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Measurement Analysis 1:
Measurement Uncertainty and Propagation
You should read Sections 1.6 and 1.7 on pp. 12–16 of the Serway text before this activity. Please note that while attending the MA1 evening lecture is optional, the MA1 assignment is NOT optional and must be turned in before the deadline for your division for credit. The deadline for your division is specified in READ ME FIRST! at the front of this manual.
At the end of this activity, you should:
- Understand the form of measurements in the laboratory, including measured values and uncertainties.
- Know how to get uncertainties for measurements made using laboratory instruments.
- Be able to discriminate between measurements that agree and those that are discrepant.
- Understand the difference between precision and accuracy.
- Be able to combine measurements and their uncertainties through addition, subtrac- tion, multiplication, and division.
- Be able to properly round measurements and treat significant figures.
1 Measurements
1.1 Uncertainty in measurements
In an ideal world, measurements are always perfect: there, wooden boards can be cut to exactly two meters in length and a block of steel can have a mass of exactly three kilograms. However, we live in the real world, and here measurements are never perfect. In our world, measuring devices have limitations.
The imperfection inherent in all measurements is called an uncertainty. In the Physics 152 laboratory, we will write an uncertainty almost every time we make a measurement. Our notation for measurements and their uncertainties takes the following form:
(measured value ± uncertainty) proper units
where the ± is read ‘plus or minus.’
9.794 9.796 9.798 9.800 9.802 9.804 9.
9.801 m/s^2
m/s^2
Figure 1: Measurement and uncertainty: (9. 801 ± 0 .003) m/s^2
Consider the measurement g = (9. 801 ± 0 .003) m/s^2. We interpret this measurement as meaning that the experimentally determined value of g can lie anywhere between the values 9.801 + 0.003 m/s^2 and 9. 801 − 0 .003 m/s^2 , or 9.798 m/s^2 ≤ g ≤ 9 .804 m/s^2. As you can see, a real world measurement is not one simple measured value, but is actually a range of possible values (see Figure 1). This range is determined by the uncertainty in the measurement. As uncertainty is reduced, this range is narrowed.
Here are two examples of measurements: v = (4. 000 ± 0 .002) m/s G = (6. 67 ± 0 .01) × 10 −^11 N·m^2 /kg^2 Look over the measurements given above, paying close attention to the number of decimal places in the measured values and the uncertainties (when the measurement is good to the thousandths place, so is the uncertainty; when the measurement is good to the hundredths place, so is the uncertainty). You should notice that they always agree, and this is most important:
— In a measurement, the measured value and its uncertainty must always have the same number of digits after the decimal place.
Examples of nonsensical measurements are (9. 8 ± 0 .0001) m/s^2 and (9. 801 ± 0 .1) m/s^2 ; writing such nonsensical measurements will cause readers to judge you as either incompetent or sloppy. Avoid writing improper measurements by always making sure the decimal places agree.
Sometimes we want to talk about measurements more generally, and so we write them without actual numbers. In these cases, we use the lowercase Greek letter delta, or δ to represent the uncertainty in the measurement. Examples include:
(X ± δX) (Y ± δY ) Although units are not explicitly written next to these measurements, they are implied. We will use these general expressions for measurements when we discuss the propagation of uncertainties in Section 4.
1.2 Uncertainties in measurements in lab
In the laboratory you will be taking real world measurements, and for some measurements you will record both measured values and uncertainties. Getting values from measuring
for a measurement decreases, the percentage uncertainty δZZ × 100% decreases, and so the measurement deviates less from perfection. For example, a measurement of (2 ± 1) m has a percentage uncertainty of 50%, or one part in two. In contrast, a measurement of (2. 00 ± 0 .01) m has a percentage uncertainty of 0.5% (or 1 part in 200) and is therefore the more precise measurement. If there were some way to make this same measurement with zero uncertainty, the percentage uncertainty would equal 0% and there would be no deviation whatsoever from the measured value—we would have a “perfect” measurement. Unfortunately, this never happens in the real world.
1.4 Implied uncertainties
When you read a physics textbook, you may notice that almost all the measurements stated are missing uncertainties. Does this mean that the author is able to measure things perfectly, without any uncertainty? Not at all! In fact, it is common practice in textbooks not to write uncertainties with measurements, even though they are actually there. In such cases, the uncertainties are implied. We treat these implied uncertainties the same way as we did when taking measurements in lab:
— In a measurement with an implied uncertainty, the actual uncertainty is writ- ten as ± 1 in the smallest place value of the given measured value.
