An Introduction to Error Analysis

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1 An Introduction to Error Analysis Introduction The following notes (courtesy of Prof. Ditchfield) provide an introduction to quantitative error analysis: the study and evaluation of uncertainty in measurement. Before doing your first lab write-up, you will put some of these concepts to work by analyzing some data in a homework assignment. Your answers to this assignment should be handed in along with other assigned problems for the first class assignment. To the extent possible, a quantitative error analysis should be included in each lab write-up. Your experience in the laboratory to date has likely shown you that no measurement, no matter how carefully it is made, can be completely free of uncertainties. Thus, the usefulness of numerical scientific results is severely limited unless their uncertainty is known and communicated. There is no single "correct" way to do this, but there are lots of wrong ways. These notes describe some conventional ways of measuring and reporting experimental uncertainty. Estimating Uncertainties Single Measurements Frequently you will have only one or two measurements of some experimental observable. In such cases, you must simply make estimates of the reasonable uncertainty in each of the operations that went into the measurements. For example, typical uncertainties associated with some commonly used equipment are: Mettler balance pipette burette etc g 0.01 ml 0.0 ml In some cases, your estimate of an uncertainty will simply be an educated guess, but in others it may be worthwhile to do some experiments to determine the uncertainties of some of the devices used. The trick is to estimate the uncertainty large enough to avoid unrealistic representation of results, but at the same time to estimate it small enough not to wipe out important scientific conclusions. This process is actually a very important part of the skill of a good scientific investigator. Repeatable Measurements In general, if the uncertainty in an experimental measurement is random it can be estimated more reliably if the measurement can be repeated several times. Suppose, for example, we measure the time (t blue ) for the blue color to disappear in the iodination of cyclohexanone (an experiment that many of you did in the Chemistry 6 laboratory) and find 3.5 min. From this Error Analysis Page 1

2 single measurement we can't say much about the experimental uncertainty. However, if we repeat the experiment and get 3.6 min, then we can say that the uncertainty is probably of the order of 0.1 min. If a sequence of four timings gives the following results (in minutes) 3.5, 3.6, 3.7, 3.6, then we can begin to make more realistic estimates. The first step is to calculate the best estimate of t blue as the average or mean value, 3.6 min. As a second step, we make a reasonable assumption that the correct value of t blue lies between the lowest value, 3.5 min and the highest value 3.7 min. Thus, we might reasonably conclude that the best estimate is 3.6 min with a probable range of 3.5 to 3.7 minutes. These results can be expressed in the compact form t blue = 3.6 ± 0.1 min. In general, the result of any measurement of a quantity x is stated as (measured value of x) = x ± δx, where x denotes the mean value of x, and δx is an estimate of the uncertainty in x. When a measurement can be repeated several times, the spread in the measured values gives a valuable indication of the uncertainty in the measured quantity. In a later section of these notes, statistical methods for treating such repeated measurements are outlined. Such methods can give a more accurate estimate of uncertainty than that presented above, and, moreover, give a much more objective value of the uncertainty. Precision and Accuracy Precision refers to the amount of scatter in a set of numbers presumed to measure the same quantity. For example, suppose you repeatedly placed the same coin on an analytical balance. The scatter in your results would define the precision of the measurement. Accuracy refers to the degree to which the set of numbers represents the "true" value of the quantity. If, in the above example, the balance were not properly leveled, it might give good precision, but the value could be wrong, thereby leading to poor accuracy. Random and Systematic Errors The scatter of results which leads to the concept of precision is attributed to random errors which are presumed to originate from external influences that: 1. are large in number,. are unknown and therefore uncontrollable and unpredictable, Error Analysis Page

