Error Analysis (NonCalculus)These notes are designed to supplement the treatments in any freshman physics laboratory manual. The level of presentation does not use calculus, and is suitable for freshman. A. UNCERTAINTIES (ERRORS) OF MEASUREMENTConsistent with current practice, the term "error" is used here as a synonym for "experimental uncertainty."No measurement is perfectly accurate or exact. Many instrumental, physical and human limitations cause measurements to deviate from the "true" values of the quantities being measured. These deviations are called "experimental uncertainties," but more commonly the shorter word "error" is used. What is the "true value" of a measured quantity? We can think of it as the value we'd measure if we somehow eliminated all error from instruments and procedure. This is a natural enough concept, and a useful one, even though at this point in the discussion it may sound like circular logic. We can improve the measurement process, of course, but since we can never eliminate measurement errors entirely, we can never hope to measure true values. We have only introduced the concept of true value for purposes of discussion. When we specify the "error" in a quantity or result, we are giving an estimate of how much that measurement is likely to deviate from the true value of the quantity. This estimate is far more than a guess, for it is founded on a physical analysis of the measurement process and a mathematical analysis of the equations which apply to the instruments and to the physical process being studied. A measurement or experimental result is of little use if nothing is known about the probable size of its error. We know nothing about the reliability of a result unless we can estimate the probable sizes of the errors and uncertainties in the data which were used to obtain that result. That is why it is important for students to learn how to determine quantitative estimates of the nature and size of experimental errors and to predict how these errors affect the reliability of the final result. Entire books have been written on this subject.[1] The following discussion is designed to make the student aware of some common types of errors and some simple ways to quantify them and analyze how they affect results.
A warning: Some introductory laboratory manuals still use oldfashioned terminology, defining experimental error as a comparison of the experimental result with a standard or textbook value, treating the textbook value as if it were a true value. This is misleading, and is not consistent with current practice in the scientific literature. This sort of comparison with standard values should be called an experimental discrepancy to avoid confusion with measures of error (uncertainty). The only case I can think of where this measure is marginally appropriate as a measure of error is the case where the standard value is very much more accurate than the experimental value. B: DETERMINATE AND INDETERMINATE ERRORSExperimental errors are of two types: (1) indeterminate and (2) determinate (or systematic) errors.1. Indeterminate Errors.[2] Indeterminate errors are present in all experimental measurements. The name "indeterminate" indicates that there's no way to determine the size or sign of the error in any individual measurement. Indeterminate errors cause a measuring process to give different values when that measurement is repeated many times (assuming all other conditions are held constant to the best of the experimenter's ability). Indeterminate errors can have many causes, including operator errors or biases, fluctuating experimental conditions, varying environmental conditions and inherent variability of measuring instruments. The effect that indeterminate errors have on results can be somewhat reduced by taking repeated measurements then calculating their average. The average is generally considered to be a "better" representation of the "true value" than any single measurement, because errors of positive and negative sign tend to compensate each other in the averaging process. 2. Determinate (or Systematic) Errors. The terms determinate error and systematic error are synonyms. "Systematic" means that when the measurement of a quantity is repeated several times, the error has the same size and algebraic sign for every measurement. "Determinate" means that the size and sign of the errors are determinable (if the determinate error is recognized and identified). A common cause of determinate error is instrumental or procedural bias. For example: a miscalibrated scale or instrument, a colorblind observer matching colors. Another cause is an outright experimental blunder. Examples: using an incorrect value of a constant in the equations, using the wrong units, reading a scale incorrectly. Every effort should be made to minimize the possibility of these errors, by careful calibration of the apparatus and by use of the best possible measurement techniques. Determinate errors can be more serious than indeterminate errors for three reasons. (1) There is no sure method for discovering and identifying them just by looking at the experimental data. (2) Their effects can not be reduced by averaging repeated measurements. (3) A determinate error has the same size and sign for each measurement in a set of repeated measurements, so there is no opportunity for positive and negative errors to offset each other.
C. PRECISION AND ACCURACYA measurement with relatively small indeterminate error is said to have high precision. A measurement with small indeterminate error and small determinate error is said to have high accuracy. Precision does not necessarily imply accuracy. A precise measurement may be inaccurate if it has a determinate error.
