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Measurements & Uncertainties
Experiment AA
Objective
In this experiment you will make measure- ments and explore aspects of determining how well those measurements were made.
Introduction
What justifies the faith scientists place in the theories, mathematical models, or “laws” they invent to describe natural phenomena? Ultimately, experimentation is the only way to establish the validity and importance of such constructions. Quantitative measure- ments and mathematical descriptions are the bricks and mortar with which science is built. A theory should make predictions about the outcomes of measurements. A set of measure- ments should be able to test those predictions.
The simplest question to ask is: Is the theo- retical model in agreement with the measure- ments designed to test it? If the answer is no, the theory has to be abandoned or at least modified. If theory and experiment do agree, the theory is not proven correct, it is simply validated. The degree to which the theory is validated depends on the quality of the mea- surements used to test it. To know to what de- gree the theory is validated, one must consider the quantities being measured, the measuring instruments, and the conditions under which the measurements were taken. One must also consider the theoretical model and the condi- tions under which it is expected to be valid.
Uncertainties, random and systematic errors, precision and accuracy, noise, calibrations and standards, significant figures, propagation of errors, and linear regression are concepts and tools used to interpret measurements. In this first experiment, the measurement process itself is explored. The second exper- iment focuses on aspects of statistical analy- sis used in interpreting measurements and in comparing measurements and predictions.
Measurements
A measurement is an attempt to discover a quantity’s ideal or “true” value. Generally, the true value is unknown. We will use the subscript u (for unknown) to denote true val- ues and the subscript m to denote measured values. Taking the length y of a carefully ma- chined rod as an example, yu would represent its true length and ym would represent a mea- surement of its length. A measurement ym made with a real instru- ment is never expected to give the exact, true value. The measurement might be a little low or it might be a little high. The difference ∆y (∆ is the capital Greek letter delta) between the true value and the measured value is the measurement error.
∆y = yu − ym (1)
∆y cannot be determined without already knowing yu. However, knowing how big ∆y might be is very important; it would allow
AA 1
AA 2 Introductory Physics Laboratory
2.63 cm
cm
Figure 1: Measuring the length of a rod with a meter stick.
you to determine a range of true values that are consistent with the measured value. This range could then be compared with theoretical predictions.
Random Errors; Uncertainties
Random errors are measurement errors arising in just about all measurements. They are just as likely to be positive as negative. When ran- dom errors are likely to be small, one says the measurements have a high precision or they are very precise. A very precise instrument is capable of distinguishing very small changes in the quantity it measures. Consider a measurement ym of a rod using a simple meter stick (see Fig. 1). The scale’s smallest marked division is 0.1 cm. We line up one end with the 0 cm mark. (Though not true in practice, to simplify the discussion we will assume this can be done perfectly.) Then, a reading is made at the other end. The fig- ure shows the true length appears to be be- tween 2.6 and 2.7 cm. Since it’s closer to the 2.6 cm mark, you might report the length as ym = 2.6 cm. As is obviously the case in this example, one can and should estimate one ex- tra digit. It would be reasonable to record the length as ym = 2.63 cm — estimating the end is about three-tenths of the way between the marks. There is seldom, if ever, any justifi-
cation for estimating two extra digits because one is usually not certain that even the single estimated digit is correct. Suppose by critically examining the mea- surement process you estimate you could be high or low by about 0.03 cm. The measure- ment would be recorded as (2. 63 ± 0 .03) cm and indicates you believe the true value is probably between 2.60 and 2.66 cm. The ± 0 .03 cm is called the random uncertainty (or simply the uncertainty) of the measurement. It gives a range for the measurement error; ∆y is probably in the range from -0.03 to 0.03 cm. We will use the symbol σy for such an esti- mate of the uncertainty. (σ is the Greek letter sigma.) Thus, the notation (2. 63 ± 0 .03) cm represents two numbers: ym = 2.63 cm and σy = 0.03 cm. Note that:
- σy is not the measurement error ∆y; it is an indicator of possible ∆y’s.
- σy should not give the maximum possible ∆y. There should be about 32% proba- bility (around 1 chance in 3) that the er- ror ∆y will actually be outside the range between −σy and +σy. Why not make σy extra large and report your measurement as (2. 63 ± 0 .05) cm? Do- ing so would give you a better chance of be- ing “within error” since you would be giving a larger range (2.58 to 2.68 cm) within which the true value can lie and still be in agree- ment with your measurement. The only thing wrong with this argument is that when you specify an uncertainty you must also specify its level of confidence. The confidence level is an estimate of the probability that the true value will lie within the range specified by the measurement value and its uncertainty. Were you to use the larger range, you should be hon- est and say that you are closer to 90 or 95%
Rev. 1.