For example, if you read g = 9.80146 m/s^2 in a textbook, you know this measured value has an implied uncertainty of 0.00001 m/s^2. To be more specific, you could then write (g ± δg) = (9. 80146 ± 0 .00001) m/s^2.
1.5 Decimal points — don’t lose them
If a decimal point gets lost, it can have disastrous consequences. One of the most common places where a decimal point gets lost is in front of a number. For example, writing .52 cm sometimes results in a reader missing the decimal point, and reading it as 52 cm — one hundred times larger! After all, a decimal point is only a simple small dot. However, writing 0.52 cm virtually eliminates the problem, and writing leading zeros for decimal numbers is standard scientific and engineering practice.
2 Agreement, Discrepancy, and Difference
In the laboratory, you will not only be taking measurements, but also comparing them. You will compare your experimental measurements (i.e. the ones you find in lab) to some theoretical, predicted, or standard measurements (i.e. the type you calculate or look up in a textbook) as well as to experimental measurements you make during a second (or third...) data run. We need a method to determine how closely these measurements compare.
To simplify this process, we adopt the following notion: two measurements, when com- pared, either agree within experimental uncertainty or they are discrepant (that is, they do
9.790 9.800 9.
m/(s*s)
g exp
g std
a: two values in experimental agreement
9.790 9.800 9.
m/(s*s)
g exp
g std
b: two discrepant values
not agree). Before we illustrate how this classification is carried out, you should first recall that a measurement in the laboratory is not made up of one single value, but a whole range of values. With this in mind, we can say,
Two measurements are in agreement if the two measurements share values in common; that is, their respective uncertainty ranges partially (or totally) overlap.
Figure 3: Agreement and discrepancy of gravity measurements
For example, a laboratory measurement of (gexp ± δgexp) = (9. 801 ± 0 .004) m/s^2 is being compared to a scientific standard value of (gstd ± δgstd) = (9. 8060 ± 0 .0025) m/s^2. As illustrated in Figure 3(a), we see that the ranges of the measurements partially overlap, and so we conclude that the two measurements agree.
Remember that measurements are either in agreement or are discrepant. It then makes sense that,
Two measurements are discrepant if the two measurements do not share values in common; that is, their respective uncertainty ranges do not overlap.
Suppose as an example that a laboratory measurement (gexp ± δgexp) = (9. 796 ± 0 .004) m/s^2 is being compared to the value of (gstd ± δgstd) = (9. 8060 ± 0 .0025) m/s^2. From Figure 3 (b) we notice that the ranges of the measurements do not overlap at all, and so we say these measurements are discrepant.
1 2 3 a: neither accurate nor precise
1 2 3 b: precise, but not accurate
1 2 3 c: both accurate and precise
Precision describes the degree of certainty one has about a measurement. Accuracy describes how well measurements agree with a known, standard mea- surement.
Let’s first examine the concept of precision. Figure 4(a) shows a precise target shooter, since all the shots are close to one another. Because all the shots are clustered about a single point, there is a high degree of certainty in where the shots have gone and so therefore the shots are precise. In Figure 5(b), the measurements on the ruler are all close to one another, and like the target shots, they are precise as well.
Accuracy, on the other hand, describes how well something agrees with a standard. In Figure 4(b), the “standard” is the center of the target. All the shots are close to this center, and so we would say that the targetshooter is accurate. However, the shots are not close to one another, and so they are not precise. Here we see that the terms “precision” and “accuracy” are definitely not interchangeable; one does not imply the other. Nevertheless, it is possible for something to be both accurate and precise. In Figure 5(c), the measurements are accurate, since they are all close to the “standard” measurement of 1.5 cm. In addition, the measurements are precise, because they are all clustered about one another.
Note that it is also possible for a measurement to be neither precise nor accurate. In Figure 5(a), the measurements are neither close to one another (and therefore not precise), nor are they close to the accepted value of 1.5 cm (and hence not accurate).
Figure 5: Examples of precision and accuracy in length measurements. Here the hollow headed arrows indicate the ‘actual’ value of 1.5 cm. The solid arrows represent measurements.
