UNIVERSITY PHYSICS LABORATORY
Western Kentucky University
MEASUREMENTS AND UNCERTAINTIES
Purpose
The purposes of this experiment are (1) to learn how to correctly use measuring instruments such as a meter
stick, a vernier caliper, a micrometer caliper and a hanging pan mechanical balance, (2) to learn how to
calculate from a set of measurements the average value and the uncertainty using the mean, standard deviation
and standard deviation of the mean, and (3) to learn how to determine the uncertainty in a value that is
calculated from two or more measured quantities by propagation of uncertainty.
Introduction
In this experiment you will compute the density (ρ) of a block of aluminum by dividing its mass by its volume
ρ=
m
.
V
A. Measurements
The mass (m) will be measured using a hanging-pan mechanical balance which is depicted in Figure 1 below.
The balance has four beams, each with a sliding mass, as shown in the figure. Each mass has different
corresponding sizes. One slides along a scale with a notch at every 100 grams, the second slides along a scale
with a notch at every 10 grams, the third slides along a scale with a notch every 1 gram, and the fourth slides
along a scale with 0.1 gram main divisions and 0.01 gram subdivisions. The object whose mass is to be
determined is placed on the pan, the masses are then adjusted for balance, and the readings associated with
each slider are added to determine the unknown mass. You should always make sure the scale reads zero when
no mass is on the pan before using the scale.
1. Sample Pan
2. Hanging Assembly
3. 100 – 400 g Poise
4. 10 – 100 g Poise
5. 1 – 10 g Poise
6. 0 – 1.0 g Poise
7. Zero Readout
8. Base
9. Beam
10. Specific Gravity Platform
11. Zero Readout Pointer
Figure 1 – Hanging Pan Mechanical Balance
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Measurements and Uncertainties
The volume (V) will be computed as the product of the block’s length, width and thickness. The length (L)
of the block will be measured using a meter stick. The proper use of a meter stick is depicted below in Figure
2. One should avoid using the end of the meter stick since it may have become worn due to extended use.
Instead place the object next to the meter stick and record the left edge position and the right edge position.
When reading the centimeter scale on a meter stick the measurements should be made to the nearest hundredth
of a centimeter (0.01 cm). This requires that you estimate between the millimeter marks on the meter stick.
For example, in Figure 2 you would have to ask yourself "is the left edge at 13.65 cm, 13.66 cm, or 13.67
cm?" and make a judgement. Not everyone will answer this question with exactly the same value.
Figure 2 – Use of a Meter Stick
The width (W) of the aluminum block will be measured using a vernier caliper. The special feature of a vernier
caliper, which is depicted in Figure 3, is its vernier scale. The vernier scale makes it possible to determine
hundredths of a cm position with much more certainty than is possible with the estimation that has to be done
when using a meter stick. In fact, the calipers you will be using can be read to the nearest 0.002 cm.
Figure 3 - The Vernier Caliper
The large jaws on the caliper are used to measure the outside dimension of an object. If you have an object
with an opening the small jaws can be used to determine the length of this inside dimension. The depth of a
hole can be measured with the depth gauge on the end of the caliper. In each case one moves the jaws on the
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Measurements and Uncertainties
caliper until they snuggly fit into or around the object to be measured. The reading from the vernier is then
determined by reading the vernier scale. The calipers you will use can be read in inches (along the top) or in
centimeters (along the bottom).
Figure 4 shows a close up of the vernier scale which slides along the main scale as the caliper jaws are moved.
The zero of the vernier scale in Figure 4 is between 2.4 and 2.5 cm. To determine the next two significant
digits one looks at the vernier scale and estimates which tick mark lines up the best with a tick mark above it
on the main scale. Again, this is somewhat of a judgement call. Different people may get very slightly
differing results when reading a vernier caliper. The vernier scale on the calipers that you will use has five
tick marks between each number. Each of these tick marks represents 0.002 cm. In Figure 4 one could declare
very easily that the tick mark on the vernier scale that lines up most directly with one above it corresponds to
0.062 cm. Thus the reading on the vernier shown in Figure 4 is 2.400 cm + 0.062 cm = 2.462 cm. The
uncertainty in this reading would be on the order of one tick mark or ± 0.002 cm.