D. STANDARD WAYS FOR COMPARING QUANTITIES1. Deviation.When a set of measurements is made of a physical quantity, it is useful to express the difference between each measurement and the average (mean) of the entire set. This is called the deviation of the measurement from the mean. Use the word deviation when an individual measurement of a set is being compared with a quantity which is representative of the entire set. Deviations can be expressed as absolute amounts, or as percents. 2. Difference. There are situations where we need to compare measurements or results which are assumed to be about equally reliable, that is, to express the absolute or percent difference between the two. For example, you might want to compare two independent determinations of a quantity, or to compare an experimental result with one obtained independently by someone else, or by another procedure. To state the difference between two things implies no judgment about which is more reliable. 3. Experimental discrepancy. When a measurement or result is compared with another which is assumed or known to be more reliable, we call the difference between the two the experimental discrepancy. Discrepancies may be expressed as absolute discrepancies or as percent discrepancies. It is customary to calculate the percent by dividing the discrepancy by the more reliable quantity (then, of course, multiplying by 100). However, if the discrepancy is only a few percent, it makes no practical difference which of the two is in the denominator.
E. MEASURES OF ERRORThe experimental error [uncertainty] can be expressed in several standard ways:1. Limits of error Error limits may be expressed in the form Q ± ΔQ where Q is the measured quantity and ΔQ is the magnitude of its limit of error.[3] This expresses the experimenter's judgment that the "true" value of Q lies between Q  ΔQ and Q + ΔQ This entire interval within which the measurement lies is called the range of error. Manufacturer's performance guarantees for laboratory instruments are often expressed this way. 2. Average deviation[4] This measure of error is calculated in this manner: First calculate the mean (average) of a set of successive measurements of a quantity, Q. Then find the magnitude of the deviations of each measurement from the mean. Average these magnitudes of deviations to obtain a number called the average deviation of the data set. It is a measure of the dispersion (spread) of the measurements with respect to the mean value of Q, that is, of how far a typical measurement is likely to deviate from the mean.[5] But this is not quite what is needed to express the quality of the mean itself. We want an estimate of how far the mean value of Q is likely to deviate from the "true" value of Q. The appropriate statistical estimate of this is called the average deviation of the mean. To find this rigorously would involve us in the theory of probability and statistics. We will state the result without proof.[6] For a set of n measurements Q_{i} whose mean value is <Q>, [7] the average deviation of the mean (A.D.M.) is:
The vertical bars enclosing an expression mean "take the absolute value" of that expression. That means that if the expression is negative, make it positive. If the A.D.M. is quoted as the error measure of a mean, <Q>_{exp}, this is equivalent to saying that the probability of <Q>_{exp} lying within one A.D.M. of the "true" value of Q, Q_{true}, is 58%, and the odds against it lying outside of one A.D.M. are 1.4 to 1. As a rough rule of thumb, the probability of <Q>_{exp} being within three A.D.M. (on either side) of the true value is nearly 100% (actually 98%). This is a useful relation for converting (or comparing) A.D.M. to limits of error.[8] 3. Standard Deviation of the mean. [This section is included for completeness, and may be skipped or skimmed unless your instructor specifically assigns it.] The standard deviation is a well known, widely used, and statistically wellfounded measure of error. For a set of n measurements Q_{i} whose mean value is <Q>, the standard deviation of the mean is found from:
The sum is from i = 1 to n. This form of the equation is not very convenient for calculations. By expanding the summand it may be recast into a form which lends itself to efficient computation with an electronic calculator:
[Note that the n<Q>^{2} is a separate term in the numerator, it is not summed over.] The calculation of the standard deviation requires two summations, one a sum of the data values (to obtain <Q>), and one a sum of the squares of the data values. Many electronic calculators allow these two sums to be obtained with only one entry of each data value. This is a good feature to have in a scientific calculator. When n is large, the quantity n(n1) becomes approximately n^{2}, further simplifying the work. The use of the standard deviation is hardly justified unless the experimenter has taken a large number of repeated measurements of each experimentally determined quantity. This is seldom the case in the freshman laboratory. It can be shown that when the measurements are distributed according the "normal" ("Gaussian")[11] distribution, average deviations and standard deviations are related by a simple formula:[12]
[average deviation] = 0.80 [standard deviation] This is a useful "rule of thumb" when it is necessary to compare the two measures of error or convert from one to the other.