You may have noticed that we have already developed techniques to measure precision and accuracy. In Section 1.3, we compared the uncertainty of a measurement to its measured value to find the percentage uncertainty. The calculation of percentage uncertainty is actually a test to determine how certain you are about a measurement; in other words, how precise the measurement is. In Section 2, we learned how to compare a measurement to a standard
or accepted value by calculating a percent discrepancy. This comparison told you how close your measurement was to this standard measurement, and so finding percent discrepancy is really a test for accuracy.
It turns out that in the laboratory, precision is much easier to achieve than accuracy. Precision can be achieved by careful techniques and handiwork, but accuracy requires ex- cellence in experimental design and measurement analysis. During this laboratory course, you will examine both accuracy and precision in your measurements and suggest methods of improving both.
4 Propagation of uncertainty (worst case)
In the laboratory, we will need to combine measurements using addition, subtraction, mul- tiplication, and division. However, measurements are composed of two parts—a measured value and an uncertainty—and so any algebraic combination must account for both. Perform- ing these operations on the measured values is easily accomplished; handling uncertainties poses the challenge. We make use of the propagation of uncertainty to combine measurements with the assumption that as measurements are combined, uncertainty increases—hence the uncertainty propagates through the calculation. Here we show how to combine two mea- surements and their uncertainties. Often in lab you will have to keep using the propagation formulae over and over, building up more and more uncertainty as you combine three, four or five set of numbers.
- When adding two measurements, the uncertainty in the final measurement is the sum of the uncertainties in the original measurements:
(A ± δA) + (B ± δB) = (A + B) ± (δA + δB) (1)
As an example, let us calculate the combined length (L ± δL) of two tables whose lengths are (L 1 ± δL 1 ) = (3. 04 ± 0 .04) m and (L 2 ± δL 2 ) = (10. 30 ± 0 .01) m. Using this addition rule, we find that
(L ± δL) = (3. 04 ± 0 .04) m + (10. 30 ± 0 .01) m = (13. 34 ± 0 .05) m
- When subtracting two measurements, the uncertainty in the final measurement is again equal to the sum of the uncertainties in the original measurements:
(A ± δA) − (B ± δB) = (A − B) ± (δA + δB) (2)
For example, the difference in length between the two tables mentioned above is
(L 2 ± δL 2 ) − (L 1 ± δL 1 ) = (10. 30 ± 0 .01) m − (3. 04 ± 0 .04) m = [(10. 30 − 3 .04) ± (0.01 + 0.04)] m = (7. 26 ± 0 .05) m
Notice that the final values for uncertainty in the above calculation were determined by multiplying the product (lw) outside the bracket by the sum of the two percentage uncertainties (δl/l + δw/w) inside the bracket. Always remember this crucial step! Also, notice how the final measurement for the area was rounded. This rounding was performed by following the rules of significant figures, which are explained in detail later in Section 5. Recall our discussion of percentage uncertainty in Section 1.3. It is here that we see the benefits of using such a quantity; specifically, we can use it to tell right away which of the two original measurements contributed most to the final area uncertainty. In the above example, we see that the percentage uncertainty of the width measurement (δw/w) × 100% is 5%, which is larger than the percentage uncertainty (δl/l) × 100% ≈ 1 .3% of the length measurement. Hence, the width measurement contributed most to the final area uncertainty, and so if we wanted to improve the precision of our area measurement, we should concentrate on reducing width uncertainty δw (since it would have a greater effect on the total uncertainty) by changing our method for measuring width.
- When dividing two measurements, the uncertainty in the final measurement is found by summing the percentage uncertainties of the original measurements and then multiplying that sum by the quotient of the measured values:
(A ± δA) (B ± δB)
( A
B
) [ 1 ±
( δA A
δB B
)] (5)
As an example, let’s calculate the average speed of a runner who travels a distance of (100. 0 ± 0 .2) m in (9. 85 ± 0 .12) s using the equation v = D/t, where ¯v is the average speed, D is the distance traveled, and t is the time it takes to travel that distance.
¯v =
D ± δD t ± δt
( D
t
) [ 1 ±
( δD D
δt t
)]
( 100 .0 m 9 .85 s
) [ 1 ±
(
- 2
- 0
)]
= 10 .15 [1 ± (0.002000 + 0.01218) ] m/s = 10.15 [1 ± (0.01418) ] m/s = (10. 15 ± 0 .1439) m/s ≈ (10. 2 ± 0 .1) m/s
In this particular example the final uncertainty results mainly from the uncertainty in the measurement of t, which is seen by comparing the percentage uncertainties of the time and distance measurements, (δt/t) ≈ 1 .22% and (δD/D) ≈ 0 .20%, respectively. Therefore, to reduce the uncertainty in (v ± δv), we would want to look first at changing the way t is measured.