3 3. cause relatively small individual effects, and 4. act independently. Defects in accuracy are often caused by systematic errors in a set of data. These refer to a consistent shift of all values away from the "true" value, such as might be caused by a faulty measuring device or other undetected external influence which acts in the same direction on all measurements. For example, suppose the clock used to measure t blue was running consistently 10% slow. Then, all timings made with it will be 10% too small and repetitive measurements (with the same clock) will not reveal this deficiency. The distinction between random and systematic error is rarely sharp, and much "random error" can be attributed to inadequate experimental design. In some experiments it is worth considerable effort to determine whether the "random error" is indeed random. Significant Figures Several basic rules for reporting uncertainties are worth emphasizing. Because the quantity δx is an estimate of the uncertainty it should not be stated with too much precision. It would be inappropriate to state the rate constant for iodination of cyclohexanone, for example, in the form k measured = (1.9 ± ) 10 M 1 min. The uncertainty in the measurement cannot possibly be known to four significant figures. In the above case, the result should be reported as k measured = (1.9 ± 0.1) 10 M 1 min A rough indication of precision is given by the use of significant figures, following these rules. 1. Write a number with all digits known to be correct, plus one doubtful figure.. In addition and subtraction, the number of significant figures in the result is limited by the component term with the largest absolute uncertainty: (5.5 ± 0.04) + (6.386 ± 0.001) = ± In multiplication and division, the result is limited by the term with the largest relative or fractional uncertainty: (5.0 ± 0.5) (6.0 ± 0.01) = 30 ± 3. In this case, the relative or fractional uncertainty in the first term is (0.5/5.0), i.e., 5 parts in 50, which is a relative uncertainty of 10%. The relative uncertainty in the second term is a little less than %. Thus, here, the relative uncertainty in the product cannot be less than 10% and is dominated by that in the first term. A systematic approach for treating the combination of such uncertainties is presented in the later section on the propagation of errors. Error Analysis Page 3

4 Statistical Analysis of Data As mentioned above, the reliability of an estimate of uncertainty in a measurement can be improved if the measurement is repeated many times. The first problem in reporting the results of many repeated measurements is to find a concise way to record and display the values obtained. This is a problem that you may have encountered previously. For example, suppose you measured the masses of all the pennies minted since 198. Clearly, you would not always find the same value, since there would be some variation in alloy composition, stamping pressure, care of handling, etc. One way to display the results is to construct a bin histogram as shown in Figure 1 for a sample of 5 pennies. Here, one divides the range of values into a convenient number of intervals or "bins" of width Δ (equal to 0.01 g in this example), and counts the number of values in each "bin." One then plots the data in such a way that the fraction of measurements that fall in each bin is indicated by the area of the rectangle drawn above the bin. That is the height P(k) of the rectangle drawn above the k th bin is chosen so that rectangular area = P(k) Δ = fraction of measurements in the k th bin. For example, the shaded area above the interval from mass =.50 to.51 g has area = 0., indicating that one fifth of the 5 masses fell in this interval. = P(k) Mass/g Figure 1 Such a plot gives us a visual representation of how the masses of pennies are distributed. In most experiments, as the number of measurements increases, the histogram begins to take on a definite simple shape, and as the number of measurements approaches infinity, their distribution approaches some definite, continuous curve, the so-called limiting distribution. If a measurement is subject to many small sources of random error and negligible systematic error, the limiting Error Analysis Page 4

5 distribution will have the form of the smooth bell-shaped curve shown in Figure. This curve will be centered on the true value of the measured quantity. P(x) Figure x In the general case, this limiting distribution defines a function which we will call P(x). From the symmetry of the bell-shaped curve, we see P(x) is centered on the average value of x. Thus, if we knew the limiting distribution we could calculate the mean value, given the symbol μ, that would be found after an infinite number of measurements. This is defined as μ = lim x 1 + x + x x = lim 1 x i (1) where x i is the i th of measurements (,, 3,, ) and the Σ symbol is the standard summation notation. If the data represent repeated measurements of the same quantity (such as the mass of one penny), then μ represents the true value, but only in the absence of systematic errors. If the measurement of interest can be made with high precision, the majority of the values obtained will be very close to the true value of x, and the limiting distribution will be narrowly peaked about the value μ. In contrast, if the measurement of interest is of low precision, the values found will be widely spread and the distribution will be broad, but still centered on the value μ. Thus, we see that the breadth of the distribution not only provides us with a very visual representation of the uncertainty in our measurement, but also, it defines another important measure of the distribution. How do we quantify this measure of the distribution? The spread of values about μ is characterized by the standard deviation σ, defined for an infinite number of measurements as Error Analysis Page 5