Figure 4 – The Vernier Scale
The thickness (T) of the aluminum block will be measured using a micrometer caliper which is depicted in
Figure 5. Micrometer calipers are often used to measure dimensions when one desires even greater precision
than is possible with a vernier caliper. Direct reading of most micrometers (at least the ones we will use) will
give a value in mm rather than in cm.
The special feature of a micrometer caliper is a rotating scale which moves one of the caliper's jaws by means
of a precisely machined screw. The divisions on the rotating scale are spaced so that turning the scale through
100 divisions moves the scale and the movable jaw 1 mm. On the calipers that we will be using this
corresponds to two complete revolutions of the rotating scale. There will be tick marks above and below the
reference line. One revolution of the thimble advances the rotating scale through one tick mark on the reference
line or 0.5 mm. If you force the scale too far, you may dent the object being measured or strip the threads of
the screw. To avoid this, always turn the scale by means of the "Ratchet Knob," which is designed to slip
at a certain maximum twisting force.
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Measurements and Uncertainties
Figure 5 – The Micrometer Caliper
Figure 6 shows an example reading from a micrometer very similar in design to the ones that you will use in
the laboratory. Study the figure and make sure that you understand how the micrometer is used to measure
a dimension to the nearest ± 0.005 mm.
Figure 6 – An Example Micrometer Reading
B. Uncertainties in Measurements
To determine the “best” value of a set of several measurements one calculates the mean or average value. The
expression for the average value of N measurements of the length of an object is
L=
L1 + L2 + " + L N
1
=
N
N
N
∑L
i
.
i =1
It is almost as important to correctly express the uncertainty in a measurement as it is to correctly represent
the average value. In the example Data Table 1 (page MU-9) all the values of the length L lie between 10.57
cm and 10.61 cm, a range of 0.04 cm. Suppose another experimenter’s measurements of L all lie between
10.55 cm and 10.63 cm, a range of 0.08 cm. Clearly, you would feel less sure about his values of L; they are
more uncertain than those in Data Table 1.
In the WKU University Physics Laboratory we will use the sample standard deviation to indicate the
uncertainty in a set of measurements. As you learned in the introductory laboratory session, to calculate the
standard deviation of the set of N measurements of the length one would compute
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Measurements and Uncertainties
1 N
σL =
(Li - L )2 =
∑
N - 1 i=1
(L − L ) + (L
2
1
2
− L ) + " + ( LN − L )
2
N −1
2
.
Alternatively, you could simply use the spreadsheet function STDEV(…)which calculates the above formula.
If one made a subsequent measurement of the length it would likely lie within one standard deviation of the
mean value. More exactly, if the only errors associated with the measurement are random in nature then we
can expect our results to follow the normal (or Gaussian) distribution. We would expect that a subsequent
measurement to have a 68 % probability of lying within one standard deviation of the mean, a 89 % probability
of lying within two standard deviations of the mean and a 96 % probability of lying within three standard
deviations from the mean.
As we continue to make more measurements we expect that the mean value we compute will become closer
to the true mean value. This concept is shown by computing the standard deviation in the mean (sometimes
called SEM for standard error of the mean) which is given by
σ L = SEM =
σL
.
N
Clearly, as the number of measurements (N) increases the standard deviation of the mean will decrease as one
would expect. In the WKU University Physics Laboratory we will use the standard deviation of the mean as
the best estimate of the uncertainty associated with the mean of a set of measurements. Also, we will always
round this uncertainty to one significant digit and then round the average value to the same precision
as the uncertainty. For the data in Table 1 the uncertainty is calculated as 0.007483 cm which we round to
0.007 cm, since uncertainty is always expressed as one significant figure. The final result would then be
indicated as
L ± σ L = (10.584 ± 0.007) cm.
