address statistical components of the NYS Integrated Algebra Regents level course with accommodations
for anticipated impact of the Common Core Standards.
Project – Cup Stacking
Outline
Project and data collection
Frequency and Cumulative Frequency Histograms
Box and Whiskers Plots
Scatter Plot
Line of Best Fit
Correlation
Central Tendency
Materials –
16 plastic stackable/disposable cups per student group, 3 to 5 students per group
Two different color markers, for each group to write on the cups
Stop watch or other method of timing to the nearest second for each group
Tape measure (reach across student’s arm-span)
Note to the Teacher –
On page 6 questions 16 and 18, you may wish to have a discussion with your students that graphical scales or
intervals can be a number other that one. For example, a histogram interval of only one is not usually a good
choice and when large numbers are involved, a vertical unit of one is impractical.
Standards – There is emphasis in the Common Core Mathematics Standards on deeper thinking and communication. To
this end students are often asked to explain their reasoning throughout this unit.
The two example histograms and the example scatter plot are borrowed from old NYS Regents exam questions.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Statistics Project
Part 1
Name: __________________________________
Section: ____
Date: __________
Page 1
Conduct experiments to collect the data:
Your group needs to collect data to fill in the following table.
Gender Arm Span Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Total
Average
Student Name
m/f
Inches
seconds seconds seconds seconds seconds Seconds
Time
1) Fill in the names, genders and arm spans. Arm span is fingertip to fingertip with arms outstretched to the
sides.
2) Number 16 cups, 1 to 16, using one color for even numbers and one color for odd numbers. Write the
numbers several times so they are visible on the sides of the cups no matter how the cup is turned. An
alternative to this is to use two different cup colors, what is important is being able to identify that only one
cup at a time is being moved and when all the cups have been moved.
3) Take turns being: a stacker, a timer, and a spotter. Stacker: holds the cups. Timer: operates the timer and
records the data. Spotter: makes sure stacker performs correctly.
4) Stand up and hold the stack of cups. Take one cup from the bottom of the stack with one hand and put it in
the top of the stack, then use your other hand to take the new bottom cup and put it at the top of the stack.
5) You must alternate hands throughout the experiment. You are done when the cup which started at the
bottom goes all the way through one cycle, returning to the bottom.
6) The stacker must be standing up.
7) The spotter is responsible for making sure every number 1-16 goes by. Specifically that only one cup at a
time is being moved and when the end is reached.
8) Mistakes must be corrected by undoing and repeating the action correctly. Do not start a new trial.
9) The timer stops as soon as the original cup is back at the bottom of the stack and the stack is complete.
10) Make sure the stacker goes through this experiment five times in a row, then everyone in the group
switches to a new position.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Statistics Project
Part 1
Name: __________________________________
Section: ____
Date: __________
Page 2
Analysis
How would you describe the relationships you see in your data?
Justify your answer from number 1 above.
How is your description helpful to someone who has not seen your data?
What else might that person want or need to know about your data?
Discuss these questions, don’t write answers yet:
1) How much time does it usually take to stack the cups?
2) Are some of the cup stacking times more or less typical than others?
3) How spread out are the cup stacking times?
4) Does practice improve the cup stacking times? If so, how much?
5) Are there any relationships between gender, arm span and cup stacking times?
6) Could you predict some data items given other data items?
7) What other questions might you ask?
During this statistical unit, we are going to use the data you collected from your experiment to answer the
above questions.
As you do your work you may think of additional questions. Think about how you might use the statistical
tools that we are going to learn about to help answer those questions. If the statistical tools we use don’t
help to answer your questions, think about how you might create a different tool to answer the question.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Statistics Project
Part 1
Name: __________________________________
Section: ____
Date: __________
Page 3
Histogram – You are going to create frequency histograms to communicate your data graphically. Two example
histograms are discussed: a regular histogram on this page and then a cumulative histogram on the next page.
Notice how the vertical axis always represents frequency.
What does “frequency mean”?
Write a sentence describing exactly what the fourth gray bar, above
190-199, for Student Heights means.
Height
Interval
Total
Cumulative
Total
Using the above histogram, complete this table.
160-169
Discuss your table answers. Should the “Total”
entry for 170-179 be 2 or 4 or 6? Why?
170-179
180-189
190-199
200-209
Should the “Cumulative Total” entry for 170-179 be 2 or 4 or 6?
Why?
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Statistics Project
Part 1
Name: __________________________________
Section: ____
Date: __________
Page 4
What does ”cumulative” mean?
Why do the bars on the Cumulative Test
Scores histogram keep getting higher?
Why is the fifth gray bar six units higher than
the fourth gray bar?
Describe exactly what the fourth gray bar, above 41-80, for CumulativeTest Scores means.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Statistics Project
Part 1
Name: __________________________________
Section: ____
Date: __________
Draw a Cumulative Student Height histogram based on the data in the Student Heights histogram.
