TEACHING PHYSICS IN LAB WITH DUE IMPORTANCE TO UNCERTAINTY IN MEASUREMENT D A Desai E-mail id: ddesai14@ gmail.com Abstract Major changes in definitions of some concepts about uncertainty in measurement and the relevant vocabulary have been introduced in metrology in the past decade. How should the change be effected in teaching labs is a question that requires a serious thought. After discussing some basic concepts about uncertainties in measurement (which may not be in consonance with those in the documents of the new approach), a brief outline of the changes recommended by BIPM are given below. Some suggestions regarding the scope of uncertainty analysis suitable for introductory lab courses are also made. For solving physics problems with numerical data the focus is on correct reasoning in applying relevant principles of physics and mathematical techniques. The students accept the numerical data without questioning whether they really represent the real world situation and solve the problems to get the results which have uncertainty only due to the rounding of last digits in the given data. Rules of significant figures are used to take care of that uncertainty and in most cases those rules are enough to get unique answers. So, checking with the answers provided with the problems, it is possible for the students to see whether they have acquired the required knowledge and skills for problem solving. In the laboratory work the focus is on entirely different aspects. The expected results of the experiments are known beforehand in most cases. The necessary formulae needed for the calculations are also known. What is of importance in experimental work is correct reasoning to see that the data collected is as free of errors as possible in representing the quantities measured and drawing correct inferences from the data. Wide and deep knowledge of principles of physics covering many fields is required to detect systematic errors and to estimate the uncertainties in the measurements. Techniques for data analysis such as graph plotting are also needed.
Depending upon the objective of an experiment the admissibility of the uncertainty in the final result may change. To get answers to such questions like What should be the least count of the measuring instrument?, or What should be the magnitude of the quantity? an analysis, based on the formula for propagation of errors, provides guidance. Such analysis is called sensitivity analysis. Unlike the data in numerical problems, the data collected from measurements can have much larger uncertainties than those due to rounding the last digit of the numbers. When students measure the value of a physical quantity, they believe that the quantity has a unique true value. The first thing that must be brought to their notice is that the measuring scale is with discrete units and any reading of the scale as coinciding with the extent of the measured quantity is uncertain by an amount equal to the rounding of the last digit. This rounding uncertainty is of magnitude of half the least count of the measuring instrument and since a measurement consists of subtraction of one reading from another the intrinsic uncertainty in every measurement is equal to the least count of the measuring instrument. This uncertainty is an essential consequence of measurement procedure and is intrinsic to every measurement. This is the minimum uncertainty in the data recorded from measurement. The true value of the measured quantity is, therefore, uncertain by one unit of the measuring instrument. But there can be many other causes of uncertainty such as calibration of the measuring instrument or variations in the behaviour of the measuring instruments due to changes in environment and so on. These uncertainties are to be evaluated on the basis of the knowledge about them. How close should the experimental result be to the expected one to have the confidence that the student has acquired the skills required for experimental work can be decided only after evaluating the uncertainty in the result. Therefore, uncertainty analysis is an integral part of experimental work. In the absence the analysis, the instructors arbitrarily decide the limits of acceptability of the result and the students remain ignorant of the tools to analyse the success or failure of their experimental work.
