"A computer should make both calculations and graphs. Both sorts of output should be studied; each will contribute to understanding... Most kinds of statistical calculations rest on assumptions about the behavior of the data. Those assumptions may be false, and then the calculations may be misleading. We ought always to try to check whether the assumptions are reasonably correct; and if they are wrong we ought to be able to perceive in what ways they are wrong. Graphs are very valuable for these purposes."
F. J. Anscombe, "Graphs in Statistical Analysis," American Statistician,
vol. 27 no. 1 (1973), pp. 1721.
Preparation
Due 12:00 PM before start of lab
Prelab assignment 2 (*.html), 1 point
Equipment
rulers/metersticks (12", 1 m, 2 m)
whiteboards, markers
wood boards
superballs (various)
laboratory laptop, Microsoft Excel
Big Ideas
Curvefitting of experimental data can mathematical models that can be used to make interpolated or extrapolated predictions. Data should be graphed first, in order to visually determine the appropriate type of curvefitting model to be applied.
Goals
Students work in (selfassigned?) groups to mathematically model the experimental rebound height of a superball that is dropped from various initial heights, and to apply this mathematical model to make verifiable predictions.
Students formulate their own systematic stepbystep procedure to determine the data to be collected and calculations to be made.
Students are introduced to graphing and curvefitting software, and best practices for data collection and graphical analysis.
Tasks
(Record your lab partners' names on your worksheet for tasks 12, to be turned in at the end of today's lab for randomly selected grading for your group.)
1. CurveFitting to Raw Data The data set (a) below lists the values of an independent variable "x" and its dependent variable "y."
Data set (a)
x  y 
10.0  8.04 
8.0  6.95 
13.0  7.58 
9.0  8.81 
11.0  8.33 
14.0  9.96 
6.0  7.24 
4.0  4.26 
12.0  10.84 
7.0  4.82 
5.0  5.68 
Use a software package (such as Excel, instructions below, which may be slightly different depending on the version) to graph this data set, and to apply a linear trendline.
 Manually enter (or cutandpaste linebyline) the x and y values as adjacent columns in the spreadsheet.
 Under "Insert" on the top bar menu, click on "Scatter," and in the popup window select "Scatter with only Markers" (plotting data points only, no connecting lines or curves). You should now see the data points plotted on the graph.
 Rightclick on any of the data points in the graph, then select "Add Trendline...," and in the popup window, select "Linear." Click on the "Options" tab, check both "Display Equation on chart" and "Display Rsquared value on chart" options. Then click "OK" when done.
Print out one copy of this graph (to be attached to the one report from your group to be selected later for grading today).
Record the raw output for the equation of the linear trendline and Rsquared value, and then truncate these results to the proper number of significant figures (in this case, two, as this was the lowest precision of some of the values in this data set).
y = __________x + __________.
y = __________x + __________ (proper sig figs).
R^{2} = __________.
R^{2} = __________ (proper sig figs).
 Repeat this process for data set (b) below.
Data set (b)
x  y 
10.0  9.14 
8.0  8.14 
13.0  8.74 
9.0  8.77 
11.0  9.26 
14.0  8.10 
6.0  6.13 
4.0  3.10 
12.0  9.13 
7.0  7.26 
5.0  4.74 
(Print out one copy of the graph, and record the results below).
y = __________x + __________.
y = __________x + __________ (proper sig figs).
R^{2} = __________.
R^{2} = __________ (proper sig figs).
 Then repeat this process for data set (c) below.
Data set (c)
x  y 
10.0  7.46 
8.0  6.77 
13.0  12.74 
9.0  7.11 
11.0  7.81 
14.0  8.84 
6.0  6.08 
4.0  5.39 
12.0  8.15 
7.0  6.42 
5.0  5.73 
(Print out one copy of the graph, and record the results below).
y = __________x + __________.
y = __________x + __________ (proper sig figs).
R^{2} = __________.
