DENSITY of the U.S. Penny
Significant Figures, Accuracy, Precision and Data Analysis
Adapted from Joseph Bularzik, Purdue University,: J.Chem. Ed., 84, 1456-1459 (September 2007)
The Lincoln penny of 1909 commemorated the centennial of Abraham Lincoln's birth. It was the first
regular-issue U.S. coin to bear the portrait of an actual American. In 1943, during World War II, a wartime
copper shortage prompted the introduction of a zinc-coated steel penny. The coin, however, proved so
unpopular that the U.S. Mint resumed production of the copper cent. Pennies minted in 1944 and 1945 were
struck from copper salvaged from spent ammunition cartridges. From 1959 through to 1981, the U.S. Mint
produced more than 130 billion pennies. Then, in 1982 the Mint introduced a penny that is 97.6 percent zinc
and only 2.4 percent copper with a density of 7.2 g/mL. By comparison, the old pre-1982 copper pennies were
95 percent copper and 5 percent zinc with a density of 8.8 g/mL. The copper-plated zinc penny is identical in
size and appearance to its copper-rich counterpart, but the current cent is 19% lighter. For example, a
5000-piece bag of copper pennies weighs 34 pounds, whereas the same size bag of zinc-alloy pennies tips the
scales at 28 pounds. The penny itself is only worth about six-tenths cent, but has increased recently due to the
rising cost of raw copper. The old copper-rich pennies cost about $0.008 to make. So, the Federal Government
actually makes money by selling pennies. For example, penny production in 1989 netted the Treasury about
$42.4 million. Since 1989, the copper-plated, zinc alloy penny is currently the only one-cent piece being
produced.
Discussion & Review: In science, it is extremely important to be able to make measurements correctly.
Reliable measurements allow for factual knowledge about the world to be expressed clearly and concisely.
Almost all scientific studies involve some kind measurement. As you continue your studies in chemistry and
science you will learn by performing experiments, that important properties of matter, such as density, depend
on specific measurements.
Numbers and Measurement. There are two general categories of numbers:
Exact numbers: numbers of absolute accuracy, having infinite precision
Measured numbers: numbers of finite accuracy with limited precision
Exact numbers are absolute values because there is no uncertainty in the value. For example, counted
numbers are exact numbers: the number of people in your lab section or the number of fingers on your hand.
Defined numbers such as 12 inches per foot, 16 ounces per pound are also exact numbers. To say there are
28.5 people in your lab section is meaningless. Anything other than 16 ounces is not a pound. These are exact
numbers. The accuracy and precision of any calculation does not depend on exact numbers. Measured
numbers are the numbers that are determined by a measurement during an experiment. For examples
measured numbers can be the length of a material measured by a linear scale, or the mass of a substance
measured by a balance. Several measurements should be made to verify measured numbers. The accuracy of a
measured number is calculated from the average of the measurements, how close the average is to the “true
value”. In the absence of a determinate error, or systematic error, the accuracy of a group of measurements
should be very high. The precision of a measurement depends upon the uncertainty of the individual
measurements. The precision of a measured number may be large if the measured values vary, but the accuracy
could be very high if the average comes close to the “true value”. On the other hand, the precision of a
measured number may be small if all the measurements have similar values, but the accuracy could be very low
if the average of those similar numbers is far from the “true value.”
Every measurement has two parts: the numerical value and the unit. Both must appear when quoting or
recording a measurement. The numerical value provides the accuracy of the measurement while the unit tells
us the dimension or property which has been measured. Without both parts, the unit and the value, the
measurement can be confusing or meaningless.
Measurement and the 10% Rule. The number of significant figures (SF) in a measurement always includes
one estimated digit when reading the measured value on a calibrated scale. We include one estimated digit
because it is standard practice in making a measurement to complete the measurement by reading or estimating
0.1 times (or 10%) of the calibrated separation between the nearest adjacent calibrations of the measuring
device. The estimated digit represents the uncertainty of the measurement. When reading the measurement on
a digital output, the instrument will usually list the uncertainty of the instrument. For example, a milligram
balance could present the measurement of 8.097 g. It is assumed that the uncertainty is + 0.001 g, an
uncertainty of + 1 for the last digit.