Draw a Test Scores histogram based on the data in the Cumulative Test Scores histogram.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Page 5
Statistics Project
Part 1
Name: __________________________________
Section: ____
Date: __________
We are now going to draw a frequency histogram based on the data you collected using the Total Time.
15 What does vertical mean?
16 What measure or label should be on the vertical (y) axis?
17 What does horizontal mean?
18 What measure or label should be on the horizontal (x) axis?
The width of each histogram bar represents an interval of your data. In our case this will be in seconds.
19 What does interval mean?
Your interval should be greater than or equal to one depending on your data.
20 What interval will you choose for your bars?
21 Why did you choose that interval?
22 How did you calculate it?
23 How many bars will you have?
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Page 6
Statistics Project
Part 1
Name: __________________________________
Section: ____
Date: __________
Page 7
On each axis there is a scale.
24 What does scale mean?
34 Complete this table based on your cup
stacking data.
25 Write the scale you will use on your horizontal axis.
35 Will you need to fill in every line in this
table?
26 Why did you choose that scale?
36 How many bars did you say you would have?
Interval
Tally Your
Entries
Total
In addition to the scale, we label each axis.
27 Why do we label the axis?
28 What label will you use on your horizontal axis?
29 How will you choose a scale for the vertical axis?
30 Write the scale you will use on your vertical axis.
31 What label will you use on your vertical axis?
32 Why do we give the histogram a name or description?
33 What will you name your histogram?
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Cumulative
Total
Statistics Project
Part 1
Name: __________________________________
Section: ____
Date: __________
37 Draw a histogram for the cup stacking data.
38 Draw a cumulative histogram for the cup stacking data.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Page 8
Statistics Project
Part 1
Name: __________________________________
Section: ____
Date: __________
Page 9
Cup Stacking Project Rubric for Part 1
Rubric
Gathered the
Data
0
1
2
Not
Done
Table has a combination of
missing data, difficult to read
and/or fewer than three team
members.
Some data is missing
or the table is difficult
to read or there are
fewer than three team
members.
Portions of complete
and fully labeled
histograms are
missing or difficult to
read or inaccurate.
Portions of complete
and fully labeled
histograms are
missing or difficult to
read or inaccurate.
Most questions are
answered in a neat,
accurate and
thoughtful manner
Properly
graphed 13
and 14 on
page 5
Not
Done
Properly
graphed 37
and 38 on last
page
Not
Done
Remaining 34
questions
Not
Done
The histograms have
incompleteness, inaccuracies
and/or are difficult to read.
The histograms have
incompleteness, inaccuracies
and/or are difficult to read.
Few questions are answered
and/or many answers are
difficult to read and/or most
answers are not thoughtful
and accurate.
3
A neat and complete data
table with Arm Length,
Gender and five trials
each for at least three
team members.
A frequency histogram
and a cumulative
frequency histogram are
completed, neat, fully
labeled and accurate.
A frequency histogram
and a cumulative
frequency histogram are
completed, neat, fully
labeled and accurate.
All or almost all questions
are answered. Answers
are thoughtful and
accurate, and answers
are neat and easy to
read.
Total
Possible deduction for being late past the due date
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Points
3
3
3
3
12
Statistics Project
Part 2
Name: __________________________________
Section: ____
Date: ___________
Page 1
Box and Whiskers
A sample Box and Whiskers plot might look like this:
Five numbers are used to create a Box and Whiskers plot. First put your data in order (sort) from smallest to largest.
1) The smallest number.
In the above plot, the smallest number is ______________.
2) The middle number in the lower half of your sorted data. This is called the first or lower quartile. Quartile
sounds like quarter which is
1
or 25% of all the data points. If two numbers are in the middle then take
4
their average by adding them up and dividing by two. If there is an overall odd number of data values then
do not include the median in the upper or the lower half when determining quartiles.
In the above plot, the first quartile is ______________.
3) The middle number in all of your sorted data. This is called the median; think of median as middle. For
example, a highway or a wide street often has a median running down the middle to separate cars going in
different directions. The median is the same as the second quartile. Half (50%) of the data comes before
the median and half (50%) of the data comes after the median. Like the lower quartile, if there are an even
number of data points so that two points are in the middle, then average them to find the median.
In the above plot, the median is ______________.
4) The middle number in the higher half of your sorted data. This is called the third or upper quartile.
In the above plot, the third quartile is ______________.
5) The largest number.
In the above plot, the largest number is ______________.
Notice how in the above Box and Whiskers plot the quartiles or vertical lines are not evenly spaced.
6) How far apart are each of the five numbers?
_________
Low to first
quartile
____________
First to Second
quartiles
____________
Second to Third
quartiles
__________
Third to High
quartile
7) What does this tell you about how the data are spread out?