R^{2} = __________ (proper sig figs).
2. CurveFitting Best Practices Suggested guidelines for measuring, graphing, and analyzing data are listed below:
 The maximum value of the independent variable should be at least five times (10 times is better) its minimum value.
 The usual rule of thumb for the minimum number of different independent variable data points is 10.
 If the data changes rapidly, then data collection should concentrate in those regions of a graph.
 If the accuracy of the measurements is low (graph appears "noisy"), each data point used to make a graph should consist of a number of observations that have been averaged.
 Outlying data point(s) should be reviewed/remeasured/replaced, or as the situation warrants, removed from the data set.
 The choice of proper trendline fit type (linear, power, exponential, etc.) should not be arbitrarily made, but only after inspection of the data graph, or if the relationship is known or expected.
Were these best practices followed for data set (a)? If "yes," then merely state so. If "maybe" or "no," give a brief explanation.
 Minimummaximum data range.
[ yes  maybe  no ]
 Has minimum number of data points.
[ yes  maybe  no ]
 Concentrated data in rapidly changing portions of graph.
[ yes  maybe  no ]
 Variable data points should be an average of repeated measurements.
[ yes  maybe  no ]
 Outlying data points should be replaced/removed.
[ yes  maybe  no ]
 Proper choice of trendline fit type.
[ yes  maybe  no ]
 Repeat the evaluation of best practices for data set (b).
 Minimummaximum data range.
[ yes  maybe  no ]
 Has minimum number of data points.
[ yes  maybe  no ]
 Concentrated data in rapidly changing portions of graph.
[ yes  maybe  no ]
 Variable data points should be an average of repeated measurements.
[ yes  maybe  no ]
 Outlying data points should be replaced/removed.
[ yes  maybe  no ]
 Proper choice of trendline fit type.
[ yes  maybe  no ]
 Repeat the evaluation of best practices for data set (c).
 Minimummaximum data range.
[ yes  maybe  no ]
 Has minimum number of data points.
[ yes  maybe  no ]
 Concentrated data in rapidly changing portions of graph.
[ yes  maybe  no ]
 Variable data points should be an average of repeated measurements.
[ yes  maybe  no ]
 Outlying data points should be replaced/removed.
[ yes  maybe  no ]
 Proper choice of trendline fit type.
[ yes  maybe  no ]
 For data set (a) (assuming that this is the only data available), use the raw output for the equation of the linear trendline to predict the value of the dependent variable y if the independent variable is x = 20. Show your work, and then truncate these results to the proper number of significant figures.
For x = 20, predicted value for y = __________.
 For data set (b), redo the graphical analysis by selecting a secondorder polynomial, and print out one copy of this graph. Record the revised raw output for the equation of the polynomial trendline and Rsquared value, and then truncate these results to the proper number of significant figures (in this case, two, as this was the lowest precision of some of the values in this data set).
Revised y = __________x^{2} + __________x + __________.
Revised y = __________x^{2} + __________x + __________ (proper sig figs).
Revised R^{2} = __________.
Revised R^{2} = __________ (proper sig figs).
 The Rsquared value (also known as the coefficient of determination) is a value that measures how close the data is fitted with a trendline, where a value of 0 is no correlation to the data, and a value of 1 is a perfect fit to the data. Compare the original (linear) Rsquared for data set (b) to the polynomial Rsquared value. Write a concluding statement discussing which Rsquared value is higher, and specifically why. Include the specific relevant numbers in this statement, such that it can be read (and cited) on its own without referring to the above calculations and numbers.
Brief concluding statement:
 Use the raw output for the equation of the polynomial trendline for data set (b) to predict the value of the dependent variable y if the independent variable is x = 20. Show your work, and then truncate these results to the proper number of significant figures.
For x = 20, predicted value for y = __________.
