Statistical Analysis of Lincoln

Learning Goals:
1. Make simple observations of data and try to hypothesize the results prior to analysis.
2. Perform simple statistical analysis of large quantities of data.
3. Perform one more complicated statistical analysis of data.

Concepts:
1. The Excel program is a powerful tool for statistical analysis.
2. When recording mass, be aware of significant digits

Vocabulary Words:
1. Standard Deviation
2. Confidence Intervals
3. t Test
4. Gaussian Distribution

This lab is typically carried out in college prep chemistry for juniors or seniors that have the math capabilities so desired for the lab. The activity can be done with as few as three students to as many as you wish. The more data points, the more information students can derive. Normally students need at minimum one full class period to record masses of pennies, then one other class period to analyze data in the computer lab. Lab report and everything else is expected to be completed outside of class time.

The only equipment you need is an electronic or triple beam balance; the choice depends on which one you want the students to use. If you use the triple beam, you may only want the students to record 20 to 30 data points instead of 100 per group.

I normally do this lab within the first couple months of school, depending on the ability of the students. One could easily adapt this to a more beginners mode by eliminating the upper level analysis or make it harder by adding more analysis.

Activity Description:

Materials:
1000 pennies of various dates
Balance (electronic or triple beam)
Microsoft Excel (or something similar)

Mechanics:
Gathering Data:

• Each lab group or student should collect and record masses for enough pennies to the nearest centigram at least so that there are 100 data points for an electronic balance or 30 points for a triple beam balance.
• Compile everyone's data into one spread sheet.
• Each column should list the masses of pennies from one calendar year. There will be two columns for 1982, one for brass & one for composite.
• Select a year other than 1982 for which you have many coins and divide them into those made in Denver (w/a "D" beneath the year) and those minted in Philadelphia (no mark beneath the year).

Discrepant Data:

• Be sure to retain at least one extra digit beyond the readable measurement on the measuring device to avoid round off errors in calculations.
• At the bottom of each column list the average and standard deviation for that year.
• Use the spreadsheet sort function and sort each column by increasing mass.
• Discard grossly discrepant masses lying >4 standard deviations from the mean in any one year.
• Recompute average and standard deviation for each column.

NOTE: Do not apply the same test again to the same data with new standard deviation; if could continue to do so you will be out of data points.

Confidence Intervals and t Test:

• Select two years in which composite coins (>1982) have the highest and lowest average mass.
• Compute the 95% and 99% confidence intervals for the masses.
• Use the t-Test to compare the two mean values at 95% and 99% confidence intervals.
• Try the same for the one year in which you segregated coins made in Philadelphia from those in Denver.

Do the Masses Follow a Gaussian Distribution?:

• Construct a bar graph of your data as in Figure 1.
• Group the data for all of the composite pennies into a single column sorted from lowest mass to highest mass. NOTE: There should be at least 100 entries in this table for an electronic balance or 50 entries for a triple beam balance.
• Divide the data into 0.01 g intervals and plot each category as a bar on the graph. EXAMPLE: The bar at 2.485 g in Fig 1 shows that 69 coins had a mass between 2.480 and 2.489 g.
• Calculate the average, median and standard deviation for the entire set of coins in the graph.
• Indicate which bars (if any) lie beyond 3 standard deviations from the mean.
• Construct the smooth Gaussian curve that has the same mean, standard deviation and area as the data set. The equation is:

y=((# of Pennies)/100) 1/(s √ 2π) 〖e^(〖-(x-x ̅)〗^2/2s)〗^2〗^
x ̅=average avlue
s = standard deviation for the whole set of pennies in the bar chart.
Carry out a Chi-Squared Test to see if the observed distribution (the bar chart in Fig 1) agrees with the Gaussian curve.
χ^2=∑▒〖(y_obs-y_calc)〗^2/y_calc
yobs = height of the bar on the chart.
ycalc = is the ordinate of the Gaussian curve (Equation 1)
At the bottom of Table 2 we see that Chi-Squared for all 21bars is 43.231.
In Table 3 we find a critical value of 31.4 for 20 degrees of freedom.
Degrees freedom = # of categories—1
Since Chi-Squared from equation 2 exceeds the critical value, we conclude the distribution is not Gaussian.
Notes: It would be reasonable to omit the smallest bars at the edge of the graph from the calculation of Chi-Squared because these bars contain the fewest observations but make large contributions to Chi-Squared. Suppose we reject bars lying >3 standard deviations from the mean. This removes the two bars at the right side of Fig 1 which give the last two entries in Table 2. Omitting these two points gives Chi-Squared = 30.277, which is still greater than the critical value of 28.9 for 18 degrees of freedom in Table 3. Our conclusion is that at the 95% confidence level the observed distribution in Figure 1 is not quite Gaussian. It is possible that exceptionally light coins are nicked and exceptionally heavy coins are dirty or corroded. You need to inspect these coins to verify this hypothesis.

For closure, have the students complete a post lab report based on the information they have obtained and calculated.

Adapted from Analytical Chemistry Course, Mayville State University, Dr. Tom Gonella. Figure 1 (Acrobat (PDF) 79kB Sep25 07) Table 2 (Acrobat (PDF) 14kB Sep25 07) Table 3 (Acrobat (PDF) 15kB Sep25 07) Equations (Acrobat (PDF) 23kB Sep25 07)

If students are experiencing great difficulty with the advanced statistical analysis, you may have to walk them through it or skip it in its entirety. I have only had one group of students thus far complete the whole project, and when finished they were quite proud of themselves.

Each student is responsible for submitting a post lab report as outlined in the rubric that is provided to each student at the beginning of the year. When I get a hold of a copy of the student's work, I will post it on this page in this spot.

1. 9-12.I.B.2—Distinguish between qualitative and quantitative data
2. 9-12.I.B.3—Apply mathematics to analyze data