lab 1 questions

7/15/2019 Lab 1 Questions

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Juhi RamchandaniChem 322

Drawer # 15Due: 02/22/13

Questions: Lab 1

1. What is the absolute standard deviat ion of a single weighing? This data

determin es the precisio n of y ou b alance. The manu facturer claims ± 0.1 mg is

the “repeatability” of the balance. Which balances are seemingly in need of repair or adjustment du e to poor l imi t ing precis ion?

The absolute standard deviation of a single weighing for balance 5 is ± 0.0000 g, or ± 0.00 mg, for the “Weigh Single Penny Ten Times” section, and it agrees with themanufacturer’s claims for the “repeatability” of the balance. The newer balances arein need of repair or adjustment because they are more variable in their massreadings, indicating a standard deviation greater than ± 0.1 mg.

2. Which resul ts appear to ref lect the p resence of sign i f icant determinate

(systematic) errors?

The results from the section involving weighing twelve pennies one time indicates agreater level of systematic errors because of the variations in the chemicalcomposition of pennies over the different years. The pennies changed from beingcomposed of mainly copper to zinc plated with copper in 19821, and the two metalshave slightly different molar masses. Also, the presence of oil, grime, and other contaminants on the pennies would have affected the weights. This would haveresulted in pennies weighing differently for different years based on the amount of contaminants that linger on them.

3. When this stud y was f i rst introduced, did you expect all pennies to weigh

exact ly the same, about the same, or w ith a distr ibut ion ? Descr ibe your

expectat ions . Does the data conf i rm yo ur ant ic ipat ion?

Initially, the pennies were expected to weigh about the same, with slight weightdifferences being due to a change in the chemical composition of the pennies andpotential contaminants adhering to older pennies. The propagation of thesesystematic errors would result in a greater level of uncertainty associated with thedata, making the mean unreflective of the actual population distribution and resultingin individual weight readings being more variable from the mean. It was

hypothesized that the change in the metal composition of pennies would cause thegreatest difference in mass between older and newer pennies, and this differencewould make the data partially normally distributed. This assumption was partiallyconfirmed by the data, which showed that the weights of the pennies wereconcentrated near 2.5 g (newer) and 3.1 g (older) but that they lied in a somewhat

1 http://www.usmint.gov/about_the_mint/fun_facts/?action=fun_facts2

http://www.usmint.gov/about_the_mint/fun_facts/?action=fun_facts2






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normal first and second standard deviations held 58.333% and 100.00% of the datarespectively.

4. Mean, median, and standard d eviat ion are character ist ic qual i t ies for

descr ibing n ormal ly distr ibuted data. Do these quant i t ies for your s et of 12

pennies adequately descr ibe you r data as being n ormal ly distr ibuted? Wh at

frac tion o f yo ur resu lts f alls w ith in t he in terval, X ± s.d., X ± 2 s.d., X ± 3 s.d.,

etc, (X is mean; “s.d.” means standard deviation)?

Mean 2.8458 # Sd Frequency %

Median 3.0413 1 7 58.333

Range 0.6241 2 12 100

Sd 1 0.2879 3 12 100

Sd 2 0.3529

Table 1 – Full Set

The quantities seem to indicate a normal distribution of pennies. For a distribution tobe considered normal, approximately 68% of the data ought to be within the firststandard deviation, and approximately 95% of the data ought to be present withinthe second deviation. Seven of the values fall within the first standard deviation,while the other five reside within the second standard deviation. In other words,58.333% of the sample lies within the first standard deviation, and 100% lies withinthe second standard deviation. Based on these percentages, it seems the data issomewhat normally distributed; increasing the sample size could potentially make ita true normal distribution.

5. For you r set of six older pennies (heavier ones), use you r X values and

standard deviat ion values to predict the distr ibut ion. For your set of s ix newer

pennies ( l ighter ones), use you r X values and standard deviat ion values to

predict the distr ibut ion and compare these two with the on e calculated from

Step 4.