AA 4 Introductory Physics Laboratory
the scale properly. (Sometimes there is a mir- ror on the scale to aid in determining where your eye should be placed.) This kind of er- ror cannot be totally eliminated and may also contribute to the random error. However, if you consistently position your eye incorrectly, it becomes a mistake which leads to incorrect results. Personal errors can always be elimi- nated with careful attention to proper proce- dure.
Systematic errors arise when using imper- fectly calibrated instruments or measurement techniques which cause the readings to be sys- tematically “off” in one direction. Thus, mea- surements with large systematic errors are al- ways above or always below the true value. For example, a parallax error from a consis- tently incorrect eye position could produce a systematic error.
Measuring instruments are never perfectly calibrated. Calibration refers to procedures — performed during manufacture, or in the laboratory — in which the instrument is veri- fied to give the expected value when measur- ing a known quantity, or standard. A standard is effectively a known value of yu that, when measured (giving the result ym) allows you to determine the value of ∆y. With this infor- mation, adjustments can be made to the in- strument or corrections can be applied to the instrument readings to make it produce cor- rect results.
For example, instrument zeroing is an im- portant (but never complete) part of a calibra- tion that can and should be performed in the laboratory whenever possible. You assure the quantity being measured is zero (e.g. no mass on a beam balance, or no current through a current meter) and then you make a zero read- ing. Then, you either adjust the instrument to give a reading of zero or subtract the zero reading from all subsequent measurements.
Accuracy refers to the level of agreement that can be expected between a measured value and a standard. An instrument with high accuracy will give measurements that are — on average — very close to the standard. The accuracy attainable by a calibration is of- ten limited by the precision of the instrument. But a very precise instrument can be quite in- accurate if it is out of calibration. You will generally be unable to calibrate the measuring instruments you use. But you should always question whether or not they are calibrated to the accuracy to which you are reading them, i.e., the precision of the measurements.
Significant Figures
In reporting quantitative values one should never bother the reader of the report by ex- pressing numerical values with more digits than are truly known or at least reasonably estimated. The numerically literate person should understand that when a newspaper ar- ticle says the population of a city is 180, people (without explicitly stating the uncer- tainty) it means that it could be anywhere be- tween 175,000 and 185,000 people. The first 0 and subsequent zeros are not really significant — they are simply place holders. The number 180,000 is understood to have two significant figures. If the article said the population was 182,000 people, it would be in- dicating a more careful census was made and the population is known to within about 500 people. The number 182,000 has three signif- icant figures. What if the reporter wanted to indicate the population is known to be 180,000 people to within about 500 people, that is, the first zero is significant? This could, of course, simply be stated in the article. Or the number could be expressed in scientific notation as 1. 80 × 105.
Rev. 1.
Measurements & Uncertainties AA 5
It would then be understood that the 0 would not have been included if it were not a mean- ingful, i.e., significant, digit. Thus, to a sci- entist, the number 180,000 when written as
- 8 × 105 has a different meaning than when it is written 1. 80 × 105 — the latter number expresses a higher degree of knowledge about the population than the first. The number
- 80 × 105 has three significant figures, the number 1. 8 × 105 , only two.
When numbers are expressed explicitly showing the decimal point, some simple logic reveals the number of significant figures. Leading zeros, as with the first six zeros in 0.000002407, are never significant, — they just indicate the exponent. Sometimes these zeros are called place holders. If the number were written in scientific notation 2. 407 × 10 −^6 ( significant figures) it is easier to see the leading zeros were not significant. Note: The sand- wiched zero is significant. Also, trailing zeros, as the last zero in 0.003470 are significant ( significant figures). The leading zeros are not significant but the final 0 is significant — oth- erwise it should not and would not be given. And in the number 180,000. — note the dec- imal! — all the zeros are significant (6 signif- icant figures) — otherwise the decimal point would not be given.
Thus, when uncertainties are not explicitly given, the values and their significant figures implicitly provide the uncertainty. The uncer- tainty is implied to be ± 1 /2 the place value of least (or last) significant digit. That is, if the least significant digit stands for the number of 1000’s in the value (as in the number 182,000), the uncertainty is implied to be 500. If it is in the tenths place (as in 42.7), the uncertainty is implied to be 0.05.