Any measured value stated without the uncertainty in it is meaningless. There can be many causes which cause deviations in the measured value. If the causes can be identified and the appropriate corrections can be made, the errors are called systematic. But if the causes are not identified then those systematic errors will distort the measurements and consequently the results of experiments. There are many situations where the causes of the systematic error can be identified, but the exact correction cannot be calculated. In such cases, generally it is possible to estimate the maximum possible error, which gives the measure of uncertainty due to those systematic errors. When the error in a measurement changes in an unpredictable manner with time, the measured values show random fluctuations if the measurements are repeated. When they are truly random, the uncertainty in the mean of a set of measured values of the same quantity can be estimated by applying statistical analysis. Such errors are called random errors. The uncertainty due to random errors is evaluated using statistical methods. For uncertainty evaluation all errors can be classified into two categories systematic and random. The various ways of classifying errors as systematic, random, instrumental, human etc., is a result of confusion about the concepts. The instrumental or human errors can either be systematic or random depending upon how they affect the measurements. Though it is possible to classify the errors in two categories, the method used to estimate the uncertainties cannot be so classified. Generally, the uncertainty due to random errors is estimated by statistical methods. But sometimes it may be more convenient to use some other procedure to estimate the uncertainty due to them. The procedures used to estimate the uncertainties due to systematic errors are generally other than statistical. But it is possible to assign some probability of occurrence of those errors and then use statistical procedures to combine them. Due to this complexity of evaluating the uncertainty in the measurement, it is now accepted that the evaluation of uncertainty needs to consider many factors. A few decades ago, the uncertainty analysis was called error analysis and error estimation was considered to be possible only for random errors. The data to be used for error analysis was to be corrected for all known systematic errors. But this limited application of error analysis was not able to account for all components that were shown to affect the measurement
from the formulae of propagation of errors. So, in an arbitrary way, those errors which did not manifest as random errors but were estimated by some other method were treated as standard deviations of normal distributions and were added in quadrature to get what was called the standard error. This procedure lacked sound logic, but was followed in the traditional error analysis in the last century. The state of affairs was not satisfactory and it became necessary to propose an uncertainty analysis more comprehensive than the error analysis of past. Guide to the Expression of Uncertainty in Measurement (GUM) and International Vocabulary of Basic and General Terms in Metrology (VIM) Because of the lack of international agreement on methods for evaluating and stating uncertainty in measurement, in 1977 the International Committee for Weights and Measures (CIPM, Comité International des Poids et Measures), the world's highest authority in the field of measurement science (i.e., metrology), asked the International Bureau of Weights and Measures (BIPM, Bureau International des Poids et Mesures), to address the problem in collaboration with the various national metrology institutes and to propose a specific recommendation for its solution. The final outcome of the project is the 100-page Guide to the Expression of Uncertainty in Measurement (referred to as GUM in short) as prepared by ISO/TAG 4/WG 3. It was published in 1993 (corrected and reprinted in 1995) by ISO in the name of the seven international organizations that supported its development in ISO/TAG 4: BIPM Bureau International des Poids ét Mesures IEC International Electrotechnical Commission IFCC International Federation of Clinical Chemistry ISO International Organization for Standardization IUPAC International Union of Pure and Applied Chemistry IUPAP International Union of Pure and Applied Physics OIML International Organization of Legal Metrology
In 1997 the Joint Committee for Guides in Metrology (JCGM), chaired by the Director of the BIPM, was formed by the seven Organizations that had prepared the original versions of the Guide to the Expression of Uncertainty in Measurement (GUM) and the International Vocabulary of Basic and General Terms in Metrology (VIM). In 2005, the International Laboratory Accreditation Cooperation (ILAC) officially joined the seven founding organizations. The following paragraph has been reproduced from the document JCGM_100_2008. This Guide establishes general rules for evaluating and expressing uncertainty in measurement that can be followed at various levels of accuracy and in many fields from the shop floor to fundamental research. Therefore, the principles of this Guide are intended to be applicable to a broad spectrum of measurements, including those required for: maintaining quality control and quality assurance in production; complying with and enforcing laws and regulations; conducting basic research, and applied research and development, in science and engineering; calibrating standards and instruments and performing tests throughout a national measurement system in order to achieve traceability to national standards; developing, maintaining, and comparing international and national physical reference standards, including reference materials. As is clear from the order in which the objectives are stated that commercial and ensuing legal difficulties are the prime concerns of the document. The objective of evaluation of uncertainty in the measurement of a single quantity in basic research and teaching laboratories can be getting the absolute limits of uncertainty rather than the probable ones. Whether the statement of uncertainty in the measurement of the magnitude of a single quantity can be different from that which is appropriate for the magnitude of any member a large group of quantities of approximately identical magnitudes as in quality control in production is a point that needs debate. The basic change in the new approach is as follows. The uncertainty of a result of a measurement is not necessarily an indication of the likelihood that the measurement result is near the value of the measurand; it is simply an estimate of the likelihood of nearness to the best value that is consistent with presently available knowledge.