- Special cases—inversion and multiplication by a constant:
(a) If you have a quantity X ± δX, you can invert it and apply the original percentage uncertainty:
1 X ± δX
X
) [ 1 ±
δX X
]
(b) To multiply by a constant,
k × (Y ± δY ) = [kY ± kδY ]
It is important to realize that these formulas and techniques allow you to perform the four basic arithmetic operations. You can (and will) combine them by repetition for the sum of three measurements, or the cube of a measurement. Normally it is impossible to use these simple rules for more complicated operations such as a square root or a logarithm, but the trigonometric functions sin θ, cos θ, and tan θ are excep- tions. Because these functions are defined as the ratios between lengths, we can use the quotient rule to evaluate them. For example, in a right triangle with opposite side (x ± δx) and hypotenuse (h ± δh), sin θ = ((xh±±δxδh)). Similarly, any expression that can be broken down into arithmetic steps may be evaluated with these formulas; for example, (x ± δx)^2 = (x ± δx)(x ± δx).
- Finding the uncertainty of a square root
The method for obtaining the square root of a measurement. uses some algebra coupled with the multiplication rule. Let (A ± δA) and (B ± δB) be two measurements. Further, assume that the square root of (A ± δA) is equal to the measurement (B ± δB). Then, √ (A ± δA) = (B ± δB) (6) Squaring both sides, we obtain
(A ± δA) = (B ± δB)^2
Using the multiplication rule on (B ± δB)^2 , we find
(A ± δA) = (B ± δB)^2
= B^2
[ 1 ±
( 2 δB B
)]
= (B^2 ± 2 BδB)
Thus,
(A ± δA) = (B^2 ± 2 BδB) which means B =
A and δB =
δA 2 B
δA 2
A
5 Rounding measurements
The previous sections contain the bulk of what you need to take and analyze measurements in the laboratory. Now it is time to discuss the finer details of measurement analysis. The subtleties we are about to present cause an inordinate amount of confusion in the laboratory. Getting caught up in details is a frustrating experience, and the following guidelines should help alleviate these problems.
An often-asked question is, “How should I round my measurements in the laboratory?” The answer is that you must watch significant figures in calculations and then be sure the number of decimal places of a measured value and its uncertainty agree. Before we give an example, we should explore these two ideas in some detail.
5.1 Treating significant figures
The simplest definition for a significant figure is a digit (0 - 9) that actually represents some quantity. Zeros that are used to locate a decimal point are not considered significant figures. Any measured value, then, has a specific number of significant figures. See Table 1 for examples.
There are two major rules for handling significant figures in calculations. One applies for addition and subtraction, the other for multiplication and division.
- When adding or subtracting quantities, the number of decimal places in the result should equal the smallest number of decimal places of any term in the sum (or difference). Examples: 51. 4 − 1 .67 = 49. 7 7146 − 12 .8 = 7133 20 .8 + 18.72 + 0.851 = 40. 4
- When multiplying or dividing quantities, the number of significant figures in the final answer is the same as the number of significant figures in the least accurate of the quantities being multiplied (or divided). Examples: 2. 6 × 31 .7 = 82 not 82. 42 5. 3 ÷ 748 = 0. 0071 not 0. 007085561
5.2 Measured values and uncertainties: Number of decimal places
As mentioned earlier in Section 1.1, we learned that for any measurement (X ± δX), the number of decimal places of the measured value X must equal those of the corresponding uncertainty δX.
Below are some examples of correctly written measurements. Notice how the number of decimal places of the measured value and its corresponding uncertainty agree.
(L ± δL) = (3. 004 ± 0 .002) m (m ± δm) = (41. 2 ± 0 .4) kg
Measured value Number of significant figures 123 3 1.23 3 1.230 4 0.00123 3 0.001230 4
Table 1: Examples of significant figures
5.3 Rounding
Suppose we are asked to find the area (A ± δA) of a rectangle with length (l ± δl) = (2. 708 ± 0 .005) m and width (w ± δw) = (1. 05 ± 0 .01) m. Before propagating the uncertainties by using the multiplication rule, we should first figure out how many significant figures our final measured value A must have. In this case, A = lw, and since l has four significant figures and w has three significant figures, A is limited to three significant figures. Remember this result; we will come back to it in a few steps.