Notice that this final result is expressed more precisely than any of the individual measurements. By taking
several measurements and finding the average value we can often improve the precision of the final result.
C. Propagation of Uncertainty
The volume of the block is V = L × W × T . Since our measurements of each of the 3 dimensions are
uncertain, the calculated volume will also be uncertain. The procedure of propagation of uncertainty allows
one to quantify the amount of uncertainty associated in a calculated value due to uncertainties in the quantities
from which it was calculated.
Consider two measured quantities u and v. Suppose the uncertainty in the mean of these two quantities are
denoted by σ u and σ v respectively. If these two quantities are added to compute w = u + v then the
uncertainty in w is given by
σ w = σ u2 + σ v2 .
It is important to realize that when combining uncertain quantities the uncertainty will never decrease. For
example, if one computes the difference in u and v, z = u – v, then the uncertainty in z is still given by
σ z = σ u2 + σ v2 .
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Measurements and Uncertainties
Sometimes one must use estimates of uncertainty (which we will express as ∆u) rather than uncertainties
calculated using the procedure described above for the uncertainty of the mean. In this circumstance it is
perfectly acceptable to simply use what is called “worse case” propagation of uncertainty. In the worst case,
the uncertainty in w and z will be given by
∆w = ∆u + ∆v .
∆z = ∆u + ∆v .
Whenever one multiplies or divides two (or more) measured quantities, say u and v,
x=auv
y=a
or
u
v
(where a is a constant) the fractional uncertainty in the either quantity will be
2
2
σx σy
σu  σv 
=
=   + 
x
y
 u   v 
On the other hand, if one simply wants to use worst-case propagation of uncertainty then the uncertainty in the
product or the quotient will be given by
∆x ∆u ∆v
=
+
.
x
u
v
Finally, if we generalize the above discussion for the uncertainty in the volume of the aluminum block then
we have
2
2
2
σ  σ  σ 
σV = V  L  +  W  +  T  .
 L  W  T 
Procedure
A. Volume of the Aluminum Block
1. Obtain an aluminum block from your instructor. Record its identification number on your data sheet.
2. Place the block next to the meter stick as shown in Figure 1. Read and record the position of the left edge
and the position of the right edge of the block each to the nearest 0.01 cm. The difference between the
readings is the length L of the block.
3. Record your measurements, as well as the resulting values of L, in a table similar to Data Table 1 (page
MU-9).
4. Repeat the measurement for a total of 5 times. Each time place the block next to the meter stick at an
arbitrary location without looking, in order not to be influenced by your previous readings.
5. Use the vernier caliper to make 5 measurements of the width W of your block, recording your
measurements in a table similar to as Data Table 1 (shown on page MU-9). Be sure to make your
measurements to within the nearest 0.002 cm.
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Measurements and Uncertainties
6. Use the micrometer caliper to make 5 measurements of the thickness T of your block, recording your
results in your Data Table 1. Notice that the micrometer will give a result in mm, to the nearest 0.001 mm.
To read the micrometer to the nearest 0.001 mm, the very last significant digit will have to be estimated.
You may only be able to judge that it is 0.000 (exactly on the line) or 0.005 mm.
7. Almost certainly your 5 values of the length L will not be exactly the same. Clearly the “best” value of
L is the average (or “mean”) value, which we will represent by L . Using the Microsoft Excel function
AVERAGE(…) calculate the average value of the length directly in your spreadsheet as is shown in Data
Table 1 (page MU-9). You should also calculate L directly on your handwritten data sheet to confirm
the correctness of your spreadsheet formula.
8. In a similar manner use the spreadsheet to determine the average value of the width and thickness of the
block.
9. Compute the standard deviation of the set of length measurements. Show this as a sample calculation on
your data sheet. The Excel function STDEV(…) will compute the sample standard deviation. Use this
formula to compute the standard deviation of the length L and compare with your value calculated by
hand. Use the spreadsheet to also calculate the standard deviation of the width W and thickness T
measurements.