It provides a realistic rather than a safe value of uncertainty based on the concept that there is no inherent difference between an uncertainty component arising from a random effect and one arising from a correction for a systematic effect. The statistical theory of errors, developed to estimate the uncertainty in the measurement due to the presence of random errors used the term error also in the sense of the estimated uncertainty. Because of this historical reason the word is still being used incorrectly and may continue to remain in use for some more years to mean uncertainty. But according to GUM recommendations the word error should not be used to mean uncertainty. The statement about sources of uncertainty in measurement from the JCGM document is reproduced below. That shows how comprehensive the procedure of evaluation of uncertainty has to be. 3.3.2 In practice, there are many possible sources of uncertainty in a measurement, including: a) incomplete definition of the measurand; b) imperfect realization of the definition of the measurand; c) non-representative sampling the sample measured may not represent the defined measurand; d) inadequate knowledge of the effects of environmental conditions on the measurement or imperfect measurement of environmental conditions; e) personal bias in reading analogue instruments; f) finite instrument resolution or discrimination threshold; g) inexact values of measurement standards and reference materials; h) inexact values of constants and other parameters obtained from external sources and used in the data-reduction algorithm; i) approximations and assumptions incorporated in the measurement method and procedure; j) variations in repeated observations of the measurand under apparently identical conditions. These sources are not necessarily independent, and some of sources a) to i) may contribute to source j). Of course, an unrecognized systematic effect cannot be taken into account in the evaluation of the uncertainty of the result of a measurement but contributes to its error. Since the uncertainties arise from different causes and not only due to random errors, they are now classified into two types A and B depending
upon whether the estimation is done by statistical methods or by some other methods. According to this classification Type A evaluation is that which uses statistical methods and Type B evaluation is that which uses other methods. Though, generally we can identify type A as due to random errors, type B consists of uncertainty due to systematic errors as well as that due to any other factors. Also it is pointed out that the words like random uncertainty and systematic uncertainty may be misleading and so should be avoided. The recommendations proposed in the basic document INC-1(1980) which form the core philosophy of the GUM are reproduced below. Recommendation INC-1 (1980) Expression of experimental uncertainties 1) The uncertainty in the result of a measurement generally consists of several components which may be grouped into two categories according to the way in which their numerical value is estimated: A. those which are evaluated by statistical methods, B. those which are evaluated by other means. There is not always a simple correspondence between the classification into categories A or B and the previously used classification into random and systematic uncertainties. The term systematic uncertainty can be misleading and should be avoided. Any detailed report of the uncertainty should consist of a complete list of the components, specifying for each the method used to obtain its numerical value. 2) The components in category A are characterized by the estimated variances s i 2, (or the estimated standard deviations s i ) and the number of degrees of freedom ν i. Where appropriate, the covariances should be given. 3) The components in category B should be characterized by quantities u j 2, which may be considered as approximations to the corresponding variances,
the existence of which is assumed. The quantities u j 2 may be treated like variances and the quantities u j like standard deviations. Where appropriate, the covariances should be treated in a similar way. 4) The combined uncertainty should be characterized by the numerical value obtained by applying the usual method for the combination of variances. The combined uncertainty and its components should be expressed in the form of standard deviations. 