F. STANDARD METHODS FOR EXPRESSING ERROR1. Absolute Error.Uncertainties may be expressed as absolute measures, giving the size of the a quantity's uncertainty in the same units in the quantity itself. Example. A piece of metal is weighed a number of times, and the average value obtained is: M = 34.6 gm. By analysis of the scatter of the measurements, the uncertainty is determined to be m = 0.07 gm. This absolute uncertainty may be included with the measurement in this manner: M = 34.6 ± 0.07 gm. The value 0.07 after the ± sign in this example is the estimated absolute error in the value 3.86. 2. Relative (or Fractional) Error. Uncertainties may be expressed as relative measures, giving the ratio of the quantity's uncertainty to the quantity itself. In general:
Example. In the previous example, the uncertainty in M = 34.6 gm was m = 0.07 gm. The relative uncertainty is therefore:
It is a matter of taste whether one chooses to express relative errors "as is" (as fractions), or as percents. I prefer to work with them as fractions in calculations, avoiding the necessity for continually multiplying by 100. Why do unnecessary work? But when expressing final results, it is often meaningful to express the relative uncertainty as a percent. That's easily done, just multiply the relative uncertainty by 100. This one is 0.2%. 3. Absolute or relative form; which to use. Common sense and good judgment must be used in choosing which form to use to represent the error when stating a result. Consider a temperature measurement with a thermometer known to be reliable to ± 0.5 degree Celsius. Would it make sense to say that this causes a 0.5% error in measuring the boiling point of water (100 degrees) but a whopping 10% error in the measurement of cold water at a temperature of 5 degrees? Of course not! [And what if the temperatures were expressed in degrees Kelvin? That would seem to reduce the percent errors to insignificance!] Errors and discrepancies expressed as percents are meaningless for some types of measurements. Sometimes this is due to the nature of the measuring instrument, sometimes to the nature of the measured quantity itself, or the way it is defined. There are cases where absolute errors are inappropriate and therefore the errors should be expressed in relative form. There are also cases where the reverse is true. Sometimes both absolute and relative error measures are necessary to completely characterize a measuring instrument's error. For example, if a plastic meter stick uniformly expanded, the effect could be expressed as a percent determinate error. If a one half millimeter were worn off the zero end of a stick, and this were not noticed or compensated for, this would best be expressed as an absolute determinate error. Clearly both errors might be present in a particular meter stick. The manufacturer of a voltmeter (or other electrical meter) usually gives its guaranteed limits of error as a constant determinate error plus a `percent' error. Both relative and fractional forms of error may appear in the intermediate algebraic steps when deriving error equations. [This is discussed in section H below.] This is merely a computational artifact, and has no bearing on the question of which form is meaningful for communicating the size and nature of the error in data and results.
G. IMPORTANCE OF REPEATED MEASUREMENTSA single measurement of a quantity is not sufficient to convey any information about the quality of the measurement. You may need to take repeated measurements to find out how consistent the measurements are.If you have previously made this type of measurement, with the same instrument, and have determined the uncertainty of that particular measuring instrument and process, you may appeal to your experience to estimate the uncertainty. In some cases you may know, from past experience, that the measurement is scale limited, that is, that its uncertainty is smaller than the smallest increment you can read on the instrument scale. Such a measurement will give the same value exactly for repeated measurements of the same quantity. If you know (from direct experience) that the measurement is scale limited, then quote its uncertainty as the smallest increment you can read on the scale. Students in this course don't need to become experts in the fine details of statistical theory. But they should be constantly aware of the experimental errors and do whatever is necessary to find out how much they affect results. Care should be taken to minimize errors. The sizes of experimental errors in both data and results should be determined, whenever possible, and quantified by expressing them as average deviations. [In some cases commonsense experimental investigation can provide information about errors without the use of involved mathematics.] The student should realize that the full story about experimental errors has not been given here, but will be revealed in later courses and more advanced laboratory work. H. PROPAGATION OF DETERMINATE ERRORSThe importance of estimating data errors is due to the fact that data errors propagate through the calculations to produce errors in results. It is the size of a data errors' effect on the results which is most important. Every effort should be made to determine reasonable error estimates for every important experimental result.We illustrate how errors propagate by first discussing how to find the amount of error in results by considering how data errors propagate through simple mathematical operations. We first consider the case of determinate errors: those that have known sign. In this way we will discover certain useful rules for error propagation, then we'll then be able to modify the rules to apply to other error measures and also to indeterminate errors. We are here developing the mathematical rules for "finite differences," the algebra of numbers which have relatively small variations imposed upon them. The finite differences are those variations from "true values" caused by experimental errors. Suppose that an experimental result is calculated from the sum of two data quantities A and B. For this discussion we'll use a and b to represent the errors in A and B respectively. The data quantities are written to explicitly show the errors:
(A + a) and (B + b)We allow that a and b may be either positive or negative, the signs being "in" the symbols "a" and "b." But we must emphasize that we are here considering the case where the signs of a and b are determinable, and we know what those signs are (positive, or negative). The result of adding A and B to get R is expressed by the equation: R = A + B. With the errors explicitly included, this is written:
(A + a) + (B + b) = (A + B) + (a + b)The result with its error, r, explicitly shown, is: (R + r):