When the uncertainty is explicitly given (as in 42. 7 ± 0 .2) one must simply not bother the reader with insignificant digits. It is NOT cor-
rect to report a value as 42. 7128 ± 0 .2. The digits 1, 2, and 8 at the end of this number cannot be significant digits. The true value might have any other digits in these three places. Including the 128 would indicate the report writer knows something the uncertainty indicates is not really known. The number,
- 7128 ± 0 .2, is not consistent with itself. It should be reported as 42. 7 ± 0 .2. It is also incorrect to report a value such as
- 7 ± 0 .002. Here, the value 42.7 is not written with enough significant figures for the specified uncertainty. Is it really 42. 700 ± 0 .002? If so, those last two zeros must be supplied. If, as is more likely the case, those last two digits were lost because you “rounded off,” you must redo the calculation to get those digits back. In later experiments, you will be obtaining quantities and their uncertainties from calcu- lations done on a calculator or using the com- puter spreadsheet program. Calculators and spreadsheets often provide many more dig- its than are really significant. When the re- sults of calculations are written into the re- port, the number of significant figures should be written to be consistent with the uncer- tainties. Suppose the calculated value of some quantity comes out 0.38746859, and its uncer- tainty comes out 0.0284176898. Uncertainties are usually not well determined, and even if they were, all the extra digits do not provide terribly useful information. One digit in the uncertainty is all that is needed. In this ex- ample, the uncertainty would be reported as 0.03. The result, including the uncertainty, should then be reported as 0. 39 ± 0 .03. When the uncertainty’s first non-zero digit is a 1, it is reasonable to keep two digits in the uncer- tainty. In the example above, had the uncer- tainty been 0.01467376, it would have been reported as 0.015, and the result would have been reported as 0. 387 ± 0 .015.
Rev. 1.
Measurements & Uncertainties AA 7
depth gauge 1
outside calipers
5
inside calipers
0 1 2 3 14 6
4 (^7 28 )
0 2 05 vernier scales
main (in) scales (cm)
reading: 1.945 cm 0
2
2 4 6 8 0
3 4
(^32 )
6 8 10
4 5 46 7 8 9 52 1 2 3 64 56
0 15 2025
1/ 77 8 9 318
(a)
(^2 34 59 6 7 84 1 2 113 4 5 6 712 8 9 5113 2344 145 6 78159 )
frame 15
anvil spindle 0
reading: 17.626 mm 15
5
10
20
ratchet 5 10 15 510 1520
sleeve
(b)
Figure 3: The vernier calipers (a) and the micrometer calipers (b). Below each figure, the scales are magnified to show a reading.
of the top, moveable scale which lines up with the bottom, main scale was determined. Ev- ery line on the top scale goes by five. Example 00, 05, 10, 15, 20... The mark that actually lines up is the 45 line. Therefore, the reading is 1.945 cm. The vernier caliper also has a set of inside calipers and a depth gauge. These are use- ful for measuring inside diameters and depths of holes. Whether measuring with the inside calipers, outside calipers or the depth gauge the reading of the scales in the same. When using the vernier calipers a zero read- ing must be made first. Then the jaws should be closed on the object and a second reading made. The object’s length is then the differ- ence in these two readings.
Micrometer
The micrometer is based on a precision screw and nut. (See Fig. 3b.) The sleeve, spindle, and screw are rigidly connected, and rotating the sleeve moves all three along the nut, which is rigidly attached to the frame. One turn of the sleeve moves the spindle by an amount equal to the pitch of the screw, i.e., the dis- tance between the threads on the screw.
Our micrometers have a pitch of 0.5 mm and the main scale on the frame is marked in mil- limeters, with markings each half-millimeter. The sleeve has 50 markings around its circum- ference. Thus the intervals on the sleeve cor- respond to 1/50 of 0.5 mm or 0.01 mm. Es- timating to the nearest tenth (interpolating) between these intervals thus corresponds to 0.001 mm. Look at the inset on Fig. 3b. The main markings on the frame determine the reading to 0.5 mm; 17.5 mm in the figure. To this partial reading is added the reading from the sleeve at the point where it intersects the cen- ter line on the main scale; 0.126 mm in the figure — the final 6 from interpolating tenths between the sleeve divisions. Thus, the read- ing in the figure is 17.626 mm. When using a micrometer, a zero reading must first be taken. Then, the spindle and anvil should be gently closed on the object to take the reading. This closing must AL- WAYS be done using the ratchet mecha- nism which may be at the end of the sleeve or wrapped around the middle of the sleeve. The ratchet will slip (continue turning, while the sleeve stops moving, producing a clicking sound) when the micrometer is properly tight-
Rev. 1.
AA 8 Introductory Physics Laboratory
zero adjust
0
pan 0
0 3
mass standards
100
10 20 1 2 7 200 300
30 40 50 60 70 4 5 6 400
80 90 8
pointer
500
100 9 10
marker
Figure 4: A triple beam balance. The object to be measured is placed on the pan and the stan- dard masses are moved along the three horizontal beams to achieve a balance as indicated by the pointer alignment.
ened around the object. Turning the sleeve too hard against an object will damage the screw/nut combination and ruin the microm- eter.
Beam Balance
The beam balance is an instrument for mea- suring masses. (See Fig. 4.) It compares an unknown mass to standard masses that are part of the balance. An unknown mass is de- termined by placing it on the pan and balanc- ing it with the standard masses. The stan- dard masses are used by sliding them along the beams. The pointer at the end of the beam lines up with the marker when all mass is bal- anced.