5) If, for particular applications, it is necessary to multiply the combined uncertainty by a factor to obtain an overall uncertainty, the multiplying factor used must always be stated. UNCERTAINTY OF TYPE A Assuming that the errors in the readings are random and are caused by the cumulative effect of a very large number of independent causes each producing only small change, it is possible to apply statistical methods to get an estimate of how close the true value of the measurand can be from the average value of the readings. Since the random errors have equal probability of increasing or decreasing the reading, if the number of measurements of the same quantity is very large, in averaging the readings most of the errors will get cancelled and the average value would be near the true value. From the theory of errors it can be shown that the uncertainty which corresponds to the difference between the true value and the average value can be expressed in terms of the standard deviation s of the sample and the number of readings N in the sample. The standard uncertainty determined this way for type A is represented by u i and it defines an interval within which the probability of locating the true value is about 68%. UNCERTAINTY OF TYPE B AND COMBINED STANDARD UNCERTAINTY The uncertainty of type B determined by methods which do not use statistical computation can be absolute or standard depending upon how it is defined. The total uncertainty in the value of the measurand is the combined uncertainty of all components of type A and type B. The procedure for combination recommended in INC-1 is statistical. For this the type B errors are assumed to have a distribution (rectangular, triangular, normal etc.) like the normal distribution for random errors. A standard uncertainty of type B
is then evaluated for the particular type of distribution and is represented by u j. The combined standard uncertainty is then given by u c 2 = Σ (u i 2 +u j 2 ). Similar formula is applicable for combining relative uncertainties. ADDING DIFFERENT UNCERTAINTY COMPONENTS The logic for adding type A and type B standard uncertainties in quadrature (Root of Sum of Squares or RSS method as termed in GUM), is based on treating different components as standard deviations of their distributions. From statistical theory, resultant variance σ 2 = σ 1 2 + σ 2 2. For making this formula applicable, appropriate standard uncertainties are defined from the absolute uncertainties of type B The procedure to combine type B uncertainties with type A uncertainties, recommended in GUM, consists of the following steps. 1. Determine type A standard uncertainty wherever random errors are detected. For this, calculate the mean and standard deviation of the sample using the formulae 1 2 N 2 N x i ( x ) i x x = 1 i= 1 and s = N N 1 and write standard uncertainty of type s A, u i = N 2. For determining uncertainty of type B, first determine the lower and upper limits of the measured quantity, within which the value of the measurand is expected to be located. Assume some distribution such as rectangular, triangular or normal and write an expression for the equivalent standard uncertainty for that distribution. If there is equal likelihood that the error can take any value within the interval, then the distribution is called uniform or rectangular. If the probability of having values near the mean increases linearly, then the distribution is called triangular. For any distribution, the half width a of the interval {X + a, X a} defined by the upper and lower limits is to be multiplied by an appropriate numerical factor to get the standard deviation. If the distribution is assumed to be normal then the limits of the interval are set at 68% probability of occurrence and the half width a = s and u j = a / N (the standard deviation of the mean).
The standard uncertainty of type B for rectangular distribution is given by u j = a/ 3. 3. The combined standard uncertainty is then calculated from 1 2 2 2 u c = u j + ui j i 4. Since the combined standard uncertainty also is with 68% confidence level, the expanded uncertainty with higher confidence level is defined with coverage factor k between 2 and 3 and is given by U = ku c. With k = 2 the confidence level is 95% and it is used by many laboratories and organizations to state the expanded uncertainty U. At k = 3, the confidence level is above 99.7%. INTRINSIC UNCERTAINTY IN A MEASUREMENT AND OTHER TYPE B UNCERTAINTIES Let us now see some most basic ideas about uncertainties in measurements. We have seen that every measurement has an intrinsic uncertainty and it arises from the procedure of measurement itself. Let us analyse the nature of this uncertainty. For measuring any quantity, we have to take two readings. For example, if we want to measure the length of a strip, we place the strip along the scale of a measuring tape with least count of 1 mm so that its two ends coincide with some marks on the scale of the tape. If the scale readings at the two ends are 10.0 cm and 35.0 cm, we conclude that the ends can neither be beyond 9.95 cm and 35.05 cm nor within 10.05 cm and 34.95 cm. In other words, each reading is rounded to the digit representing the smallest unit of the scale, viz., 0.1 cm. Since the systematic errors (or biases) at the ends included in the rounding are not known, the total uncertainty in the length of the strip becomes 0.1 cm or equal to the smallest unit of the measuring scale. Even if we hold the object with one of its ends at 0.0 cm and the other at 25.0 cm, the uncertainty is same because 0.0 may be anything between 0.05 and 0.05. [In this example since the scale has least count of 1 mm, objection may be raised that the uncertainty can be less, because eye can judge the difference much smaller than 0.5 mm. But the fraction of less than half unit cannot be estimated on the scales with smaller units such as that of a traveling