6 σ = lim 1 x i μ () The standard deviation, σ, characterizes the average uncertainty in each of the measurements x 1, x, x 3,..., x from which μ and σ were calculated. Clearly, P(x), μ and σ are related. Gauss showed that, for randomly distributed errors, the limiting distribution function (the bell-shaped curve) is related to μ and σ by the equation: P μ,σ (x) = 1 σ π exp 1 x μ σ (3) Here, the subscripts μ and σ have been added to P(x) to indicate the center and width of the distribution. Measurements whose limiting distribution is given by the Gauss function are said to be normally distributed. The significance of this function is shown by Figure 3. The fraction of measurements that fall in any small interval x to x + dx is equal to the area P μ,σ (x) dx of the white strip in Figure 3(a). P μ,σ (x) x x+dx x Figure 3(a) P μ,σ (x) b P μ,σ (x) dx a a b x Figure 3(b) More generally, the fraction of measurements that fall between any two values a and b is the total area under the graph between x = a and x = b as in Figure 3(b). This area is just the definite integral of P μ,σ (x). Thus, we have the important result that, after we have made a very large number of measurements, a b P μ,σ (x) dx = fraction of measurements that fall between x = a and x = b. (4) Error Analysis Page 6

7 Similarly, the integral a b P μ,σ (x) dx defines the probability that any one measurement will lie between x = a and x = b. Because the total probability of our measurement falling anywhere between and + must be unity, the limiting distribution function P μ,σ (x) must satisfy P μ,σ (x) dx = 1 (5) A function satisfying equation (5) is said to be normalized. Thus, the probability that any one x value lies between the limits x = μ δ and x = μ + δ is the area under the Gaussian curve between these limits. If one computes (by integration) such areas for various choices of δ, one can show that the probability of finding any one measurement of x between various limits, measured as multiples of the standard deviation, σ, is given by the data presented in Table 1: Table 1. Gaussian Probability Intervals Probability Interval 0.50 μ 0.674σ < x < μ σ 0.68 μ 1.000σ < x < μ σ 0.80 μ 1.8σ < x < μ + 1.8σ 0.90 μ 1.645σ < x < μ σ 0.95 μ 1.960σ < x < μ σ 0.99 μ.576σ < x < μ +.576σ μ 3.91σ < x < μ σ This table says that, for example, we can be 95% confident that any one measurement will lie within approximately σ of the mean (where we have approximated by ). We can thus think of the Probability column in the table as a confidence level and the Interval column as a corresponding confidence interval. Although this analysis is elegant and all looks very straightforward, it would not be unreasonable for you to feel somewhat perplexed at this stage, since we can only know μ and σ if we can make an infinite number of measurements! This would clearly make for very long lab periods! Practically, we always sample only a finite number of all possible measurements. Thus, we need to know how the mean and standard deviation for a finite number of measurements are related to μ and σ. Error Analysis Page 7

8 For any finite number of measurements, the mean of those measurements will, in general, depend on how many measurements are made. To distinguish the mean of a particular finite set of measurements from the mean of an infinite number, we will use a different symbol: x = x 1 + x + x x = 1 x i (6) For a finite number of measurements, the experimental standard deviation s is defined as s = 1 1 x i x (7) (ote that s is defined with a factor 1 in the denominator rather than. As approaches, 1 approaches, but for finite, one uses 1 simply because the calculation of x has used up one independent piece of information. Initially, all the x i values are independent of each other; they were made as independent measurements. But once we compute x, we lose one independent piece of information in the sense that we can calculate any one x i given x and the 1 other data.) Thus, our goal is to determine how x is related to μ and how s is related to σ. Consider a finite number () of measurements of x with the results: x 1, x,..., x. The problem we confront is to determine the best estimates of μ and σ based on these measured values. If the measurements follow a ormal distribution P μ,σ (x), and if we knew the parameters μ and σ, we could easily calculate the probability of obtaining the values x 1, x,..., x that were actually measured. The probability of finding a value of x within a small interval dx 1 of x 1 is given by: Prob(x between x 1 and x 1 + dx 1 ) Prob(x 1 ) = 1 σ π exp 1 x 1 μ σ dx 1 (8) Similarly, the probability of finding a value within a small interval dx of x is given by: Prob(x between x and x + dx ) Prob(x ) = 1 σ π exp 1 x μ σ dx. (9) Since these probabilities are uncorrelated, the simultaneous probability of finding x 1 in the range x 1 x 1 + dx 1, x in the range x x + dx, x 3 in the range x 3 x 3 + dx 3, etc. is given by: Prob(x 1, x, x 3,..., x ) = Prob(x 1 ) Prob(x )... Prob(x ) (10) or Prob μ,σ (x 1,x,..., x ) 1 σ e x i μ /σ (11) Error Analysis Page 8