Example: Consider the line segment below in relation to an arbitrary measuring scale. The ruler is calibrated
to 0.1 units thus 10% would be 0.01 units. The uncertainty in a measurement using the ruler shown below is
then ± 0.01 units. We can say the line segment is 0.3 units long. But as you can see, the segment is closer to
0.31 units long than 0.30 or 0.32 units. We can measure 0.1 units exactly and estimate one-tenth of the smallest
unit by applying the 10% Rule. The measurement 0.3 units is accurate to 1 significant figure (SF) while 0.31
units (by using the 10% Rule) is accurate to 2 SF. The observation of 0.31 resulted from a judgment of the
observer. The accuracy of any measurement depends on the ability to take an accurate measurement and to
judiciously apply the 10% rule.
Line
Segment
Ruler
0.0
0.1
0.2
0.3
0.4
Volume Measurements. The volume of a sample is the total amount of space occupied by the sample. When
cooking, liquid volumes are measured in units of teaspoons, tablespoons and cups. In the laboratory, liquid
volumes are typically measured by using graduated cylinders or volumetric glassware. Such glassware is read
by observing the bottom of the meniscus level of the liquid and reading to 0.1 times (the 10% Rule) of the
smallest calibrated mark.
Mass Measurements and Weight. Mass is measured in the laboratory by using a balance. Electronic
balances can be tared when performing mass measurements. The term weight refers to the resulting force of
gravity on the mass of a body. However, we commonly use the term weigh to mean “determine the mass of
something.” To tare a balance means to set the diplay equal to zero while the container is on the balance. Then
the mass of the matter being weighed can be directly read from the balance, without having to subtract the mass
of the container.
Average Results and Average Deviation. Consider a set of valid experimental results. Set means a group of
results which are related in that each result refers to the same quantity. Valid means that each result is
acceptable and has been obtained by performing measurements correctly within the limits of the experimental
accuracy. Rather than report the entire set of values, we prefer to report the average result for each set. The
average result is the sum of all individual results in the set divided by the number of results.
Error analysis can sometimes be complicated. When given several sets of data, each of the individual sets must
be interpreted before an overall average result can be determined. An example of determining the average and
average deviation for an experimental data set is shown.
[NOTE: The Average Deviation does not capture the distribution of data. A statistical probability of data
falling within a certain range is described by the Standard Deviation. It is more commonly used. and will be
introduced shortly in class lecture. See page 3 of the Chemistry 120 ordinary Lab Manual (The Penny Lab).]
Average
Deviation
n
1

X =
n
Di =
Xi - X
Xi
i=1
Average Deviation
n
1

D =
n
Xi - X
i=1
4.32mL
4.41mL
4.28mL
4.94mL
Average = [4.32 + 4.41 + 4.28 + 4.94 ] / 4 = 4.49
Deviations:
For 4.32 D = 4.32 – 4.49
= 0.17
For 4.41
D = 4.41 – 4.49
= 0.08
For 4.28
D = 4.28 – 4.49
= 0.21
For 4.94
D = 4.94 – 4.49
= 0.45
Average Deviation = [0.17 + 0.08 + 0.21 + 0.45] / 4 = 0.23
Reported Value: 4.49 +/- 0.23 mL
Standard Deviation
Experimentally Determining Density
Significant Figures, Accuracy, Precision and Data Analysis
Complete the form on the opposite side of this page: one per group with each group member’s name is to
be turned-in when finished.
1. Calculate the density of a metal cylinder using your linear measurements and the mass. Be sure to
have the correct number of significant figures. Identify the metal from its density and show your
results to Dr. R.
2. Using the same cylinder or a cylinder made of the same metal, record it’s mass and measure its
volume by displacement using a graduated cylinder and water. Enter your data on the form.
Calculate the density. Record the value and provide a brief discussion of which method is more
accurate.
A group of students at Purdue University applied these methods to “new” and “old“ pennies. (NOTE:
Both the “new” and “old“ pennies that they used in the experiment were in public circulation and not in
mint condition.)
A group of students followed the following four different procedures for one set of “new” pennies and
another group repeated the measurements with “old” pennies.
Method 1:
The mass, height and diameter of a stack of ten pennies were measured respectively using an
analytical balance and calipers. The student data is provided in Table 1.