microscope. The example with tape of least count of 1 mm is given to help visualize the measurement procedure to arrive at a general conclusion.] In analogue meters as well as in meters with digital display, the same principle is applicable. Even the zero reading of the meter is uncertain by half the smallest unit of the selected range of the meter. So, any measurement of the meter is uncertain by an amount equal to its least count which is its limit of resolution. In the case of time measurement the situation is slightly different. Suppose the smallest unit of the watch is 1 s. When we read the watch the reading lasts for one second before it changes to the next second. For example, suppose we start counting the oscillations of a pendulum at time equal to 10 s on the watch and stop the watch when the count is 20 oscillations. Suppose the final reading of the watch is 98 s. What is the uncertainty in the time interval? Suppose the uncertainty at the start is negligible, but the uncertainty in the last reading is 1 s. Then the time interval could be 89 s though the subtraction of the readings gives 88 s. The other extreme case could be that starting time was just short of 11 s and the final reading has negligible uncertainty. In this case the interval would be 87 s instead of the recorded value 88. This shows that in time readings though the uncertainty in every reading is only positive and equal to smallest unit of the clock, the uncertainty in time interval is still ± 1 unit of the clock. When the smallest unit of the measuring instrument is a multiple or a fraction of the unit in which the reading is expressed, the uncertainty is equal to the unit of the instrument and not the unit of the quantity. For example the scale on the dial of an ammeter may have smallest unit of 0.5 ma or another may have 2 ma. Though the current measured in both cases may be expressed in unit ma, in the former case the uncertainty would be 0.5 ma whereas in the latter case it would be 2 ma. When the unit of the quantity is different from the amount of uncertainty, it is mandatory to express the measurement with the uncertainty, otherwise the data would be misleading as regards the accuracy. For example, 15.0 ± 0.5 mm and 15.0 mm do not mean the same thing. In the second case, it will be presumed that the uncertainty is 0.1 mm, when in reality it is 0.5 mm. Now we can make a general statement regarding the uncertainty in the data measured using some measuring instrument. Every measured quantity has uncertainty at least equal to the smallest unit of the measuring
instrument. Why do we say it is the least uncertainty? The uncertainty equal to one unit of the measuring instrument is the unavoidable uncertainty arising due to the process of measurement. Let us call it the intrinsic or essential uncertainty in a measurement. This uncertainty is not due to some external factor affecting the measurement and hence cannot be overcome. It is an inherent aspect of measurement and so may be called essential. As discussed earlier this intrinsic uncertainty includes systematic error equal to the unmeasured fraction of the smallest unit of measuring instrument. There may be other external factors which may add to the uncertainty of the reading of the instrument. For example, in resonance tube experiment the length of the air column in the tube for which maximum sound is obtained due to resonance is measured. If somebody measures the length of the resonating column with a metre scale and says that the measurement is uncertain by an amount equal to 1 mm, the least count of the scale, then it may not be correct. It may be difficult to determine within a range of few millimeters for which length of the column the sound intensity is maximum. This can be easily seen by varying the length by small amount. The uncertainty in the length measurement is about judging the variation in loudness of sound critically by ear. In such cases we may find that the length is uncertain by the amount in which the change in loudness is undetectable by ear. As an example, suppose for a particular frequency the loudness of sound is found to be same for lengths from 15.0 cm to 15.6 cm. Then the correct length must be somewhere in this range. We may take the length as the mean value equal to 15.3 cm with the uncertainty of 0.3 cm. There can be many situations where similar reasoning is applicable. Here the uncertainty is much larger than the intrinsic uncertainty of the measuring instrument. It is possible to reduce such uncertainties by using better detectors. For measuring time intervals with greater precision one may use a stop watch of least count of 0.1 s. The intrinsic error would then be reduced to 0.1 s. With practice, one may be able to measure time intervals with uncertainty of this order. But measurements with stop watch of least count 0.01 s or less with manual operation bring in uncertainties which are due to unpredictable time lags in the operations of starting and stopping the watch. The readings will appear to vary randomly about some value over some range. These readings are with additional errors arising from some random factors. Hence they are called random errors. These random factors are present when we operate manually stop watch of least count of 1 s or 0.1 s