9 In equation (11), the numbers μ and σ are not known; we want to find the best estimates for μ and σ based on the given observations x 1, x,..., x. We might start by guessing values of μ and σ (call them μ' and σ') and computing the probability Prob μ',σ' (x 1, x,..., x ). The next step would be to guess new values μ'' and σ'' and compute the probability Prob μ'',σ'' (x 1, x,..., x ). If Prob μ'',σ'' (x 1, x,..., x ) was larger than Prob μ',σ' (x 1, x,..., x ), μ'' and σ'' would be better estimates of μ and σ the best estimates of μ and σ are those values for which the observed x 1, x,..., x are most likely. Continuing in this way, we would select different values for μ and σ to make the probability Prob μ,σ (x 1, x,..., x ) as large as possible. That is, we wish to maximize the value of Prob μ,σ (x 1, x,..., x ) with respect to variations in μ and σ. Using this approach, we can easily find the best estimate for the true value μ. From equation (11), the probability Prob μ,σ (x 1, x,..., x ) is a maximum when the sum in the exponent is a minimum. That is, the best estimate for μ is that value of μ for which x i μ /σ (1) is a minimum. To locate this minimum, we differentiate equation (1) with respect to μ and set the derivative equal to zero, giving x i μ = x i μ = 0 (13) or (best estimate for μ) = 1 x i (14) Thus, we have shown that the best estimate for the true mean μ is simply the mean of our measurements, x = 1 x i. Proceeding in a similar manner, we obtain (best estimate for σ) = 1 x i μ. (15) Since the true value of μ is unknown, in practice, we have to replace μ in equation (15) by our best estimate for μ, namely the mean value x. Because the calculation of x has used up one independent piece of information, the factor of in the denominator of equation (15) must also be replaced by 1. Thus, in practice, Error Analysis Page 9

10 (best estimate for σ) = 1 1 x i x = s (16) At this point, it is probably worthwhile to summarize our progress to date. If the measurements of x are subject only to random errors, their limiting distribution is the Gaussian or ormal distribution P μ,σ (x) centered on the true value μ, and with width parameter σ. The width σ is the 68% confidence level, in that there is a 68% probability that any measurement of x will fall within ± σ of the true value μ. In practice, neither μ nor σ is known. The data available are the measured values x 1, x, x 3,..., x, where is as large as our time and patience (and research budget!) allow. Based on these measured values, we have shown that the best estimate of μ is the mean value x, and the best estimate of σ is the standard deviation s. Several other questions remain. As mentioned above, the standard deviation, s, characterizes the average uncertainty in each of the measurements x 1, x, x 3,..., x. We may also ask, what is the uncertainty in the mean value x? How is this uncertainty related to s? This question can be answered by considering the following experiment. We start by weighing a penny times and determine x and s for this set of measurements. ow suppose that we repeat this experiment M times. For each of the M data sets we would compute a value of x and s, and, in general, each of these M values of x would be different. We could now average the values of x to give a mean of means, but this value would be the same number that would result from analyzing the combined data sets ( M values) from the M experiments. We could also compute the standard deviation of the means. This number, which we will call s m, can be shown to have the following simple relationship to s, which characterizes the uncertainty in each experimental measurement: s m = s (17) That is, s characterizes the uncertainty associated with each experimental measurement, while s m characterizes the uncertainty associated with the mean of any one set of measurements. Clearly, the more times we measure a quantity, the better we know its mean, but as the equation above shows, the uncertainty decreases only as the square root of the number of measurements. We now see how we might report the results of our experimental measurements of x. If we wish to report a value of x at the 68% confidence level, we might report it as: (value of x) = x ± s m. (18) On the other hand, if we wish to report a value for x at the 95% confidence level frequently used in scientific reports, we might report it as: (value of x) = x ± s m. (19) Error Analysis Page 10