(R + r) = (A + B) + (a + b)The error in R is therefore: r = a + b. We conclude that the determinate error in the sum of two quantities is just the sum of the errors in those quantities. You can easily work out for yourself the case where the result is calculated from the difference of two quantities. In that case the determinate error in the result will be the difference in the errors. Summarizing:
(R + r) = (A + a)(B + b) = AB + aB + Ab + ab or: r = aB + Ab + abThis doesn't look promising for recasting as a simple rule. However, when we express the errors in relative form, things look better. If the error a is small relative to A, and b is small relative to B, then (ab) is certainly small relative to AB, as well as small compared to (aB) and (Ab). Therefore we neglect the term (ab) (throw it out), since we are interested only in error estimates to one or two significant figures. Now we express the relative error in R as
This gives us a very simple rule:
The approximation made in the next to last step was to neglect b in the denominator, which is valid if the relative errors are small. So the result is:
A consequence of the product rule is this: One illustrative practical use of determinate errors is the case of correcting a result when you discover, after completing lengthy measurements and calculations, that there was a determinate error in one or more of the measurements. Perhaps a scale or meter had been miscalibrated. You discover this, and fine the size and sign of the error in that measuring tool. Rather than repeat all the measurements, you may construct the determinateerror equation and use your knowledge of the miscalibration error to correct the result. As you will see in the following sections, you will usually have to construct the error equation anyway, so why not use it to correct for the discovered error, rather than repeating all the calculations?
I. PROPAGATION OF INDETERMINATE ERRORSIndeterminate errors have unknown sign. If their distribution is symmetric about the mean, then they are unbiased with respect to sign. Also, if indeterminate errors in different quantities are independent of each other, their signs have a tendency offset each other in computations.[11]When we are only concerned with limits of error (or maximum error) we must assume a "worstcase" combination of signs. In the case of subtraction, A  B, the worstcase deviation of the answer occurs when the errors are either +a and b or a and +b. In either case, the maximum error will be (a + b). In the case of the quotient, A/B, the worstcase deviation of the answer occurs when the errors have opposite sign, either +a and b or a and +b. In either case, the maximum size of the relative error will be (a/A + b/B). The results for the operations of addition and multiplication are the same as before. In summary, maximum indeterminate errors propagate according to the following rules:
A consequence of the product rule is this:
These rules apply only when combining independent errors, that is, individual errors which are not dependent on each other in size or sign. It can be shown (but not here) that these rules also apply sufficiently well to errors expressed as average deviations. The one drawback to this is that the error estimates made this way are still overconservative in that they do not fully account for the tendency of error terms associated with independent errors to offset each other. This, however, would be a minor correction of little importance in our work in this course. Error propagation rules may be derived for other mathematical operations as needed. For example, the rules for errors in trig functions may be derived by use of trig identities, using the approximations: sin ß = ß and cos ß = 1, valid when ß is small. Rules for exponentials may be derived also. When mathematical operations are combined, the rules may be successively applied to each operation, and an equation may be algebraically derived[12] which expresses the error in the result in terms of errors in the data. Such an equation can always be cast into standard form in which each error source appears in only one term. Let x represent the error in x, y the error in y, etc. Then the error r in any result R, calculated by any combination of mathematical operations from data values X, Y, Z, etc. is given by:
r = (c_{x})x + (c_{y})y + (c_{z})z ... etc.This may always be algebraically rearranged to:
r/R = {C_{x}}(x/X + {C_{y}}(y/Y) + {C_{z}}(z/Z) ... etc.The coefficients (c_{x}) and {C_{x}} etc. in each term are extremely important because they, along with the sizes of the errors, determine how much each error affects the result. The relative size of the terms of this equation shows us the relative importance of the error sources. It's not the relative size of the errors (x, y, etc), but the relative size of the error terms which tells us their relative importance. If this error equation was derived from the determinateerror rules, the relative errors in the above might have + or  signs. The coefficients may also have + or  signs, so the terms themselves may have + or  signs. It is therefore possible for terms to offset each other. If this error equation was derived from the indeterminate error rules, the error measures appearing in it are inherently positive. The coefficients will turn out to be positive also, so terms cannot offset each other. It is convenient to know that the indeterminate error equation may be obtained directly from the determinateerror equation by simply choosing the worstcase, i.e., by taking the absolute value of every term. This forces all terms to be positive. This step is only done after the determinateerror equation has been fully derived in standard form. The error equation in standard form is one of the most useful tools for experimental design and analysis. It should be derived (in algebraic form) even before the experiment is begun, as a guide to experimental strategy. It can show which error sources dominate, and which are negligible, thereby saving time one might spend fussing with unimportant considerations. It can suggest how the effects of error sources might be minimized by appropriate choice of the sizes of variables. It can tell you how good a measuring instrument you need to achieve a desired accuracy in the results. The student who neglects to derive and use this equation may spend an entire lab period using instruments, strategy, or values insufficient to the requirements of the experiment. And he may end up without the slightest idea why the results were not as good as they ought to have been. A final comment for those who wish to use standard deviations as indeterminate error measures: Since the standard deviation is obtained from the average of squared deviations, equation (7) must be modified—each term of the equation (both sides) must be squared:
(r/R) = (C_{x})^{2}(x/X) + (C_{y})^{2}(y/Y) + (C_{z})^{2}(z/Z)This rule is given here without proof.