also, but as long as the least count is larger their contribution the random errors are not observable. The intrinsic uncertainty of measurement includes the systematic error equal to the unmeasured fraction of the unit of measurement and also random errors which are much less than least count of the instrument. The intrinsic uncertainty in every measurement is the maximum uncertainty due to the systematic error equivalent to the unmeasured fraction of the unit of the measuring instrument. Hence it is of type B and when we write a measured length as (48 ± 1) mm, we mean that we are 100% sure that the length is neither less than 47 mm nor larger than 49 mm. This uncertainty, therefore, defines an interval with 100% confidence level of the measurement. When a quantity such as radius of a cylinder varies along its length and we have to substitute a single value for it in the formula, we measure its values at various points along its length in various directions and take the average of all as the best value. Though the values of radius differ from point to point all these values are not of the same radius and so the set of measured values is not strictly a sample consisting measurements of the same quantity. One way to estimate the uncertainty in the radius is by treating it as of type A. But it can also be evaluated by a much simpler method of type B as suggested in GUM. We know that approximately 2/3 of the number of the values in the set should be within a range of two standard deviations. So, if the number of measurements is 6, the middle four should give us a range of 2s. From this we can determine the standard uncertainty which is the standard deviation of the mean, as equal to s/ 6. This uncertainty does not cover the intrinsic uncertainty of measurement, which is equal to least count. [This is clear because with increasing number of readings N, the standard uncertainty tends to zero, but the same systematic error being in all readings, the average also has the same error.] How should these two components of uncertainty be combined to get the total uncertainty in the value of radius of the cylinder? THE PROBLEM OF COMBINING COMPONENTS OF UNCERTAINTY The traditionally accepted standard error as determined by statistical method gives a probabilistic statement of uncertainty, whereas other uncertainties mostly are in absolute terms. The formula for propagation of errors shows the uncertainties should be added linearly and at elementary
level this procedure is still followed when random errors do not appear in the observations. Wherever random errors are present, the correct procedure to estimate the uncertainty in the mean is the statistical one of calculating variances, standard deviations and then standard deviations of the means. But because of the probabilistic interpretation of the standard error, it cannot be added to absolute errors. A way out was to treat the other components as if they are standard deviations of some distributions and add all terms in quadrature to obtain the standard error in the measurand. This procedure was followed widely in the last century. But the procedure lacked sound logical basis and appeared to be based on postulates whose plausibility could not be convincing. [The problem of estimating uncertainty due to random errors is clearly in the realm of statistics. But the basis of the belief that estimating any uncertainty is also a problem to be tackled using statistics is not clear to me. The idea of maximum uncertainty in a measurement is easy to grasp but obtaining combined uncertainty due to systematic errors using statistical methods is not always clear.] The clarifications, explanations and examples given in the GUM are of much help in understanding the difficulties of arriving at a consensus as regards evaluation and expression of uncertainty in measurements. Still there are some points about which diverse views are expressed and debates are going on. But on the whole the GUM has provided much clarity about evaluating uncertainty. One problem facing the teaching labs would be whether to follow the procedure recommended in GUM right from the introductory lab courses or to use simpler procedures which could be explained to the students easily. The applicability of concepts such as Bayesian probability, different types of distributions with a priori probabilities etc., would be difficult to explain to students who have no knowledge of probability theory. But they must be made aware of the basic concepts about uncertainties in measurements. Using correct formulae given in GUM for evaluating uncertainty, without the explanation how the formulae are arrived at, would prove counterproductive, because that will stop students from thinking and they will evaluate mechanically the uncertainty without understanding its significance in the experimental work. (Unfortunately, the same thing happens when students use the formula for the standard error without understanding true significance of evaluation of uncertainty.) Perhaps at the introductory level it would be better if we use the procedure of simple addition of various uncertainty components.