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Statistics Project
Part 2
Name: __________________________________
Section: ____
Date: __________
Page 2
8) The Interquartile Range shows how spread out the middle half of the data is. The Interquartile Range is
calculated by subtracting the first quartile from the third quartile. What is the Interquartile Range for the
box and whiskers plot on the previous page?
Create a Box and Whisker Plot
Sort your whole team’s cup-stacking data from shortest Trial Time to longest Trial Time.
Find your five numbers using the sorted Trial Times:
1) Lowest:
___________
2) Lower quartile: ___________
3) Median:
___________
4) Third Quartile : ___________
5) Highest:
___________
The Box and Whiskers plot is drawn above part of a number line which is the scale for the plot.
What scale will you choose?
Why did you choose that scale?
How did you calculate that scale?
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Statistics Project
Part 2
Name: __________________________________
Section: ____
Date: __________
Page 3
Draw your cup-stacking Box and Whisker plot. Be sure to include a title, the scale, the box around the three middle
numbers, vertical lines on the first, third and fifth numbers and connect the whiskers with horizontal lines.
How does the spread of the data from your Cup Stacking Box and Whiskers plot compare with the spread of the data
from the sample Box and Whiskers plot that we have been using?
Scatter Plots
A sample scatter plot for the maximum height and speed of some roller coasters is shown in the table below and
graphed with a scatter plot.
The scatter plot has a data point graphed for every piece of data. Although the above table is sorted by height, it does
not need to be. The points are not connected with lines. Both axis and the graph are labeled. Each axis has a scale and
a label. The whole graph has a label. If there was more than one point with the same height value, they would each be
graphed. Notice how the scatter plot does not use intervals; this is different from a histogram.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Statistics Project
Part 2
Name: __________________________________
Section: ____
Date: __________
Page 4
Describe the relationship you see between Max Height and Max Speed:
A scatter plot is useful for seeing relationships between two variables. Do you think there is a relationship between Arm
Length and Average Trial Time in your data?
Explain your reasoning for the answer above.
Create a scatter plot with Average Trial Time on the horizontal and Arm Length on the vertical. Join with one or two
other groups so that you will have at least ten points to plot.
What scale will you use on the horizontal axis?
How did you determine the scale for the horizontal axis?
What scale will you use on the vertical axis?
How did you determine the scale for the vertical axis?
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Statistics Project
Part 2
Name: __________________________________
Section: ____
Date: __________
Complete your scatter plot.
Is there a relationship between Average Trial Time and Arm Length?
Explain your above answer.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Page 5
Statistics Project
Part 2
Name: __________________________________
Section: ____
Date: __________
Page 6
Remember the Roller Coaster Scatter Plot? Here it is again, drawn twice. The one on the right has a “line of best fit”.
A line of best fit is all of the following:
 It is always straight
 It may go through some of your points or it may not, however it is as close as possible to as many of
your points as possible while still being straight.
 If does not have to go through the origin (0, 0).
If your data is close to the line of best fit, then we can say there is a relationship or correlation between the data on the
x-axis and the data on the y-axis. The closer your data is to the line, the stronger the correlation is. The correlation is
either positive (positive slope), negative (negative slope) or zero (no slope, no correlation).
Go back to the previous page and add a line of best fit to your scatter plot.
What kind of correlation does your line show?
Describe how the correlation strengthens, weakens or changes your prior answer about the relationship between
Average Trial Time and Arm Length.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Statistics Project
Part 2
Name: __________________________________
Section: ____
Date: __________
Page 7
Cup Stacking Project Rubric for Part 2
Rubric
0
Box and whiskers
pages 1 and 2
Not
Done
Box and Whiskers
Graph
Not
Done
Scatter Plot
preparation on
page 4
Scatter plot
Graph on page 5
and work on page
6
Not
Done
Not
Done
1
Many answers are
missing, not neat or
inaccurate.
Multiple mistakes
and/or omissions
and/or not neat
and/or no written
analysis.
Few answers are
completed, neat and
thoughtful.
Multiple mistakes
and/or omissions
and/or not neat
and/or no written
analysis.
2
3
Most answers are
completed, neat and
accurate.
Minor mistakes or
omissions or not
neat. A written
analysis is provided.
All answers are completed,
neat and accurate.
Most answers are
completed, neat and
thoughtful.
Minor mistakes or
omissions or not
neat. Written
analysis is provided.
All answers are completed,
neat and thoughtful.
A correct box-and-whiskers is
drawn with a labeled scale and
everything is neat. A written
analysis is provided.
A correct scatter plot is drawn
with a labeled scale and
everything is neat. A written
analysis is provided and a
correlation is described.
Total Possible
Possible deduction for being late past the due date
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Reasoning from Data and Chance
Summer 2012
Points
3
3
3
3
12