1 Measurement Uncertainties

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1 1 Measurement Uncertainties (Adapted stolen, really from work by Amin Jaziri) 1.1 Introduction No measurement can be perfectly certain. No measuring device is infinitely sensitive or infinitely precise. For example, knowledge of length determined with a meter stick or ruler divided into centimeters and millimeters is limited to fractions of a millimeter. Such measurement is an estimate of the position between millimeter lines on the scale, but even this is less than certain because the reading also depends on exactly how you align the measured object with the scale and your vantage point while reading it. Repeated measurements that span the true value (however this is known) are said to be accurate. Those that have very little spread among them are said to be precise. High precision makes it more difficult to be accurate. Precision is related to statistical, or random, uncertainties, and accuracy is related to systematic uncertainties. We will discuss both sorts of uncertainty later in this note. 1.2 Reporting measured values Science, in contrast to almost every other branch of knowledge, attempts to quantify the degree of uncertainty associated with any statement of fact. 1 Stating the uncertainty provides a sense of the range in which the true value of a quantity being measured probably lies. A measurement should always be accompanied by the uncertainty of the measurement, both labeled with appropriate units. Thus, scientific measurements include appropriately precise numerical values, uncertainties of matching precision, and units to label what in fact was measured. Results lacking any of these are useless, because they are uninterpretable. As students in this course, you are required to provide all of them every time you state a result. Note that an uncertainty is not a mistake, which is the result of inattention or other carelessness on the part of the experimenter, but rather an inadvertent and inevitable part of the measurement process. Mistakes (sometimes referred to as experimenter error, for that is what they are) are correctable in real time and therefore inexcusable: it is not an acceptable explanation of results. You must be careful, pay attention, and check your results as you go along. It is a universal convention in scientific work to report a numerical result with the number of significant figures, digits, or decimal places up to, and including, but no more than, the first uncertain one. This immediately tells the reader the approximate level of uncertainty. To return to our example: in reading a meter stick or ruler on which the smallest scale divisions are millimeters, measurements (in cm) would be reported 1 Another name for uncertainty is error, and the two terms are often used interchangeably. This instructor prefers the former to the latter. 1

2 as, say, cm or 5.30 cm or cm, etc. If the reading is exactly seven centimeters, it is reported as 7.00 cm and not as 7 cm or cm. If the edges of an object were particularly uneven such that trying to read the ruler to a tenth of a mm is hopeless, and the best estimate can be made only to the nearest whole mm, a measured value would be reported as 48.2 cm or 1.3 cm, etc. This indicates an uncertainty at the level of tenths of a cm (whole mm s) rather than at the hundredth of a cm level as the scale itself might indicate. Thus, the scale s precision is only one factor determining the precision of a measurement. What to report is a judgment call, and one of the purposes of this course is to help you develop the capacity to judge. This appropriate use of significant figures tells only order of magnitude of the uncertainty (to the tenths, hundredths, or whatever). It indicates that the next-to-last figure is the one in which we can have considerable confidence, while the last one is uncertain, although it represents the best estimate we can make. The size of the uncertainty is not conveyed by this convention, but must be estimated and reported separately. 1.3 Precision of significant figures When reporting a measurement, you express its relative precision in terms of the number of digits or significant figures, in the sense that the fractional uncertainty in the last figure becomes smaller as the number of significant figures increases. The place of the least significant digit gives the absolute precision. If, for example, the length of a cylinder is reported to be cm, the claim is that the length of the cylinder is known to the level of tenths of a millimeter; this is the absolute precision of the measurement. In this case, because there are four significant figures, the relative precision is therefore a few parts in a couple of thousand. Even if you wrote this number in terms of kilometers as km, it still has the same precision and number of significant figures. The zeros preceding the 2 are used only to indicate the position of the decimal point. The zero between the 2 and 6 is a significant figure, but the other zeros are not. If you reported a length as 0.78 cm, you still claim to know it to within tenths of a millimeter but to just two significant figures. This measurement has been made to the same absolute level of precision, but the relative precision is less. Clearly, it takes better precision to know a larger measurement to a certain level of uncertainty than to know a smaller measurement to the same level: while the fractional uncertainty in the cm measurement is a few parts in a couple of thousand, the fractional uncertainty in 0.78 is no better than a few parts in less than a hundred. Writing numbers in scientific notation helps remove some of the ambiguity of zeros while emphasizing the relative precision of a number. For example, cm can be rewritten as cm and km can be written as km. Now, in both cases, it s easy to see that there are four figures considered significant. Similarly, 0.78 cm can be written as cm, and it becomes immediately obvious that is has two significant figures, and is relatively less precise than the former. 2