Let me consider one simple example to clarify some of the points which are discussed above. EXAMPLE To determine the density of a cubical body, its side is measured using vernier callipers and is found to be 2.00 cm and its mass is measured to be 16.0 g. Let us evaluate the uncertainty in the density of the body by traditional as well as the method recommended in GUM. m Using d = we get the formula for propagation of uncertainty as 3 l Δm Δl Δd = d + 3 giving maximum possible uncertainty and m l 1 2 2 2 Δm Δl Δd = d 3 + m l giving the standard uncertainty (traditional approach) Assuming uniform distribution the standard relative uncertainties are obtained by dividing relative uncertainties by 3. Then 1 2 2 2 3 3 3 Δm Δl Δd = d + m l giving the combined uncertainty (GUM). The uncertainties are all of type B and are 0.01cm in 2.00 cm and 0.1g in 16.0 g. Therefore, the maximum uncertainty is 0.0425 g/cm 3, the standard uncertainty is 0.0325 g/cm 3 and the combined uncertainty is 0.0188 g/cm 3. When we express the result as (2.00 ± 0.04) g/cm 3 we mean that the value is in the interval [1.96, 2.04] g/cm 3 and we have a confidence level of 100%. Expressing the value with standard uncertainty as (2.00 ± 0.03) g/cm 3 means the probability of the value being in the interval [1.97, 2.03] g/cm 3 is 0.68. This is said to give 68% confidence level, but such statement is of no use whether the measured value is within the interval or not. To get that information either the value is stated as (2.00 ± 0.06) g/cm 3 with confidence level of 95% or (2.00 ± 0.09) g/cm 3 with confidence level of greater than 99.7%. In this method it is impossible to have 100% confidence. But as can be seen from the above example, the uncertainty for 99.7% confidence level is even larger than the maximum uncertainty evaluated earlier by simple
method. This shows that this traditional method that was being in use does not serve any useful purpose. Expressing the result with the expanded combined uncertainty with k =2 for 95% confidence level, as is usually done, we write (2.00 ± 0.04) g/cm 3. This shows that even at 95% confidence the uncertainty is almost equal to the maximum uncertainty. Comparing the three results it seems the simple method of adding uncertainties is satisfactory for such cases. The additional statistical exercise does not give any advantage. On the contrary, it may create some confusion in the minds of the students. Here are some more points from the document JCGM 100_2008 which must be given some thought. Related to what we call the intrinsic uncertainty of measurement there is mention of an uncertainty due to finite instrument resolution or discrimination threshold. But from the statement 3.2.4 It is assumed that the result of a measurement has been corrected for all recognized significant systematic effects and that every effort has been made to identify such effects, it is not clear whether its origin in rounding the last digit is of nature of systematic error, as explained earlier, is taken note of. This is important because from 3.4.1 If all of the quantities on which the result of a measurement depends are varied, its uncertainty can be evaluated by statistical means shows that the impossibility of reducing the intrinsic uncertainty in a measured value of a single quantity is not accepted. Most likely, the statement is made in the context of measurements for quality control and similar situations, where the uncertainty is to be stated for the value of any member from the set. WHAT IS THE INTRINSIC UNCERTAINTY IN THE MEASURED VALUE? F.2.2.1 The resolution of a digital indication One source of uncertainty of a digital instrument is the resolution of its indicating device. For example, even if the repeated indications were all identical, the uncertainty of the measurement attributable to repeatability would not be zero, for there is a range of input signals to the instrument spanning a known interval that would give the same indication. If the resolution of the indicating device is δx, the value of the stimulus that produces a given indication X can lie with equal probability anywhere in the interval X δx/2 to X + δx/2. The stimulus is thus described by a rectangular
probability distribution of width δx with variance u 2 = (δx) 2 /12, implying a standard uncertainty of u = 0,29δx for any indication. Thus a weighing instrument with an indicating device whose smallest significant digit is 1 g has a variance due to the resolution of the device of u 2 = (1/12) g 2 and a standard uncertainty of u=(1/ 12)g=0,29g. From the above we see that GUM considers the half width of uncertainty in the measured value equal to half the least count (resolution), whereas we have seen it is half width of uncertainty in a reading on the instrument originating in the rounding of the last digit, and the half width of uncertainty in the measured value is double of that i.e. equal to the least count. The assumption that any stimulus in the interval would give the same reading and so it must be treated as having a uniform distribution is also not true as can be demonstrated by simply considering how rounding includes a definite fraction of the unit in the uncertainty. When the voltage across a dry cell is measured on a digital voltmeter on ranges of 2, 20, 200 and 1000 volts, the readings are, 1.527 V, 1.53 V, 1.5 V and 2 V respectively. These values clearly show how the rounding uncertainty includes the systematic error in the measurement. So, the idea of uniform distribution is irrelevant and calculation based on that misconception is unjustified. JCGM takes note of the fact that the reliability of type A evaluation from samples with small number of readings is poor. See the following paragraph. 4.3.2 The proper use of the pool of available information for a Type B evaluation of standard uncertainty calls for insight based on experience and general knowledge, and is a skill that can be learned with practice. It should be recognized that a Type B evaluation of standard uncertainty can be as reliable as a Type A evaluation, especially in a measurement situation where a Type A evaluation is based on a comparatively small number of statistically independent observations. NOTE: If the probability distribution of q is normal, then σ[s(q)]σ(q), the standard deviation of s(q) relative to σ(q), is approximately [2(n 1)] 1/2. Thus, taking σ [s(q)] as the uncertainty of s(q), for n = 10 observations, the relative uncertainty in s(q) is 24 percent, while for n = 50 observations it is 10 percent.