With the change in metal composition of pennies, it was expected that there wouldbe two weights around which the data would be concentrated, and these weightswould correspond to the manufacture year of the pennies. This was confirmed by

Newer Pennies

Mean 2.5208

Median 2.5178

Range 0.0446

Sd 1 0.0168

Sd 2 0.0171

RelativeSd 1 0.666%

RelativeSd 2 0.680%

Table 4 – Newer Pennies

Older Pennies

Mean 3.0779

Median 3.0859

Range 0.0814

Sd 1 0.0298

Sd 2 0.0310

RelativeSd 1 0.968%

RelativeSd 2 1.01%

Table 3 – Older Pennies

Full Set (Bal. 5, Set E)

Mean 2.8458

Median 3.0413

Range 0.6241

Sd 1 0.2879

Sd 2 0.3529

RelativeSd 1 10.12%

RelativeSd 2 11.60%

Table 2 – Full Set



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evaluating the differences in mean and standard deviations for the three data sets – older pennies, newer pennies, and older and newer pennies combined (full set).The full set had a standard deviation and relative standard deviations that wereapproximately ten times higher than the older and newer penny sets. Therefore,since standard deviation is a measure of population variation, the individual weightsin the full set had a greater variability from the mean weight. Thus, moreuncertainty, or error, was associated with the full penny set data.

When separated, both older and newer pennies had standard deviations that weresignificantly smaller, so the two populations fell highly close to their calculated meanvalues. Thus, dividing the full set into older and newer pennies seems to havenormalized the distribution. This is reflective in the frequency histograms of thethree sets, particularly in the overall shapes of the graphs. The full penny sethistogram has two distinct peaks, whereas the older and newer penny sets havesomewhat bell-shaped frequency distributions. Below are the frequency histogramsfor the three data sets.

Figure 1 – Frequency Histogram of Full Penny Set (Balance 5, Set E)

Figure 2 – Frequency Histogram of Older Pennies separated from Set E (Balance 5)

0 0

5

0

7

0 0 0

Full Penny Set

0 0

2

1

3

1

0 0

Older Pennies



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Figure 3 – Frequency Histogram of Newer Pennies separated from Set E (Balance 5)

6. Calculate the means and standard d eviat ion o f 36 pennies (your set and oth er

two sets from two o ther students) , and predict the distr ibut ion. Compare this

distr ibut ion with tho se calculated is Step 4 and 5. What have you fo und?

12 Penn ies, Weighed One Time – Sets E & D, Balance 5

Mean 2.8200

Median 3.0413

Range 0.7066

Sd 1 0.2928

Sd 2 0.3699

Relative Sd 1 10.38%

Relative Sd 2 12.16%

Below is the frequency histogram of the two different sets of pennies.

Figure 4 – Frequency Histogram of Sets D & E, weighed on Balance 5

0 0

1

2

1 1

0 0

Newer Pennies

0 0

8

3

11

2

0 0

Frequency Histogram (#6)



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From this histogram, it is possible to conclude that the data does not fall under anormal distribution, because there are two distinct frequencies that are predominantin the data. These two frequencies may correspond to the two different metalcompositions of the pennies. The data is similar to the full set, in that it has higher standard deviation and relative standard deviation values and a more variable meanin comparison to the separated older penny and newer penny data sets. Thus, thisdata set is somewhat normally distributed, and further increasing the sample size or subdividing according to manufacture year could make it a true normal distribution or could make the distribution more representative of the pennies.

7. Have you learned anything new abou t pennies? Have you learned anything

new abou t stat is t ics and th eir appl icat ion?

It was interesting to see how variable pennies can be in their weights, and it wasalso a creative way to learn how to use an analytical balance. Some of thedetermining factors that affect the weight include manufacturing year, which reflectschemical composition, and contaminants, like dust particles or oily residue from the

hands. Regarding statistics, it seems that many aspects of statistics can be appliedtoward every-day purposes, and statistics can be quite handy in monitoring subtlechanges or trends, especially the mean and standard deviation functions.

lab 1 questions

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