3 It s usually a simple matter to determine the greatest possible precision that can be recorded for a measurement (if, for example, your meter stick is ruled in millimeter divisions, then your precision is at best a fraction of a millimeter), but you must take into consideration all aspects involved in the making of the measurement to determine a realistic level of precision. Furthermore, additional difficulties may arise when measurements are used in calculations, which can produce a large number of figures that might seem significant but really aren t. Calculators, in particular, often prove a bane to understanding, because they produce all kinds of figures which, unfortunately, tend to be written down without consideration. Calculators don t cause errors (assuming all the numbers have been correctly entered), but mindlessly recording all the figures of the result gives a physically incorrect answer. No mathematical computation may produce a result whose absolute precision is greater than that of the quantities used; the result can be no more precise than the least precise quantity that went into the calculation. As a general rule, though, it is better when computing to carry too many figures than not enough, and then to round-off to the appropriate precision later. This is more or less straight forward when adding or subtracting, since all the numbers involved have to be the same sort of quantities, that is they have to have the same dimensions, such as length, time, mass, or volume. Simply enter the experimental values (being sure that they are all in the same units) to the absolute precision known and then round to that of the least precise quantity. For example, if you combined four masses, g, g, g, and g, you would be adding four numbers each with four significant figures. But plugging these values into your calculator leads to something that has eight figures. The worst absolute precision is a tenth of a gram, so you must round to this, giving a total of g. In this case, the worst relative precision is around a part in a thousand, and simple rounding produced an answer that roughly matches both senses of precision. Multiplication and division are not so obvious because they may involve unrelated quantities. The least precise result still determines the precision of the final result, but it s not so clear what the absolute precision of the final result is, because its units may be different from those entering into the computation. Suppose your measurements for the sides of a rectangle were 38.2 cm and 21.4 cm, respectively. By the convention, the 2 in the first number and the 4 in the second number are the uncertain figures. If we want to determine the area of the rectangle, we multiply the two measurements together. A calculator would give the value of and the units would be cm 2. But the 7, 4, and final 8 of the result each involves an uncertain number (for example, the 8 results from the multiplication of the uncertain 2 of the first side by the uncertain 4 of the second). Any result in which an uncertain number is involved must itself be uncertain. Here, since the 7 is uncertain, any figure following it must be completely meaningless. If you reported cm 2 as the area of the rectangle, you would be misleading the reader by implying that the final 8 was the first uncertain figure and all the others were certain. This would be untrue. The truth of the situation would be conveyed by reporting the result as 817 cm 2 and 3

4 dropping the other figures. In this way, the relative precision of the outcome roughly matches that of the least (relatively) precise factor. Yet, even this is inadequate. We know by looking that the last digit is uncertain, but unless told explicitly the magnitude of this uncertainty, we are still in the dark about the meaning of the result. We discuss ways to clarify this question in what follows. 1.4 Statistical and Systematic Uncertainties Repeated measurements, even of the same quantity, tend to vary. Variations that distribute indeterminately both in magnitude and sign about a central value and thus average to zero are known as statistical or random uncertainties. A basic tenant of probability and statistics is that if only random uncertainties arise, then the average of more measurements is a better estimate of the actual value than the average of fewer measurements. The quantity given of statistical uncertainty should indicate the range of results that would be obtained from multiple measurements. Typically, this quantity characterizes the measuring device(s) and process. The most common method for estimating this quantity is simply to repeat the measurement many times. If measurements are all shifted in both magnitude and direction from the true value, the uncertainty is called systematic. Repeating measurements, without identifying and rectifying the cause of the shift, will not improve knowledge of an actual value. The difficulty, of course, is that the true value is usually unknown (why else measure it?), and so this uncertainty is difficult to quantify. The most common methods for doing so are to recheck the calibration of the measuring device after the measurement and to alter slightly and in carefully controlled ways aspects of the measuring process. Almost all measurements suffer from both random and systematic uncertainties. When these can be estimated independently, they should be quoted independently. 1.5 Uncertainty in Individual Measurements An irreducible source of uncertainty is the precision or resolution of the measuring instrument whose values are typically indicated on a dial or scale. The coarseness of the divisions on an indicator limits the absolute precision with which a value can be determined. One ends up guessing to a fraction of a division, and the variation of guesses is typically random. This, then, is a source of statistical uncertainty. Common practice assigns an uncertainty due to the instrument of half the smallest division on the indicator. But this is probably just one component of the total uncertainty. Consider the case of a digital stop watch, which gives readings to 0.01 second. The inherent uncertainty of the watch would ordinarily be estimated to be second, but human reaction time associated with starting and stopping the watch is roughly 0.05 second, also a statistical effect. Thus, the uncertainty associated with a stop-watch measurement must be at least 0.05 second, rather than second as the scale 4