The zero adjust on the balance can be used to zero it, but a container is often useful to hold the object to be measured or to center it on the pan. Rather than zero the balance with the adjustment knob, it is simpler to take a zero reading with the container on the pan. Then a second reading is made with both the object and the container on the pan. The zero reading is subtracted out of this final reading with the object.
Procedure
In the first part of this experiment you will determine the length and/or mass of various objects — in some cases with more than one measurement instrument. The purpose is not just to determine the length or mass of the object but also to assess various contributions to the measurement uncertainty. For example, all measurements require you first to make a zero reading. You cannot do this perfectly; therefore there is an uncertainty in this step. There are always uncertain- ties in zeroing an instrument or in mak- ing a zero reading. This zeroing uncertainty will have some effect on the uncertainty of the object’s measured value ym. If we let yzr represent the zero reading and let yr represent the reading with the object in place, then the object’s measured value, ym, is: ym = yr − yzr (2) For example: If we line an object on a ruler at the 1.00 cm line (assume we think the un- certainty in placing the object right on a mark- ing is ± 0 .02 cm) and it ends at 10.44 cm line (assume the estimated uncertainty in this reading is ± 0 .03 cm), then the value or actual length of the object is 10.44 cm − 1 .00 cm =
- 44 cm. What is its uncertainty in this length? Let σzr represent the uncertainty in the zeroing, i.e., the uncertainty in yzr (which will always be non-zero even if yzr is zero). Let σr represent the uncertainty in the mea- surement when the actual object is in place, i.e., the uncertainty in yr. We state without proof that σy, representing the uncertainty in ym, is given by
σy =
√ (σzr)^2 + (σr)^2 (3)
For the example above the uncer- tainty in the value would be σy =
Rev. 1.
AA 10 Introductory Physics Laboratory
should get the same reading each time (to within the reproducibility of the in- strument). If the sphere is not perfectly round you would get different readings. If your readings fluctuate, you might still call it a reproducibility problem — you do not get the same reading each time. Or you might call it a noise problem — the value being measured is not well de- fined and “jumps” about. Do not take a formal sample of repeated measurements. Just get a feel for the range of readings. Then take a rough average as “the” read- ing, and for the reading uncertainty use about half the range.
Symbol Definitions
yu True value of a quantity y. ym Measured value of a quantity y. ∆y The measurement error: ∆y = yu − ym. σy The uncertainty in the determination of a measured value ym. lr Size corresponding to smallest marked division. σmin Uncertainty in interpolating (estimating) between smallest marked divisions. yzr Zero reading; instrument reading corresponding to zero. σzr Uncertainty in yzr, the zero reading. yr Instrument reading with object to be measured in place. σr Uncertainty in yr, the instrument reading with the object in place.
- Measure the diameter d′^ of the dented sphere using the micrometer. In this case, the reproducibility or noise problem is even more pronounced, which will raise your uncertainty in the reading.
- Measure the mass m of the smooth sphere. Place the washer at the center of the balance pan. It will be used to keep the sphere centered on the pan. You do not have to zero the balance first because the reading with the washer can be taken as the zero reading.
- Measure the mass m′^ of the dented sphere.
Rev. 1.
Experiment AA Measurements & Uncertainties
Title Sheet
Name: Date:
Partner: SSN:
Course: Section:
Instructor:
Comments on the experiment or write-up.
Experiment AA Measurements & Uncertainties
Comprehension Questions
- How do the two ruler measurements illustrate that an uncertainty does not just depend on the measuring instrument? Why are the uncertainties for these measurements different? Be specific.
- We agreed to take the least reading lr on the vernier caliper from the markings on the moveable scale (as opposed to the main scale). Unlike lr for our other instruments, you cannot interpolate between these least readings. In general, if you cannot interpolate between least readings, how should σmin be related to lr?
- Make and report below at least three zero readings on the micrometer. How reproducible are your micrometer zero readings? So, how does the best case uncertainty σmin for the micrometer compare with σzr? Specifically, if yzr is very reproducible, should σzr be less than σmin?
Experiment AA Measurements & Uncertainties
- Considering all three measurements of the smooth sphere’s diameter (ruler, vernier caliper, and micrometer), which of the three would you report as the best estimate of the true diameter of the smooth sphere and its uncertainty? Explain your reasoning. Are any of the measurements more than two of their own uncertainties from this value?
- Compare the uncertainties in the micrometer measurements of the smooth and dented sphere’s diameter and explain why they are different. If your uncertainties are not different then explain why, in general, they should be and why yours are not.
- What determines the uncertainty of mass measurements made with the balance? If you consider it balanced as long as the pointer is anywhere within about two millimeters of the center line, what would be the precision of the mass readings? You must actually test it with your beam balance. Explain how you tested the precision.
Experiment AA Measurements & Uncertainties
