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
IEC
IFCC
ISO
IUPAC
IUPAP
OIML
Bureau International des Poids ét Mesures
International Electrotechnical Commission
International Federation of Clinical Chemistry
International Organization for Standardization
International Union of Pure and Applied
Chemistry
International Union of Pure and Applied
Physics
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 si2, (or the estimated “standard deviations” si) and the number of
degrees of freedom νi. Where appropriate, the covariances should be given.
3) The components in category B should be characterized by quantities uj2,
which may be considered as approximations to the corresponding variances,
the existence of which is assumed. The quantities uj2 may be treated like
variances and the quantities uj 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 ui 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
uj. The combined standard uncertainty is then given by
uc2 = Σ (ui2 +uj2).
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 σ2 = σ12 + σ22. 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
⎛
⎜ ∑ ( xi − x )
⎜
and s = ⎜ i =1
N −1
⎜
⎜
⎝
N
N
x=
∑x
i
1
N
A, u i =
2
1
2
⎞
⎟
⎟
⎟ and write standard uncertainty of type
⎟
⎟
⎠
s
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 uj = a /√N (the standard deviation of the mean).
The standard uncertainty of type B for rectangular distribution is
given by uj = a/√3.
3. The combined standard uncertainty is then calculated from
⎛
2
2⎞
u c = ⎜⎜ ∑ u j + ∑ u i ⎟⎟
i
⎝ j
⎠
1
2
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 =
kuc. 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
UNCERTAINTY
OF
COMBINING
COMPONENTS
OF
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 = 3 we get the formula for propagation of uncertainty as
l
Δl ⎞
⎛ Δm
Δd = d ⎜
+ 3 ⎟ giving maximum possible uncertainty and
l ⎠
⎝ m
⎛ ⎛ Δm ⎞ ⎛ Δl ⎞
Δd = d ⎜⎜ ⎜
⎟ + ⎜3 ⎟
m
⎝
⎠ ⎝ l ⎠
⎝
(traditional approach)
2
2
1
2
⎞
⎟ giving
⎟
⎠
the
standard
uncertainty
Assuming uniform distribution the standard relative uncertainties are
obtained by dividing relative uncertainties by √3. Then
1
2
⎛ ⎛ Δm ⎞ ⎛ Δl ⎞ ⎞
Δd = d ⎜⎜ ⎜
⎟ ⎟⎟ giving
⎟ + ⎜3
⎝ ⎝ 3m ⎠ ⎝ 3l ⎠ ⎠
(GUM).
2
2
the
combined
uncertainty
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/cm3, the standard
uncertainty is 0.0325 g/cm3 and the combined uncertainty is 0.0188 g/cm3.
When we express the result as (2.00 ± 0.04) g/cm3 we mean that the value is
in the interval [1.96, 2.04] g/cm3 and we have a confidence level of 100%.
Expressing the value with standard uncertainty as (2.00 ± 0.03) g/cm3 means
the probability of the value being in the interval [1.97, 2.03] g/cm3 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/cm3 with confidence
level of 95% or (2.00 ± 0.09) g/cm3 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/cm3. 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 u2 = (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 u2
= (1/12) g2 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 Uc = Uj + 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