11 ote that the multipliers of s in equations (18) and (19) are taken from Table 1. This raises one last issue to address in our discussion of statistical methods. The data in Table 1 allow us to relate a confidence level to some multiple of σ, which we do not know, rather than to some multiple of s, which we can determine. That is, the values given in Table 1 are only valid for the limiting distribution resulting from an infinite set of measurements. For a finite number of measurements, what is the correct relationship between the confidence level and s? The connection between the confidence level and s was made by W. S. Gosset, a mathematician who worked in the quality control department of a British brewery. Apparently, the company realized the importance of Gosset s work to both the general scientific community and to their own business, because they allowed him to publish his results, but only under a pseudonym! The pseudonym Gosset used was A. Student, and the critical quantity of his analysis, which he denoted by the symbol t in his papers, is to this day known as Student s t. Gosset found a function (which cannot be evaluated analytically, but is tabulated from the result of numerical calculations) that lets us compute the one number we need to relate a measured, experimental standard deviation, s, to a confidence level. In short, Gosset found that the true value of x fell in the interval x t s < (true value of x) < x + t s (0) where t is found, at various confidence levels, from Table below. ote that t depends not only on the confidence level, but also on, and note as well that as approaches, t approaches the confidence interval values tabulated earlier from the Gaussian function. In this course, we will use the common 95% confidence level (i.e., the t values in bold in Table ), and we will approximate t for any value of >15 by t =.0. Error Analysis Page 11

12 Table. Table of Student s t factors Confidence Level Adapted from Handbook of Mathematical Functions, Edited by M Abramowitz and I. A. Stegun, Dover Publications, Inc., ew York, 197 Propagation of Uncertainties Suppose the observable an experiment is designed to determine is not measured directly, but is derived from other measured experimental variables through some explicit functional relationship. For example, the Bomb Calorimetry experiment does not measure the molar internal energy of combustion of a hydrocarbon directly, but derives a value for it from measurements of the temperature rise produced on combustion, the heat capacity of the apparatus, and the weight of hydrocarbon used in the experiment the so-called raw data. How is the uncertainty in the derived quantity related to the uncertainties in the raw data? In general, suppose an experimental quantity, C, depends on other variables (the raw data) A, B,... via the relationship C = ƒ(a, B,...). The values of {A, B,...} are obtained through measurement, and each has an associated uncertainty, δ A, δ B,... that has been estimated in some way. For example, we may have used a statistical analysis to estimate δ A as δ A = ± ts A /, or we may simply have made an educated guess to estimate δ A. How is the uncertainty in C, δ C, dependent on the uncertainties δ A, δ B,...? The simple "significant figure" rules referred to earlier are very approximate attempts to account for uncertainties in derived quantities. However, for Error Analysis Page 1

13 complicated functional dependencies, such as logarithmic, exponential, and trigonometric dependencies, that appear in many equations in chemistry and physics, these approximate ideas are of little value. Sometimes a tiny uncertainty in a measurement will produce a huge uncertainty in a derived quantity the exponential function is notorious for this. Clearly, we need a more general approach for propagating uncertainties. If one assumes that the results of many duplicate measurements would produce a ormal or Gaussian distribution about the mean, then statistical theory provides a mechanism for estimating the uncertainty s C in the derived quantity C = ƒ(a,b). When the uncertainties δ A and δ B are both small and uncorrelated, statistical arguments show that the propagated uncertainty δ C is given by δ C = δ A ƒ A B + δb ƒ B A where the derivatives are evaluated using the best (i.e., the mean) values of A and B. When C depends on more than two quantities, for example C = ƒ(a, B, D,...), the formula is extended: δ C = δ ƒ A A B, D,... + δ ƒ B B A, D,... + δ ƒ D D A, B, ote that all of the quantities inside the square root are positive; random errors in uncorrelated variables tend to add in the calculated uncertainty. To help reduce the effort in the analysis of error propagation, a short list of error propagation formulae for some common functional relationships is given below. When a calculation requires several of these operations, these formulae may be combined according to the rules of differentiation. In complicated cases, you may find it easier to do the overall calculation in stages, obtaining the uncertainties in intermediate results as you go along. Error Propagation Formulae A and B are measurements with associated uncertainties δ A and δ B, respectively. C is a derived quantity with associated uncertainty δ C. 1. Addition of an exact (constant) number β: C = A + β. δ C = δ A. Multiplication by an exact number β: C = β A δ C = β δ A 3. Addition (or subtraction): C = A ± B ± D ±... Error Analysis Page 13