J. EXAMPLESExample 1: A student finds the constant acceleration of a slowly moving object with a stopwatch. The equation used is s = (1/2)at^{2}. The time is measured with a stopwatch, the distance, s, with a meter stick.s = 2 ± 0.005 meter. This is 0.25%. t = 4.2 ± 0.2 second. This is 4.8%.What is the acceleration and its estimated error? We'll use capital letters for measured quantities, lower case for their errors. Solve the equation for the result, a. A = 2S/T^{2}. Its indeterminateerror equation is:
The factor of 2 in the time term causes that term to dominate, for application of the rule for errors in quantities raised to a power causes the 4.8% error in the time to be doubled, giving over 9.5% error in T^{2}. The 1/4 percent error due to the distance measurement is clearly negligible compared to the 9.5% error due to the time measurement, so the result (the acceleration) is written: A = 0.23 ± 0.02 m/s^{2}. Example 2: A result is calculated from the equation R = (G+H)/Z, the data values being:
G = 20 ± 0.5 The ± symbol tells us that these errors are indeterminate. The calculation of R requires both addition and division, and gives the value R = 3.40. The error calculation requires both the addition and multiplication rule, applied in succession, in the same order as the operations performed in calculating R itself. The addition rule says that the absolute errors in G and H add, so the error in the numerator is 1.0/36 = 0.28. The division rule requires that we use relative (fractional errors). The relative error in the numerator is 1.0/36 = 0.028. The relative error in the denominator is 1.0/106 = 0.0094. The relative error in the denominator is added to that of the numerator to give 0.0374, which is the relative error in R. If the absolute error in R is required, it is (0.0374)R = 0.0136. The result, with its error, may be expressed as:
Example 3: Write a determinateerror equation for example 1. We follow the same steps, but represent the errors symbolically. Let N represent the numerator, N=G+H. The determinate error in N is then g+h. The relative error in the numerator is (g+h)/N. The relative error in the denominator is z/Z. The relative error in R is then:
This equation is in standard form; each error, g, h, and z appears in only one term, that term representing that error's contribution to the error in R. Example 4: Derive the indeterminate error equation for this same formula, R = (G+H)/Z. Here's where our previous work pays off. Look at the determinate error equation of example 3 and rewrite it for the worst case of signs of the terms. That's equivalent to making all of the terms of the standard form equation positive: Example 5: Rework example 2, this time using the indeterminate error equation obtained in example 4. Putting in the values:
This is less than 4%. Example 6: A result, R, is calculated from the equation R = (G+H)/Z, with the same data values as the previous example. After the experiment is finished, it is discovered that the value of Z was 0.05 too small because of a systematic error in the measuring instrument. The result was obtained from averaging large amounts of data, and the task of recalculating a correction to each value is daunting. But that's not necessary Use this information to correct the result. Look at the determinate error equation: The 0.05 error in Z represents a relative error of 0.05/106 in Z. Assuming zero determinate error in G and H, we have: r/R = (z/Z) = (0.05/106) So: r = (0.05/106)(0.338) = 0.0001594 Example 7: The density of a long copper rod is to be obtained. Its length is measured with a meter stick, its diameter with micrometer calipers, and its mass with an electronic balance.