We may now use the multiplication rule to calculate the area:
(A ± δA) = (l ± δl) × (w ± δw)
= (lw)
[ 1 ±
( δl l
δw w
)]
= (2. 708 × 1 .05)
[ 1 ±
)] m^2
= (2.843) [1 ± (0.001846 + 0.009524)] m^2 = 2 .843 (1 ± 0 .011370) m^2 = (2. 843 ± 0 .03232) m^2
Notice that in the intermediate step directly above, we allowed each number one extra significant figure beyond what we know our final measured value will have; that is, we know the final value will have three significant figures, but we have written each of these intermediate numbers with four significant figures. Carrying the extra significant figure ensures that we will not introduce round-off error.
We are just two steps away from writing our final measurement. Step one is recalling the result we found earlier—that our final measured value must have three significant figures. Thus, we will round 2.843 m^2 to 2.84 m^2. Once this step is accomplished, we round our uncertainty to match the number of decimal places in the measured value. In this case, we round 0.03233 m^2 to 0.03 m^2. Finally, we can write
(A ± δA) = (2. 84 ± 0 .03) m^2
This page is deliberately left blank.
0 1 2 3 cm
A B^ C D
Measurement Analysis Problem Set MA
Name Lab day/time
Division GTA
Write your answers in the space provided. For full credit, show all essential intermediate steps, include units and ensure that measured values and uncertainties agree in the number of decimal places.
- Using the above figure, write the measured values and uncertainties for the following:
(a) A = ( ± ) cm. (b) B = ( ± ) cm. (c) C = ( ± ) cm. (d) D = ( ± ) cm. (e) How did you determine these uncertainties?
- In the laboratory, your partner uses a digital balance to find the mass of a small object. He/she tells you (correctly) that the digital readout shows 43 g.
(a) Write the correct mass measurement and its uncertainty. (M ± δM ) = ( ± ) g. (b) What is the percentage uncertainty of this measurement?
- For a laboratory exercise, you determine the masses of two airtrack gliders as (m 1 ± δm 1 ) = (484. 9 ± 0 .3) g and (m 2 ± δm 2 ) = (314. 2 ± 0 .1) g. Determine the following combinations of the measurements, and show your work.
(a) (m 2 ± δm 2 ) − (m 1 ± δm 1 ) = ( ± ) g.
(b) Determine the precisions of the measurements (m 1 ± δm 1 ) and (m 2 ± δm 2 ). Which calculated precision is the larger? Using this information, determine which is the better measurement.
(c) (m 1 ± δm 1 ) + (m 2 ± δm 2 ) = ( ± ) g.
- (a) Laboratory measurements performed upon a rectangular steel plate show the length (l ± δl) = (2. 41 ± 0 .25) cm and the width (w ± δw) = (8. 30 ± 0 .10) cm. Determine the area of the steel plate and the uncertainty in the area. A ± δA = ( ± ) cm^2.
(b) As Director of the National Science Foundation, you must decide what is the best means to improve the precision of the area measurement of the steel plate. You can spend money on a space gizmotron to better measure length, or on a superconducting whizbang to better measure width, but not both. On which measurement should you spend the money? Justify your decision with numbers.
- During another airtrack experiment, you determine the initial position of a glider to be (xi ± δxi) = (0. 482 ± 0 .001) m and its final position to be (xf ± δxf ) = (0. 633 ± 0 .003) m, with an elapsed time (t ± δt) = (0. 48 ± 0 .01) s during the displacement.
(a) Find the total displacement of the glider (D ± δD) = (xf ± δxf ) − (xi ± δxi). (D ± δD) = ( ± ) m.
(b) Find the average speed of the glider (¯vglider ± δ¯vglider) = (D ± δD)/(t ± δt). ¯vglider ± δ¯vglider = ( ± ) m/s.
(c) Calculate the percentage uncertainties for the two quantities (D±δD) and (t±δt). Based on these precisions, determine which of the two quantities contributes most to the overall uncertainty δ¯vglider.
(d) If we could change the apparatus so as to measure either distance or time ten times more accurately (but not both), which should we change and why?