5 resolution might suggest. If the watch were fast or slow compared to a standard clock, then a systematic uncertainty would have to be cited, as well. Careful consideration, you should see, is often necessary to identify the dominant sources of uncertainty in a measurement. 1.6 The RMS Deviation Perhaps the most common way of specifying statistical uncertainty is with the quantity called the RMS deviation, where RMS stands for Root Mean Squared. In words, this quantity measures a kind of average discrepancy around the mean, or average, of values. Given a set of N values of x i, the average, or mean, value x is defined as x 1 N N x i. (1) i=1 The RMS deviation of this distribution, denoted σ, is then defined as σ 1 N (x i x) N 2. (2) i=1 If the variation of values happens to be normally distributed (like a so-called bell curve) about the mean, then sigma is called the standard deviation. If N is small (< 30), then this quantity actually gives too small an estimation of the uncertainty and the factor 1/(N 1) is used instead of 1/N in Equation 2 and the symbol typically used is s rather than σ. s 1 N 1 N (x i x) 2. (3) You will show in your homework that an expression for σ equivalent equivalent to Equation 2 is given by: σ = = i=1 x 2 x 2 (4) ( ) 2 1 N x 2 i N 1 N x i. N i=1 As an example, consider the following two sets of measurements taken with two different devices (arbitrary units): Set 1: Set 2: i=1 5

6 The average for both sets is the same (3.00 in arbitrary units). The RMS deviation for the first set is 0.65 (0.73) in arbitrary units, while the RMS deviation for the second is 0.12 (0.14) in arbitrary units [the values in parenthesis, more appropriate in this case, use Equation 3 instead of Equation 2]. The second device is said to have greater precision than the first. If the number of measurements N is increased, the values of x and σ tend toward limiting values which are independent of N. The limiting value of x is presumably that of the physical quantity being measured. The values of the uncertainties depend on the measurement technique employed. The implication of this is that the more measurements we make, the more confident we become of the central value and the uncertainty of an individual measurement: after many measurements, neither the mean nor the RMS will change substantially even if many more measurements are made. Thus, as the number of measurements increases, confidence in the value of the central value increases regardless of the size of the uncertainty of an individual measurements. We indicate the level of confidence by reporting not RMS as the statistical uncertainty of the result, but rather a quantity called the RMS of the mean, σ x. You will prove in a homework assignment is that this is given by σ x = σ x N, (5) where σ x is the RMS deviation of the individual measurementsx, and N is the number of measurement. This uncertainty, σ x is the statistical uncertainty that should be reported when the central value is an average: (x ± σ x ± systematic uncertainty) in some units. (assuming there is no systematic uncertainty: Note that the standard way to report results is: central value ± statistical uncertainty ± systematic uncertainty. It is essential that the number of decimal places for the central value and for the uncertainty is the same. This must always be the case. One cannot know the uncertainty to more decimal places (absolute precision) than the central value or vice versa. If the statistical uncertainties are Gaussian, or normally distributed, as they often are, the RMS deviation, or standard deviation, has a very specific, probabilistic interpretation. Given a mean of x and a standard deviation σ x (standard deviation of the mean σ x ), the probability that an individual measurement (another determination of the mean) will differ from the mean decreases as the magnitude of this difference increases. This probability can be related to increments of standard deviations [see Table 1]. It is not statistically likely for a measured value to differ from a true value by more than 3 4σ, but not unusual for one to differ by 1 2σ. 6

7 Table 1: Probability that a value x i differs by n standard deviations from the mean x. n, Number Probability Probability of standard x i x > nσ x i x < nσ deviations (%) (%) You see in this the importance of knowing the uncertainty of a result: agreement or disagreement depends not on the absolute difference between values, but on the difference as a function of the uncertainty. Consider the results of two investigations of a certain quantity which a theory, for example, predicts [see Figure 1]. First of all, note that experiments A and B disagree with each other as to the central value (as indicated by a dot) of their results, with B s result closer to the predicted value. But since the uncertainties of the experiments (as indicated by the error bars) overlap, this disagreement is not considered significant. Experiment A, however, deviates from the prediction by less than 2σ, and therefore cannot be said to disagree significantly with the prediction, while experiment B deviates by more than 3.5σ from the prediction, which is a significant disagreement. So, even though B is closer to the prediction than A, B is said to disagree with the prediction, while A does not. Again, what matters is not the absolute difference, but the difference in units of uncertainty. 1.7 Error Propagation Frequently, the goal of an experiment requires arithmetically combining the results of different sorts of measurements. For example, determining the average velocity may require measuring the displacement and the associated time interval with different instruments and then dividing the former by the latter. We may determine the statistical uncertainty of the displacement and the change of time separately, but what then is the uncertainty of the resulting velocity? To get this, we perform what is known as error propagation. Let us say that the final result, z, depends on two, independent sets of measurements, x and y, according to some functional relationship f: z = f(x, y). (6) Knowing the functional relationship f as well as the uncertainties of x and y, σ x and σ y, respectively, we determine the uncertainty of z, σ z, by 7