L = 60.0 ± 0.1 cm (0.17%) D = 0.632 ± 0.002 cm (0.32%) [The error in D^{2} is therefore 0.64%] m = 16.2 ± 0.1 g (0.006%)The cross sectional area is πr^{2} = πD^{2}/4. So the density is = m/v = 4m/LπD^{2}. The relative error in the result (the density) should be no more than (0.17% + 0.64% + 0.006% = 0.816%) or about 0.8%. This is written: density = 8.606 ± 0.07 g/cm^{3} A reference book gives 8.87 g/cm^{3} as the density of copper. The experimental discrepancy is 0.26, indicating that something is wrong. The student who took this data may have blundered in a measurement. Maybe the material wasn't pure copper, but a copper alloy. If it is a measurement blunder, the diameter measurement is the most likely suspect.
K. THE OBJECTIVES OF LABORATORY WORKA good way to conclude this chapter is to consider what the students' objectives in laboratory ought to be. The freshman laboratory is not the same as a research lab, but we hope that the student will become aware of some of the concerns, methods, instruments, and goals of physics researchers.Experiments in freshman lab fall into several categories. In each case below, we indicate what the student's responsibility should be. 1. To measure a fundamental physical quantity. The student designs an experimental strategy to obtain the most accurate result with the available equipment. The student must understand the operation of the equipment and investigate the inherent uncertainties in the experiment fully enough to state the limits of error of the data and result(s) with confidence that the "true" values (if they were known) would not lie outside of the stated error limits. 2. To confirm or verify a wellknown law or principle. In this case it is not enough to say "The law was (or was not) verified." The experimenter must state to what error limits the verification holds, and for what limits on range of data, experimental conditions, etc. It is too easy to overgeneralize. A student in freshman lab does not verify a law, say F = ma, for all possible cases where that law might apply. The student probably investigated the law in the more limited case of the gravitational force, near the earth's surface, acting on a small mass falling over distances of one or two meters. The student should state these limitations. One should not broadly claim to have "verified Newton's law." Even worse would be to claim to have "proved Newton's law." 3. To investigate a phenomena in order to formulate a law or relation which best describes it. Here it is not enough to find a law that "works," but to show that the law you find is a better representation of the data than other laws you might test. For example, you might have a graph of experimental data which "looks like" some power of x. You find a power which seems to fit. Another student says it "looks like" an exponential function of x. The exponential curve is tried and seems to fit. So which is the "right" or "best" relation? You may be able to show that one of them is better at fitting the data. One may be more physically meaningful, in the context of the larger picture of established physics laws and theory. But it may be that neither one is a clearly superior representation of the data. In that case you should redesign the experiment in such a way that it can conclusively decide between the two competing hypotheses. The reader of your report will look very carefully at the "results and conclusions" section, which represents your claims about the outcome of the experiment. The reader will also look to see whether you have justified your claims by specific reference to the data you took in the experiment. Your claims must be supported by the data, and should be reasonable (within the limitations of the experiment). This is a test of your understanding of the experiment, of your judgment in assessing the results, and your ability to communicate.
L. CONCLUSIONError analysis is not an "afterthefact" activity; it pervades the entire experimental process from experiment design through datataking to the final analysis of the results. Nor is it a "cutanddried" procedure or set of recipes for "calculating errors." While there are statistical mathematical criteria which underlie the entire process, considerable insight and judgment and common sense must be brought to bear on the experiment to properly assess the dynamical interaction of the error sources. The experimenter must understand the physics which bears on the experiment to do a proper job of this. The experimenter must exercise judgment and common sense in choosing experimental strategies to improve results, and in choosing methods for determine the effect of experimental uncertainties. When error analysis is treated as a "mindless" calculation process, the gravest blunders of analysis and interpretation can occur.APPENDIX I. MEASURES OF UNCERTAINTYThe size of the experimental uncertainty in a set of measurements may be expressed in several ways, depending on how "conservative" you want to be.1. Limits of error. An attempt to specify the entire range in which all measurements will lie. In practice one specifies the range within which the measured values lie. 2. Average deviation. The average deviation of a set of measurements from its mean is found by summing the deviations of the n measurements, then dividing the sum by (n1). This measure describes the "spread" of the set of measurements. When one wishes to make inferences about how far an estimated mean is likely to deviate from the "true" mean value of the parent distribution, use the average deviation of the mean. To calculate it, sum the deviations of the n measurements, then divide this sum by n(n1)^{1/2}. This measure expresses the quality of your estimate of the mean. This is the measure we call the uncertainty (or error) in the mean. This last definition automatically includes two mathematical corrections, one required to make inferences about the parent distribution from a finite sample of data, and one to correct for the fact that you have used only a small sample. 3. Standard deviation. The standard deviation has become a "standard" method for expressing uncertainties because it is supported by a welldeveloped mathematical model. Unfortunately it is only appropriate when the experimenter (a) has large data samples, and (b) knows that the distribution of the data is really Gaussian, or nearGaussian. Therefore its use in the freshman lab is seldom justified—something like using a sledgehammer to crack a walnut.