 For data set (c), redo the graphical analysis by first removing the outlying data point (assuming that it can't be redone, and considered erroneous), then redoing the linear trendline to the data, and print out one copy of this graph. Record the revised raw output for the equation of the new trendline and Rsquared value, and then truncate these results to the proper number of significant figures (in this case, two, as this was the lowest precision of some of the values in this data set).
Revised y = __________x + __________.
Revised y = __________x + __________ (proper sig figs).
Revised R^{2} = __________.
Revised R^{2} = __________ (proper sig figs).
 Compare the original (outlierincluded) Rsquared for data set (c) to the revised (outlierexcluded) Rsquared value. Write a concluding statement discussing which Rsquared value is higher, and specifically why. Include the specific relevant numbers in this statement, such that it can be read (and cited) on its own without referring to the above calculations and numbers.
Brief concluding statement:
 Use the raw output for the equation of the revised linear trendline for data set (c) to predict the value of the dependent variable y if the independent variable is x = 20. Show your work, and then truncate these results to the proper number of significant figures.
For x = 20, predicted value for y = __________.
3. Superball Rebound
(Done on whiteboard only, to be worked on and presented as a group.) Drop a superball onto a flat solid surface of your choice. Make note of its rebound height. Develop an experimental trendline equation (not necessarily linear?) of how the rebound height of the superball (dependent variable) depends on its drop height (independent variable), starting from an initial drop height of 10 cm. Create a detailed, stepbystep description of evidence that needs to be collected, and calculations to be donenot just "record each rebound height and plot the data," but exactly what would someone need to do, stepbystep, to accomplish this.
 Record your data on another whiteboard, and briefly show your work for each calculation (clearly labeled). Print out one copy of your graph (with trendline equation); you do not need to draw the graph on a whiteboard. Note that you should follow the suggested bestpractice guidelines for data collection and graphical analysis:
 Minimummaximum data range.
 Has minimum number of data points.
 Concentrated data in rapidly changing portions of graph.
 Variable data points should be an average of repeated measurements.
 Outlying data points should be replaced/removed.
 Proper choice of trendline fit type.
 Write out a concluding statement on the whiteboard regarding the validity of your mathematical modeltesting it by measuring rebound heights using a drop height that is higher than your maximum experimental drop height, and comparing the percent discrepancy between the measured rebound height and predicted rebound height from your trendline equation. Include the specific relevant numbers in this statement, such that it can be read (and cited) on its own without referring to the above calculations and numbers.
 Bring up your whiteboards to the front of the class, to be presented to the instructor, which should include:
 A descriptive abstract: a brief summary of the purpose and methods used, but not of the data nor the conclusions. (Use the sample abstract below for today's lab.)
"We constructed a mathematical model of how high a superball rebounded when dropped from a range of different heights, and tested the validity of this model by predicting and measuring rebound heights for an initial drop height outside the range of the original data set."  Stepbystep procedure.
 Data table, calculations and/or results.
 Evidencebased conclusion statement.
 Documentation Rubric (tasks 12)
(Graded from randomly selected group member)
Score  Description 
3  Explanations complete and calculations correct, or very nearly so. 
2  Essentially complete; few explanations/calculations missing or incorrect. 
1  Substandard effort; substantive amount of explanations/calculations missing or incorrect. 
0  Unacceptable or no significant effort. 
 Whiteboard Rubric (task 3)
(Graded as a group, evaluated by instructor during debrief session)
Score  Description 
3  Complete, thorough, understandable, with little or no clarification needed during verbal instructor critique (can be resubmitted and presented again with requested corrections/revisions made, and still receive full credit). 
2  Minor problems; some corrections/revisions requested by instructor still needed, but not completed during time remaining in lab. 
1  Minimally acceptable effort, essential/critical revisions still needed. 
0  Unacceptable or no significant effort beyond experimental work. 
Followup
Complete this week's postlab assignment, next week's prelab assignment, and review lab instructions.
Due 12:00 PM before start of next lab
Postlab assignment 2 (*.html), 1 point