Method 2:
The mass, and volume of a stack of ten pennies were measured respectively using an analytical
balance and 100 mL graduated cylinder. The student data is in Table 2.
Method 3:
The mass, height and diameter of a single penny were measured respectively using an analytical
balance and calipers. The student data is provided in Table 3.
Method 4:
The masss, and volume of a single penny were measured respectively using an analytical balance and
10 mL graduated cylinder. The student data is in Table 4.
However, unlike diligent and meticulous DVC Chem 120 students, the Purdue students failed to record
whether the data they obtained was for “new” or “old“ pennies.
3. Your indivdual challenge is to determine in your breakout subgroup which of the data sets that
you’ve been assigned to analyze is for the “new” and which is “old“ and the precision and accuracy
of the method. You must use significant figures and knowledge of mass and volume measurements
correctly.
4. Returning to your parent group, select one of the four methods as being the best experimental
approach in obtaining the most precise and accurate results, and then ranking the four methods
respectively in increasing order of 1) precision and 2) accuracy.
#1) Calculate the density for your method
1) The stack height ranges from 14 to 15 mm and the diameter is about 19 mm. 2. Place the
stack of ten pennies into a 50 mL graduated cylinder. The volume of the stack is about 3.5 ml. 3.
Measure the thickness and width of a single penny using a vernier calipers. 4. Place a single
penny into the graduated cylinder and measure the displacement. class average density and
standard (or average) deviation demonstrate these differences. The data for 66 stacks of
pennies from several laboratory sections are shown in Table 1. More data are supplied in the
Supplemental Material.W The density standard deviations from the caliper measurements are
less than the standard deviations of the measurements using the graduated cylinder. As
expected, the graduated cylinder provides more accurate measurements. The data for six
laboratory sections are shown in Figure 1. Each section shows the expected behavior.
A. Mass Determination of New Pennies. Obtain fifteen, new, post-1982 pennies. Weigh ten of
the pennies together, record the mass, and then determine the average mass of a single penny. Fill in
the Table 3 to record your data. Weigh five individual new pennies and record their masses. Make
sure you keep track of which penny is number 1, 2, 3, 4 and 5.
B. Volume Determination through Linear Measurements of New Pennies. Use the vernier
calipers to measure the diameter and thickness (in cm) of the stack of the ten pennies. Your
instructor will show the proper techniques to use a vernier calipers. Record the total height and
diameter of the stack of pennies. From this measurement, calculate the average dimensions of a
single penny and record this in the table. Use the vernier calipers to measure the diameter and
thickness of each of the five individual pennies. Calculate the volume using the equation:
V = ( ¼ d2 ) (h) where d is the diameter in cm and h is the thickness in cm. Pay careful attention to
units.
C. Mass and Volume Determination of Old Pennies. Obtain fifteen, old, pre-1982 pennies.
Make the same mass and volume measurements on the old pennies and fill in the Table 4 for Part C
with their masses, linear measurements and calculated volumes.
D. Volume by Displacement of Water. Fill a large graduated cylinder with about 20 mL of water.
Record the exact initial volume of water added. Carefully add the same new ten pennies to the
cylinder as were measured in the stack. Record the new final volume of water. Determine the
volume (in cm3) occupied by the pennies by subtracting the initial volume from the final volume.
Calculate the volume of one average penny from this data. Fill a large graduated cylinder with about
20 mL of water. Add new penny number 1 to the cylinder and record the new volume. Repeat this
for new pennies 2 through 5. Put all this data in Table 5 Repeat the volume measurements for the
fifteen old pennies. Fill in Table 6 for the old pennies.
E. Density of a Penny. From the mass and volume measurements, calculate the average density per
penny. Density equals mass divided by volume, D = g/mL. Fill in Table 7 for Part E listing all the
density values for the pennies determined both by the linear measurements and volume measurements
for the new and old pennies.
F. Average and Average Deviation of Densities. For the five new pennies with individual
measurements, calculate an average density and average deviation of the density for both the
measurements using linear dimensions and displacement. Do this also for the five old pennies. Fill
in Table 8 for Part F with these averages and average deviations.
Discussion and Conclusion:
1) Compare the volume and average density that were measured for the stack of ten pennies by linear
measurement and displacement methods for both old and new pennies. Which method gave a larger
value?