14 δ C = δ A + δb + δd Multiplication and division: C = A B... D E... δ C = C δ AA + δ BB + δ DD Power law: C = A n (n 0; n can be fractional or negative; hence the absolute value below) δ C = n C δ A A For example: C = A : δ C = A δ A A = Aδ A C = A 1/ : δ C = (1/)A1/ δ A A = δ A A 1/ 6. Combined multiplication and power law: C = A m B n δ C = C m δ AA + n δ BB 7. Exponential relationship: C = β exp (α A) where α and β are exact numbers δ C = α C δ A = α β δ A exp (α A) 8. Logarithmic relationship: C = β ln (α A) δ C = β δ A A Error Analysis Page 14

15 Discarding Suspect Data Sometimes one result in a set seems way out of line, and it is suspected that some influence outside the usual play of random fluctuations was at work. (power surge, stuck needle, human error in reading, etc.) When should a suspect result be tossed out? There are two schools of thought: 1. ever. Sometimes Even the sometimes school does not agree on when, but a commonly used criterion is that one and only one result in a set can be tossed out, if it has less than 10% chance of being a legitimate part of the random set. The test is the "Q-test" and goes as follows. Calculate a quantity Q: Q = Q suspect Q closest Q highest Q lowest Here Q suspect is the value of the suspect result. If Q exceeds the values in the accompanying table, the value in question may be discarded with 90% confidence ( is the total number of measurements in a data set). A similar table exists for 95% confidence, etc. : Q: Least Squares Fitting One of the most common types of experiment involves the measurement of several values of two different variables to investigate the mathematical relationship between the two variables. For example, suppose that one wished to determine the rate constant, k, for the second order dimerization of butadiene: C 4 H 6 C 8 H 1. The integrated rate law for this process is 1 C 4 H 6 1 C 4 H 60 = k t (1) Here [C 4 H 6 ] is the concentration of butadiene at time t and [C 4 H 6 ] 0 is the initial concentration. One would measure the concentration of C 4 H 6 at known times t, and if this reaction is second order, a plot of 1/[C 4 H 6 ] vs. t should be linear with slope equal to twice the value of the rate constant k. How do we obtain the best value of k from such data, and what is an appropriate estimate of the uncertainty in this best value of k? We will consider the general case where two variables x and y are connected by a linear relation of the form y = A + Bx () Error Analysis Page 15

16 where A and B are constants. If y and x are linearly related, then a graph of y vs. x should be a straight line with slope B and y-intercept = A. If we were to measure different values y 1, y,..., y corresponding to values x 1, x,..., x, and if our measurements were subject to no uncertainties, then each of the points (x i, y i ) would lie exactly on the line y = A + Bx as in Figure 4 (a). In practice, there are always uncertainties, and the most we can expect is that the distance of each point (x i, y i ) from the line will be reasonable compared with the uncertainties in the data see Figure 4(b). y Intercept = A Slope = B y Figure 4(a) x Figure 4(b) x How do we find the values of A and B that give the best straight line fit to the measured data? This problem can be approached graphically, but it can also be treated analytically using leastsquares fitting. To simplify our discussion, we assume that although there is appreciable uncertainty in the measured y values, the uncertainty in our measurement of x is negligible. This assumption is often reasonable, because the uncertainties in one variable often are much larger than those in the other, which we can then safely ignore. For example, in the kinetics experiment mentioned above, uncertainties in the measured concentrations are usually much larger than uncertainties in the measured times. We also assume that the uncertainties in y all have the same magnitude. (If the uncertainties are different, then the following analysis can be generalized to weight the measurements appropriately so-called weighted least squares fitting.) Finally, we assume that the measurement of each y i is governed by the Gaussian or ormal distribution, with the same width parameter σ y for all measurements. If we knew the constants A and B, then, for any given value of x i we could calculate the true value of the corresponding y i : (true value for y i ) = A + Bx i (3) From our assumptions, the measurement of y i is governed by a ormal distribution centered on this true value, with width parameter σ y. Therefore, the probability of obtaining the observed value y i is Prob A, B (y i ) 1 σy e y i A Bx i /σy (4) Error Analysis Page 16

