Making Conjectures Based on
Graphical and Basic Statistical Analysis of Data
Jay Yohe
Susquehanna Township High School
Grade 10-12
Selected Topics
Prerequisite skills:
*Adept with TI-83 Graphics Calculator
*Basic knowledge of graphical methods
(line graphs, box plots, histograms, etc.)
*Basic knowledge of mean, median and
standard deviation.
Objectives:
| Students will be able to graphically analyze data sets and be able to use statistics to make sound conjectures. |
| Students will use technology to create appropriate graphical representations of data using EXCEL, TI-83 calculators, Fathom or other software packages. |
| Students will use EXCEL, Fathom and the TI-83 calculator to calculate statistical measurements like mean, median, standard deviation, range, quartiles, etc. Students will then use these statistical measurements to make conclusions. |
| Students will utilize the Internet and the Library Media Center to find more data and information to support and enhance their conjectures and they will create their own questions based on the results of their graphical analysis, statistical analysis and advanced research of the data. |
PA Math Standards:
Academic Standard |
Description |
| 2.4.11 Mathematical Reasoning and Connections | (B) Construct valid arguments from stated facts. |
| 2.5.11 Mathematical Problem Solving and Communication |
(B) Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas and results. |
| 2.5.11 Mathematical Problem Solving and Communication |
(D) Conclude a solution process with a summary of results and evaluate the degree to which the results obtained represent an acceptable response to the initial problem and why the reasoning is valid. |
| 2.6.11 Statistics and Data Analysis | (B) Use appropriate technology to organize and analyze data taken from the local community. |
Materials and Resources:
*TI-83 calculators
*Computers with MicroSoft EXCEL
*Access to the Internet/Library Media Center
*Attached MicroSoft WORD data sets
*Fathom (Key Curriculum Press) --desired but optional
Instruction:
Students will work together in small groups during the introductory problem and will also work together on the actual assignment. Each student will be responsible for submitting individual written results and summaries.
Procedures:
*Introduction Problem (below)
*Explanation of tasks
*Assignment of projects with explanation of assessment
Introductory Problem:
The following data set is used courtesy of
(http://lib.stat.cmu.edu/DASL/)
Estimating the Speed of Light
Reference:
| Moore, David S., and George P. McCabe (1989). Introduction to the
Practice of Statistics. Original source: Stigler, S.M., "Do robust estimators work with real data?" Annals of Statistics, 5 (1977), pp. 1055-1078. |
Description:
Simon Newcomb measured the time required for light to travel from his laboratory on
the Potomac River to a
mirror at the base of the Washington Monument and back, a total distance of about 7400
meters. These measurements were used to estimate the speed of light. The data file
contains the estimated speed of light for 66 trials.
Number of cases: 66
Variable Name:
1.Speed_of_Light: Speed of Light
(Does this measurement actually represent the
speed of light?)
The Data:
Speed_of_Light
28
22
36
26
28
28
26
24
32
30
27
24
33
21
36
32
31
25
24
25
28
36
27
32
34
30
25
26
26
25
-44
23
21
30
33
29
27
29
28
22
26
27
16
31
29
36
32
28
40
19
37
23
32
29
-2
24
25
27
24
16
29
20
28
27
39
23
Step 1. Analyze the Data using EXCEL
- Copy and Paste the Speed of Light data into a column of your EXCEL spreadsheet.
- Sort the Data.
- If we want every value from the Min to the Max to be Histogram Bin values, we need to create a list from the minimum value up to the maximum value. We can see that our data currently goes from -44 to 40. Create this list by going to a new cell and typing in -44.
- Highlight the cell with -44 and drag down rather far over the blank cells below the -44 cell and release your MOUSE button. Don't worry about dragging too far beyond the necessary number of cells. It won't matter.
From the "Edit" Menu choose "Fill" and drag down and release on "Series..."
Fill in the Series box as illustrated below:
- Look for "Data Analysis" under the "Tool" Menu (if missing follow the directions below ***)
Choose "Data Analysis" and select "Histogram" from the dialog box
Highlight the original data for the "Input Range", highlight the -44 to 40 intervals for "Bin Range" and drag over a blank region of your spreadsheet for the "Output Range" so there is room for the new chart data. Check off the "Chart Output". Click OK. Look at the illustration below:
This will produce a Histogram. You can improve it by altering various items. To remove the annoying tick marks, find the "Format Axis" box below and change it as I did:
To remove the space between bars so that your graph is a true Histogram, double click on one of your bars and change the resulting dialog box as follows (you can also remove backgrounds, change colors of bars, alter borders, alter fonts, etc.)
- With enough work, you should be able to represent the data as illustrated below (don't
forget that this process also creates new data tables)
***
Adding Data Analysis to your Tool Bar- From the "Tool" menu select "Add-Ins..."
- Check both options as illustrated below and click "OK"
 |
Step 2: Calculate Statistical Data (in EXCEL)
- Enter equations into EXCEL to determine basic statistics.
Have students calculate the following statistics:
Example:
Standard Deviation: =STDEV(C2:C67)
Step 3: Make conclusions and answer questions from the EXCEL data set and graph
- What kind of value is -44 in this data set?
- Newcomb is actually measuring the time that it took a light signal to pass from his laboratory on the Potomac River to a mirror on the base of the Washington Monument and back. So what are the units of his measurements?