8 Experimental values A B Expected value Data Points Figure 1: Data points from experiments A and B. The dots indicate central values and the lines indicate 1σ uncertainties. σ z = ( f x ) 2 σ 2 x + ( ) 2 f σ y y, 2 (7) where f x is the partial derivative of f with respect to x (that is, as if x is the only variable in f, all other terms in the function are treated as constants). And similarly for y. This expression can be extended to any number of independent measurements included in the relationship. Suppose that z = xy 2. Then, z x = y2 and z y = 2xy, so (σx ) ( ) 2 2 σ z = y 4 σx 2 + 4x 2 y 2 σy 2 = xy 2 2σy +. x y Be sure to check that you can get this answer. You will need to derive uncertainties for other functional relationships in homework, labs, and final examination. 1.8 Graphing and Linear Regression Graphs are particularly important in physics because they display clearly the relationship between interdependent quantities. If two quantities, x and y have a linear relationship, y = ax + b, a graph of y against x, or y vs. x, will be a straight line with slope a and y-intercept b, where the y- intercept is the value of y at x = 0. x is identified with the independent variable and y is identified with the dependent variable. 8

9 A theory which predicts that a certain physical quantity y depends linearly on another physical quantity x can be tested experimentally by measuring corresponding values of x and y and plotting these results. The plot will readily show whether a straight line can be drawn through them, even if random errors cause them to scatter instead of all to lie right on the line. To find the correct line amidst the scattering requires the technique of best fit or regression. You will use a computer and the Excel program to do nearly all plotting and fitting. Here, we ll simply provide you a sense of what s going on so that the computer procedure isn t a completely black box. Given a set of pairs of values, y and x, the slope of the best linear fit is found by a = N( xy) ( x)( y) N x 2 ( x) 2. (8) The intercept is computed, after determining y and x, by b = y ax (9) The discrepancy between the actual points and the line constitutes an uncertainty, the magnitude of which is computed much like the RMS, essentially for the same reasons, and is called the standard error of estimate: s r = = (y ŷ) 2 (10) N 2 [ ] { 1 N y N(N 2) 2 ( y) 2 [N xy ( x)( } y)] 2 N x 2 ( x) 2, where ŷ is the predicted value of y given a value of x according to the best-fit line. If this measure of spread around the line becomes too large, that is, if the line seems to be arbitrary in relation to the points, then it s difficult to conclude that y depends linearly on x. If this error is small, then the linearity of the relationship is indicated and predictions can readily be made. Actually, what you ll get from Excel are uncertainties on the slope and intercept. If the uncertainty on the former is large, then it is unlikely that a straight line describes the relationship well. A linear relationship is obvious when the points are plotted. Other relationships may not be so clear. Suppose the relationship is expected to be y = ax 2 +b. Plotting y vs x might result in a curve, but one that is very difficult to distinguish from higher order curves, say, y = ax 3 +b. The straight line is the only graph that is visually really obvious. There are fitting procedures, incorporated in the plotting and fitting package you will be using in this course, that allow you to make this distinction. However, Excel only supplies uncertainties for linear fits, and it is essential that uncertainties are reported along with any result. The solution is linearization. 9

10 To linearize the example equations, you could make the substitution u = x 2 and try the fit y = au + b, and similarly for u = x 3 and see which has the smaller standard error. A special case of this procedure arises when y depends exponentially on x: y = Ae ax, where e is the base of the natural logarithm, approximately equal to 2.718, and A and a are constants. Again, your computer program can fit this, but it won t supply the uncertainty. The first step to linearizationin this case is to take the natural logarithm of both sides of the prediction: ln y = ln Ae ax = ln A + ln e ax = ln A + ax, which is simply a linear equation except that ln y rather than y is plotted versus x. This plot would be linear if the exponential relationship were correct: the slope would give the value of a and the intercept would give the value of A through ln A. One can do the same thing with a power relationship y = ax n. Taking the natural logarithm of both sides, we get ln y = ln a + n ln x, and a plot of ln y versus ln x will produce a straight line with slope n. Once you ve managed to manipulate the data to get a straight line, the Excel Regression function determines the slope and intercept as well as their respective uncertainties. 10