2) Compare your average density measurements from table 8 for both the stack and individual
pennies to the average density measurements for the class.
3) By comparing to the actual density, which method, dimensions or displacement, gives a more
accurate measurement of the density of the penny? Why?
4) Which method is more accurate, the stack of ten pennies or individual pennies? Why?
5) Look at the average deviation for the class for each of the methods of measuring the density.
Compare your average deviations from table 8 to the average deviation for the class of the five
individual pennies.
6) Which method is more precise, the dimensions or the displacement? Why?
7) Which method is more precise, the stack of ten pennies or individual pennies? Why?
8) Comment on the measuring devices used in this experiment and how they effected the accuracy or
precision
9) What did you learn about accuracy and precision from this experiment?
Report the following values to the instructor to determine a class average. The class average will be
posted on Blackboard and should be included in your laboratory discussion.
You will report the values to the instructor by using an excel spreadsheet that on the computer in the
laboratory room.
The mass, height and volume for the stack of ten new pennies by linear measurement
The mass, height and volume for the five new pennies by linear measurement
The mass, height and volume for the stack of ten old pennies by linear measurement
The mass, height and volume for the five old pennies by linear measurement
The mass and volume for the stack of ten new pennies by displacement
The mass and volume for the five new pennies by displacement
The mass and volume for the stack of ten old pennies by displacement
The mass and volume for the five old pennies by displacement
The Density of a Penny: An Illustration of Significant Figures, Insignificant Numbers,
Accuracy and Precision.
Table 3 for Parts A and B
New Masses and Dimensions: By Linear Measurement
Penny
mass (g)
height (cm)
diameter (cm)
volume (cm3 or mL)
All Ten
Average of Ten
New Penny 1
New Penny 2
New Penny 3
New Penny 4
New Penny 5
Table 4 for Part C
Old Masses and Dimensions: By Linear Measurement
Penny
mass (g)
height (cm)
diameter (cm)
All Ten
Average of Ten
Old Penny 1
Old Penny 2
Old Penny 3
Old Penny 4
Old Penny 5
Tables 5 and 6 for Part D
volume (cm3 or mL)
Volume of New Pennies: by Displacement
Penny
initial V. (mL)
final V. (mL)
volume (cm3 or mL)
All Ten
Average of Ten
XXXXXX
XXXXXX
New Penny 1
New Penny 2
New Penny 3
New Penny 4
New Penny 5
Volume of Old Pennies: by Displacement
Penny
initial V. (mL)
final V. (mL)
All Ten
Average of Ten
Old Penny 1
Old Penny 2
Old Penny 3
Old Penny 4
Old Penny 5
XXXXXX
XXXXXX
volume (cm3 or mL)
Table 7 for Part E
Penny
Density
by linear dimensions
Average of Ten New
New Penny 1
New Penny 2
New Penny 3
New Penny 4
New Penny 5
Average of Ten Old
Old Penny 1
Old Penny 2
Old Penny 3
Old Penny 4
Old Penny5
by displacement
Table 8 for Part F
Pennies
Method
Stack of New
Dimensions
XXXXXXXXXX
Displacement
XXXXXXXXXX
5 New Pennies
Average Density
Average Deviation
Dimensions
Displacement
Stack of Old
5 Old Pennies
Dimensions
XXXXXXXXXX
Displacement
XXXXXXXXXX
Dimensions
Displacement
Discussion Section:
Discussion Section Continued:
14
Post Lab Questions. Show All Work and Be Specific with All Explanations.
1.
A 100 mL graduated cylinder calibrated in increments of 1.0 mL is filled with water so
that the meniscus coincides with the 9-mL mark. Explain why it is correct to say the
cylinder contains exactly 9 mL but is not acceptable to say the cylinder contains 9.00 mL
of water?
2.
Is the assumption that a penny has a regular shape valid?
3.
Because they have the same metal composition, will the masses of all the new pennies be
the same? Why would they be different?
15
4.
Why was it necessary to use only new pennies in order to determine an average density?
For example, what would happen if we used a mixture of old and new pennies?
5.
When you determined the volume of ten pennies by displacement of their volume of
water, how would the density calculation be affected if some water spilled from the
16