17 where the subscripts A and B indicate that this probability depends on the (unknown) values of A and B. Since the measurements of the y i are independent, the probability of obtaining the set of values, y 1, y,..., y is the product Prob A, B (y 1, y,..., y ) = Prob A, B (y 1 ) Prob A, B (y )... Prob A, B (y ) (5) where the exponent χ is given by 1 σ y e χ / (6) χ = y i A Bx i σ y (7) Using the approach followed in the section on the statistical treatment of data, we will assume that the best estimates for A and B based on the measured data are those values for which the probability Prob A, B (y 1, y,..., y ) is a maximum, or for which the sum of squares χ in equation (7) is a minimum hence the name least-squares fitting. Thus, to find the best values of A and B we differentiate χ with respect to A and B and set the derivatives equal to zero: χ = A B σ y y i A Bx i = 0 (8) χ = B A σ y x i y i A Bx i = 0 (9) These two equations can be rewritten as simultaneous equations for A and B: A + B x i = y i (10) A x i + B x i = x i y i (11) Equations (10) and (11), sometimes called normal equations, are easily solved for the best leastsquares estimates of A and B: Error Analysis Page 17

18 A = x i y i x i D x i y i (1) B = x i y i x i D y i (13) where the denominator D is given by D = x i x i (14) The resulting line is called the least-squares fit to the data. The next step is to calculate the uncertainties in the constants A and B. To achieve this goal we first need to estimate the uncertainty in the y values, i.e., in y 1, y,..., y. In the course of measuring the y values, we will have formed some idea of their uncertainty. onetheless, calculating the uncertainty in y by analyzing the data is important. Since the numbers y 1, y,..., y are not measurements of the same quantity, we cannot get any idea of their reliability by examining the spread in their values. On the other hand, we can easily estimate the uncertainty σ y in the values y 1, y,..., y. Since we assume that the measurement of each y i is normally distributed about its true value A + Bx i, with width parameter σ y, the deviations (y i A Bx i ) are normally distributed, all with the same central value (zero) and the same width σ y. As usual the best estimate for σ y is that value for which the probability of equation (6) is a maximum. By differentiating equation (6) with respect to σ y and setting the derivative equal to zero we obtain the following familiar-looking expression for σ y : σ y = 1 y i A Bx i (15) As you may have suspected, this estimate of σ y is not quite the end of the story. The numbers in equation (15) are the unknown, true values of the constants A and B. In practice, these numbers must be replaced by the best, least-squares estimates for A and B given by equations (1) and (13). This replacement must be accompanied by replacing in the denominator of equation (15) by. Initially, all the (x i, y i ) data pairs are independent of each other; they were made as independent measurements. But once we compute A and B we lose two independent pieces of information. Thus, our final expression for σ y is: Error Analysis Page 18

19 σ y = 1 with A and B given by equations (1) and (13). y i A Bx i (16) Having found the uncertainty σ y we can easily calculate the uncertainties in A and B. From equations (1) and (13), we see that there are well-defined functional relationships between A, B and the measured values y 1, y,..., y. Therefore, we can estimate the uncertainties in A and B using the propagation of errors ideas discussed earlier. The results are: σ A = σ y 1 x D i (17) and σ B = σ y D (18) where D is given by equation (14). The problem of least-squares fitting to a general polynomial can be approached in a completely analogous manner. Many computer programs are capable of finding the values of the parameters (A and B in the above linear model) that provide a best fit, in the least-squares sense, of the sets of data to a theoretical model, as well as the associated uncertainties in the parameters. The Excel spreadsheet application has a least-squares function built in, but it does not produce statistical uncertainties in the A and B parameters. Error Analysis Page 19

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