This means for experiments in undergraduate laboratory, where the number of measurements is rarely more than 10, and more likely about 5, for which the relative uncertainty is 36%, the type B evaluations should have much better legitimacy than type A. The recommendations of GUM are meant to be used in estimation of uncertainty for the purposes of quality control, calibration and many similar objectives. The logic used in the method of evaluation is based on certain interpretations of probability and there are voices of difference about the interpretation. Since these concepts are not simple and clear to be explained to students of introductory teaching labs it would be better to have a simple method whose logic could be understood by the students. What is more important in a teaching lab is to make the students familiar with the idea of evaluating uncertainties in measurements rather than using correct formulae without understanding the logic behind them. In GUM it is accepted that statistical methods are not the only methods for evaluating uncertainties in measurements. It is also accepted that when the number of observations is small the uncertainty in the standard deviation itself is large. When the statement about uncertainty is made in a probabilistic way and further the probability evaluated itself has large uncertainty, the legitimacy of the calculated result becomes suspect. The absolute uncertainties of type B are defined for confidence level of 100%. On the other hand, for type A uncertainty, the expanded uncertainty equal to 3s/ N gives a confidence level greater than 99.7% which can be treated as equal to 100% for all practical purposes. So, instead of combining equivalent standard uncertainty of type B to other components of both type B and type A using statistical methods, logically justified procedure would be obtaining the combined expanded uncertainty U c = U j + 3 s/ N. Anyway, the need for stating the measurement with expanded uncertainty instead of the standard uncertainty arises precisely because the probabilistic statement of the uncertainty in the magnitude of a quantity is incapable of giving definite information of the uncertainty in the specific measurement. So, though the uncertainty determined by this approach may be larger, it is with 100% confidence level and that is easy for students to understand. In most of the cases of interest, the number of terms to be combined to get the combined uncertainty is not large and so, the basis for adding them in quadrature is also not strong enough for the same reason as discussed above.
Simple addition of uncertainties appears to be adequate for introductory lab courses. Even the uncertainty due to random errors can be evaluated by type B methods as mentioned in the GUM by defining the range with limits for confidence level of 68%. The range includes two thirds of the observations. The value of standard deviation would be given by range/2. The expanded uncertainty corresponding to 100% confidence level should then be added to the other absolute uncertainty components of type B to get the combined absolute uncertainty. The rigorous study of the procedure recommended in GUM may be introduced in the lab curriculum at a higher level when the students are familiar with the basic concepts of uncertainty analysis. References 1 JCGM 100:2008 GUM 1995 with minor corrections Evaluation of measurement data Guide to the expression of uncertainty in measurement First edition September 2008 JCGM 2008 2 Measurement Good Practice Guide No. 11 (Issue 2) A Beginner s Guide to Uncertainty of Measurement Stephanie Bell Centre for Basic, Thermal and Length Metrology National Physical Laboratory August 1999 Issue 2 with amendments March 2001 National Physical Laboratory Teddington, Middlesex, United Kingdom, TW11 0LW 3 Wikipedia: Measurement uncertainty: ISO GUM as on 11 Nov. 2009 4 Practical Physics G L Squires Cambridge University Press 5 Introduction to Error Analysis John R Taylor University of Colorado