10. Calculate the standard error of the mean for the length by dividing the standard deviation by the square
root of the number of measurements. Show this as a sample calculation on your data sheet. Also compute
this using the spreadsheet and compute the standard deviation of the mean for the width and thickness
measurements as well. In each case indicate your final result as the mean value ± the standard deviation
in the mean.
11. Calculate the volume of your block using V = L × W × T .
12. Calculate the uncertainty in the volume σ V using the appropriate expression for uncertainty of a product.
13. Write your final result as V ± σ V being sure to round the result and uncertainty appropriately.
B. Mass of Aluminum Block Using the Hanging-Pan Mechanical Balance
1. Place the block on the pan of the triple-beam balance. First the 100 gram and 10 gram masses should be
adjusted and set into their appropriate notches, then adjust the smallest mass to achieve balance.
2. Record the reading M of the balance as precisely as you can. Note that the smallest division on the riderscale represents 0.01 gram.
3. Like all measurements, your measurement of the mass is uncertain. To estimate the size of the uncertainty
in the mass, we will use a method different from before -- one that does not require several measurements.
We will estimate. Starting with the object balanced, slide the smallest mass to the right until the pan is
clearly unbalanced. Record this unbalanced mass. Bring the pan into balance again. Now slide the
smallest mass to the left until the pan is again clearly unbalanced. Record this mass. Use one half of the
range of the two unbalanced measurements as the uncertainty of the block's mass.
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Measurements and Uncertainties
C. Density of the Block
1. The density ρ of an object is defined as its mass per unit volume, ρ = M/V. Calculate the density of your
block of aluminum.
2. Write a generalized equation for the uncertainty in the density σ ρ using the rules for propagation of
uncertainty.
3. Calculate the uncertainty in the density using your equation for σ ρ .
4. Report your final result as ρ ± σ ρ .
5. Calculate the percent discrepancy between your value of the density of the block and the published value
of 2.70 g/cm3. The formula for percent discrepancy is
% discrepancy =
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(measured value − published value) × 100% .
published value
MU-8
Measurements and Uncertainties
Data Table 1
Determination of the Density of Aluminum Block # 99
(SAMPLE ONLY)
Trial Number
1
2
3
4
5
Left Edge
(cm)
xL
15.62
41.27
22.32
9.46
71.38
Right Edge
(cm)
xR
26.21
51.84
32.93
20.04
81.95
Mean
Std. Dev.
SEM
Length (cm)
Width (cm)
L = xR - x L
10.59
10.57
10.61
10.58
10.57
W
5.372
5.370
5.368
5.374
5.364
Thickness
(mm)
T
??
??
??
??
??
10.584000
0.016733
0.007483
5.369600
0.003847
0.001720
??
??
??
Length =
Width =
Thickness =
10.584
5.370
??
±
±
±
Mass =
??
±
?? ??
Volume =
??
±
?? ??
Density =
??
±
?? ??
Published
Density =
% Discrepancy =
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0.007 cm
0.002 cm
?? ??
2.70 g/cm3
??
MU-9
Measurements and Uncertainties
University Physics Laboratory
Western Kentucky University
Measurements and Uncertainties
Prelab Questions
MU-01
What is the indicated measurement, in cm, on the vernier caliper depicted in the picture (random picture
similar to Figure 3) below?
MU-02
What is the reading, in mm, on the micrometer caliper depicted in the picture (random picture similar to Figure
5) below?
MU-03.
The following measurements were made of the length of a Copper bar using a vernier caliper:
________ cm, ________cm, ________cm, ________cm, ________cm, ________cm, ________cm.
What is the mean of these measurements?
____________________ cm
What is the standard deviation of these data ?
____________________ cm
What is the standard error of the mean ?
____________________ cm
MU-04
Laboratory measurements performed upon a rectangular steel plate show that the length l = ________ ±
________ cm and that the width w = ________ ± ________ cm. Determine the area of the steel plate and the
corresponding uncertainty.
A =____________________ cm2 ± ____________________ cm2
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Measurements and Uncertainties