APPENDIX II. CALCULATIONS USING STANDARD DEVIATIONSThe rules for error propagation for the elementary algebraic operations may be restated to apply when standard deviations are used as the error measure for random (indeterminate) errors:
When using standard deviations, the rules for combining average deviations are modified in this way: Instead of simply summing the error measures, you square them, sum the squares and then take the square root of the sum. This is called "summing in quadrature." Are Standard Deviations Better? Too many elementary laboratory manuals stress the standard deviation as the one standard way to express error measures. However, one can find, from standard statistical theory that when very few measurements are made, the error estimates themselves will have low precision. The uncertainty of an error estimate made from n pieces of data is
So we'd have to average 51 independent values to obtain a 10% error in the determination of the error. We would need 5000 measurements to get an error estimate good to 1%. If only 10 measurements were made, the uncertainty in the standard deviation is about 24%. This is why we have continually stressed that error estimates of 1 or 2 significant figures are sufficient when data samples are small. This is just one reason why the use of the standard deviation in elementary laboratory is seldom justified. How often does one take more than a few measurements of each quantity? Does one even take enough measurements to determine the nature of the error distribution? Is it Gaussian, or something else? One usually doesn't know. If it isn't close to Gaussian, the whole apparatus of the usual statistical error rules for standard deviation must be modified. But the rules for maximum error, limits of error, and average error are sufficiently conservative and robust that they can still be reliably used even for small samples. However, when three or more different quantities contribute to a result, a more realistic measure of error is obtained by using the `adding in quadrature' method described at the beginning of this section. Just as it's bad form to display more significant figures than are justified, or to claim more significance for results than is warranted by the experiment, so, too, it is bad form to use statistical techniques and measures of error to express results when the data does not justify those error measures nor the mathematical rules used to obtain them. This implies more quality significance to the results than may be the case, and borders on scientific fraud.
APPENDIX III. IMPORTANCE OF INDEPENDENCE WHEN USING ALGEBRAIC ERROR PROPAGATION EQUATIONSThe algebraic rules given for propagation of indeterminate errors are one way to derive correct error equations, but must be used with care. Here's an example which illustrates a pitfall you must avoid.A student wishes to calculate the error equation for the effective resistance, R, of two resistors, X, and Y, in parallel. The equation for parallel resistors is:
The student solves this for R, obtaining:
The error in the denominator is, by the sum rule, x+y. To proceed, we must use the quotient rule, which requires relative error measures. So the student converts the error in the denominator to relative form, (x+y)/(X+Y). The rest involves products and quotients, so the relative determinate error in R is found to be:
The next step requires some algebra to cast this in standard form, but let's not waste the effort, for this equation is already wrong! Why? Eq. 11 has X and Y in both numerator and denominator. Therefore the numerator and denominator are not independent. The quotient rule is not valid when the numerator and denominator aren't independent. To avoid this blunder, do whatever algebra is necessary to rearrange the original equation so that application of the rules will never require combining errors for nonindependent quantities. In fact, the form of the equation 10 is an ideal starting point, for all of its operations (+ and /) involve independent quantities. To do this correctly, begin with Eq. 10 (in which each quantity appears only once and there is no question that every operation is independent). The relative error in 1/X is, by the quotient rule, (0  x/X) which is simply x/X. The error in 1/X is therefore (x/X)(1/X) = x/X^{2}. Likewise the error in y is y/Y^{2} and in r is r/R^{2}. Finally, using the addition rule for errors, the result is:
Or, using Eq. 11, the right side can be expressed in terms of measured quantities only.