- We know that speed equals distance divided by time (d=r*t or r=d/t). Newcomb used a distance of 7400 meters. If your units are correct in the previous step, use them to calculate the speed of light based on the "mean" measurement. What do you get?
- What is the actual speed of light?
- Assuming your units are correct from part 2 above, explain how different -44 is from the mean value calculated for time in Newcomb's speed of light test data.
- How many measured data points are within one standard deviation of the mean on both sides?
- Remove -44 and redo all the data including the Histogram graph and the formulas. What does this do to the distribution of data? Explain.
- What other experiments could be conducted to validate Newcomb's test data? Can you find any other test data from other experiments that gives similar results?
Step 4: Analyze Newcomb's data using the TI-83 Graphing Calculator
- Enter the original 66 data points into L1 of your graphics calculator.
- Turn on your Stat Plot (2nd Y=). Select the bar for "Type" (histogram)
- Choose "ZoomStat" from the "Zoom" Menu (I also turned off my x/y axis)
- You may wish to alter the "Window" settings until you get a graph you like
- You should see a graph similar to the one illustrated below:
- Press the "Stat" button and arrow over to "CALC". Choose 1-Var Stats
- Observe the graphics calculator screen shots below:
- Do these results agree with those calculated in EXCEL?
- Now form Box Plots of our data. Turn on the Stat Plot (2nd Y=).
- Choose the second picture of the Box Plot in the "Type" menu.
- From the "Zoom" menu choose "ZoomStat". This Box Plot includes all outliers. Observe:
- Choose the first picture of the Box Plot in the "Type" menu.
- From the "Zoom" menu choose "ZoomStat". This Box Plot does not stretch to all outliers. Observe:
- Explore with the Trace button:
- Did you obtain any additional data to help you answer questions from Step 3 above?
- Remove -44 from L1 and observe the new Box Plot. What do you observe?
Step 5: Observe the data with Fathom (optional)
This is a screen shot of Fathom with the Speed of Light Data:
Observe the Box Plot Created in the Fathom Software Package:
Closure Activities:
A -- Alternate Data for Calculating Speed of Light (class work)
Use the following data to calculate the speed of light.
Star/Planet/Object in Space |
Time for Light to Travel from Space Entity to Earth |
| Moon | 1.28 seconds |
| Sun | 8.3 minutes |
| Communication Satellite (Geosynchronous Orbit) | 0.12 seconds |
| Jupiter | ? |
| Pluto | ? |
- To use the table above to calculate the speed of light, you will first need to find the distances from the Earth to each entity (or at least some of the entities).
- This will only give you two data points. Is that enough?
- Find the distance the satellite is from the Earth.
- How long does it take light to reach Earth from Jupiter and Pluto?
B -- Graded Project on Quarterback Statistics
Download the Quarterback File to your computer (this is a Word document)
Part 1:
Analyze 1991 Quarterback salaries across the football league using graphical and statistical analysis.
Requirements (Analyzing 1-Variable Statistics):
- You must create one graphical histogram representation on EXCEL and both the Histogram and Box Plot representation on your graphics calculator. You may substitute Fathom for either EXCEL or the graphics calculator if available. Use technology to calculate statistical formulas also.
- You must make one graph (of any type) by hand on poster board or construction paper.
- Create and answer 4 questions related to the observed data and support your results with the graphs and formulas. You should write at least 3 descriptive sentences to answer each question. Example: What do the outliers mean if there are any? What causes the outliers if there are any?
- Research the National Football League during the 1991 season. Can you find reasons for the results suggested by the data?
Part 2: Two Variable Statistics
Use other graphical means to compare quarterback salaries to team salaries, NFC vs AFC differences, quarterback salaries vs wins and losses, etc. Create and pose at least 4 questions related to this data even if you decide there is no correlation. You may also compare 1-variable quarterback salaries with 1-variable team salaries to compare similarities or vast differences. Summarize your data with graphs, charts and formulas.
Part 3: Finding your own Data Set
Using the Internet or Library Media Center, find a set of 1-variable data to analyze. Provide at least one graph and analyze this data set. If the data is relative to the local community, you can gain extra credit. You may also find additional data related to the current project from 1991 National Football League statistics.
Assessment:
| Grade | Requirements |
50 points |
All objectives for Part 1, 2 and 3 are complete. All questions/answers are supported well by the data. Students may also earn extra credit by doing extra research and by presenting creative questions (as long as they are supported by the data). All graphs and formulas have been completed. |
45 points |
One or two questions/answers are not answered adequately and/or the questions are not supported well by the data. All graphs and formulas have been completed for part 1 and 2. |
40 points |
Questions/answers are not supported well by the data and/or some questions are missing altogether. All graphs and formulas have been completed for part 1 and 2. |
35 points |
Questions/answers are not supported by the data. Questions are missing. Some graphs and formulas have not been completed. |
<35 points |
Few objectives have been met! |
References:
**Eisenhower Institute (July 2001)
**http://lib.stat.cmu.edu/DASL/ (data sets)
**http://home.earthlink.net/~ob1gui/ (some football statistics)
**http://astro.uchicago.edu/cara/outreach/resources/ysi94/light.html
(planet data)
**http://www.stolaf.edu/people/wallace/Courses/Statistics/backgroundinfo.htm
(Newcomb's summary)
A Shane/Greydon Production