EXERCISESIn the following situations, consider common sense physical principles to determine which is the most meaningful way to describe the error: as an absolute error or a fractional error, an indeterminate error or a determinate error, a precise measure or an accurate one. Support your answers by stating your reasoning.(1) A batch of plastic meter sticks is accurately manufactured, but a year after leaving the factory the plastic shrank fairly uniformly by an average amount of 2 mm. (2) The knife edges of a mechanical balance (used for weighing objects) have become blunted. (3) The fast/slow setting screw in a precision mechanical stopwatch is misadjusted. (4) The supports of the cone bearing in a mechanical electrical voltmeter have become loose so that the pointer bearing is very loosely confined. (5) The effect (small) of air drag on a measurement of the acceleration due to gravity by a falling body experiment. (6) (a) The effect of uncontrolled and unmeasured laboratory temperature on a delicate mechanical instrument which makes measurements daily over many months. (b) The effect of temperature on the instrument if the experiment took 60 seconds to complete. (7) The effect of air drag on the period of a pendulum. (8) The effect of very impure alcohol used as the liquid in the determination of density of a solid by Archimedes' principle. [The solid is weighed when immersed in the liquid and the formula for the result contains the density of the liquid.]
In the next group of exercises, assume the following data: A = 10, B = 2, C = 5, D = 20. In each case the formula for the result, R, is given. Calculate the numeric value of R. Find the determinate error equation in each case, and then use it to answer the specific question asked. (9) Equation: R = (C  B)/A. Use the determinateerror equation to find what the value of R would be if B were actually 2.1 instead of 2. Check your answer by direct calculation.
Hint: Without actually writing the whole determinateerror equation, we can write the term of that equation which gives the contribution due to error in B.
due to error in B alone. (10) Equation: R = (C/A)  C  5. Use the error equation to find R if C were changed to 4.7. Check answer by direct calculation. (11) Equation: R = (D^{2}C^{2})3/(D  A)^{2}. Find how R changes if D changes to 22, A changes to 12 and C changes to 5.3 (all at once). (12) Equation: R = D sin [(A  C)/3B]. Find how R changes if C increases by 2%. Remember that arguments of trig functions are always in radians. (13) Equation: R = exp[(C  B)/D] Find how R changes if B decreases by 2% and D increases by 4 units. This is standard notation: exp(x) means the same as e^{x}. Here e is, of course, the base of natural logarithms.
This last group of questions is more general and requires careful thought and analysis of all possibilities. Be sure to consider these in the most general context, considering all possible measures of error: indeterminate, determinate, relative and absolute. The statements might be true for one kind of error measure and false for others. If so, specify this in your answer. (14) A student says, "When two measurements are mathematically combined, the error in the result is always greater than the error of either of the measurements." Discuss this statement critically. (15) Another student says, "When two measurements have 2% error, and they are used in an equation to calculate a result, the result will have 4% error." Discuss, critically. (16) Still another student says, "When several measurements are used to calculate a result, the error in the result can never be less than the error of the worst measurement". Discuss, critically. (17) Yet another student says, "When several measurements are used to calculate a result, and the error of one is 10 times as large as the next worst one, you might as well neglect all but the worst one in the error propagation equation." Discuss, critically.
ENDNOTES1. Some of the better treatments of error analysis are:
3. The magnitude of a quantity is its size, without regard to its algebraic sign. 4. The average deviation might more properly be called the "average absolute deviation," or "mean absolute deviation," since it is a mean of the absolute values of the deviations, not of the deviations themselves. [The mean of the deviations of a symmetric distribution would be zero.] 5. In the statistical study of uncertainties, the words "average" and "mean" are not used as if they were complete synonyms. When referring to the average of a set of data measurements, the word "mean" is always used, rather than "average." When referring to other averaging processes the word "average" is preferred. Perhaps this usage distinction is to avoid generating a clumsy name like "mean deviation of the mean." 6. See Laboratory Physics by Meiners, Eppensein and Moore for more details about the average deviation, and other measures of dispersion. 7. This relatively new notation for mean values is, I think, neater and easier to read than the old notation of putting a bar over the Q. 8. For a good discussion see Laboratory Physics by Meiners, Eppenstein and Moore. There (on p. 36) you will find a sidebyside calculation of average deviation and standard deviation, and a discussion of how they compare as measures of error. 9. The Gaussian distribution, sometimes called the "normal curve of error" has the equation:
where <X> is the mean value of the measurement X, and s is the standard deviation of the measurements. C is a scaling constant. f(X) is the number of measurements falling within a range of values from X to X + x, where x is small. This is the famous "bellshaped curve" of statistics. 10. See Meiners et. al., who comment: "This means that for many purposes, we can use the average deviation...instead of the standard deviation. This is an advantage because the average deviation is easier to compute than the standard deviation." 11. Independent errors are those for which the error of one individual measurement is not dependent on the errors in other measurements. No error influences the others, or is mathematically determinable from the others. 12. Calculus may be used instead.
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