10.1 General Remarks If there is one premise basic to
instrumentation engineering, it is this: no measurement is without
error. Hence neither the exact value of the quantity being measured
nor the exact error associated with the measurement can be
ascertained. In engineering, as in physics, the uncomfortable
principle of indeterminacy exists. Yet as we have seen in our
discussion of interpolation methods (Section 9.3) uncertainties can
be useful and, like friction are often a blessing in disguise.
Slide 3
It is toward a methodical use of measurement uncertainties as a
guide to approaching true values that this chapter is addressed.
The output in most experiments is a measurement. the reliability of
the measurement depends not only on variations in controlled inputs
but also in general, on variations in factors that are uncontrolled
and perhaps unrecognized. Some of these factors that might
unwittingly affect a measurement are the experimenter the
supporting equipment and conditions of the environment..
Slide 4
Thus in addition to errors caused by the device under test, and
in addition to errors caused by variations in the quantity being
measured, extraneous factors might introduce errors in the
experiment that would cloud the results use of different measuring
equipment. Effects of those variables that are not part of the
study can be further minimized by taking observations in a random
order. This is called randomization
Slide 5
The important task of measuring the remaining significant
errors is approached by taking a member of independent observations
of the output at fixed values of the controlled input. This is
called replication Staling the above ideas in mathematical terms
each measurement x can be visualized as being accompanied by an
error a such that the interval x (10.1) will contain the true value
of the quantity being measured. The measurement error, in turn, is
usually expressed in tents of two components, a random error e and
a systematic error such that
Slide 6
10.1.1 RANDOM ERRORS When repeated measurements are taken,
random errors will show up as scatter about the average of these
measurements. The scatter is caused by characteristics of the
measuring system and/or by changes in the quantity being measured.
Random errors always will be observed as long as the readout
equipment has adequate discrimination. The term precision is used
to characterize random errors. Precision is quantified by the true
standard deviation or of the whole population of measurements or,
more often by its estimator S the precision index of the data
available.
Slide 7
These statistical teams will be defined shortly by working
equations. Now it is sufficient to understand that a large
precision index means a lot of scatter in the data, and conversely,
a small precision index means high precision.
Slide 8
10.1.2 SYSTEMATIC ERRORS Over and above the random errors
involved in all measurements there are also errors that are
consistently either too high or too low with respect to the
accepted true value. Such errors, which are termed fixed errors or
systematic errors are characterized by the term bias Systematic
error is quantified buy the true bias BATA or, more often, by B,
the estimate of the limit of the bias. When bias can be quantified
it is used as a correction factor to be applied to all
measurements. A zero bias implies that there is no difference
between the true value and the true mean of many observations.
Slide 9
However the zero-bias case is rare indeed; and experience
indicates a strong tendency to underestimate systematic errors. All
of these terms and symbols are shown geometrically in Figure
10.1.
Slide 10
Slide 11
Systematic errors can be minimized by various methods as, for
example, by calibration (Figure 10.2). Calibrations are usually
accomplished by comparing a test instrument to a standard
instrument. Since such comparisons are not always direct or
perfect, we may not succeed in totally determining the bias, that
is, the bias may have a random component, but it is essentially
fixed, and is never as random as precision errors.
Slide 12
Slide 13
10.2 STATISTICAL RELATIONS There are cases in engineering
practice, however, when we can presume that the bias is removed
that all errors are of the random type, and that hence the errors
can be treated statistically [1],[2]. In this section we overlook
for a time the fixed (bias) errors and consider only the random
(precision) errors. It is clear that, even in the absence of fixed
errors we are to be denied by the nature of things the ability to
measure directly the true value of a variable.
Slide 14
Thus it becomes our job to extract from the experimental data
at least two vital bits of information. First we must from an
estimate of the best value of the variable. This will he denoted
by. Closely coupled with this requirement, we must give an estimate
of the intervals centered on with in which the true value is
expected to lie. This will be denoted by the uncertainty margin
that we tack on to [3]
Slide 15
10.2.1 BEST VALUE AT A GIVEN INPUT When an output X is measured
many times at a given input, the mean value of X is simply 10.3
where X k is the value of the k th observation (called
interchangeably the k th reading or measurement) and N is the
number of observations in the sample. It is a mathematical fact
that the arithmetic average defined by equation(10.3) is the best
representation of the given set of X k
Slide 16
Note that when the estimated best value of X is taken as the
sum of the squares of the deviations of the data from their
estimate is a minimum. (This is essentially the least- squares
principle.) However, whereas represents an unbiased estimate of the
true arithmetic means niu of all possible values of X. There is no
assurance that is the true value that is in any actual measurement,
the bias would have to be considered. (Thus good agreement, that
is, high precision in small sample replication does not imply that
is close to that is, high accuracy.) Nevertheless, from any
viewpoint the best estimate of the true value of the population
mean at a given input is the average, of the available measurements
(Figure 10.3).
Slide 17
10.2.2 CONFIDENCE INTERVALS Having decided on the best
available value of X (which is ), we inquire next as to its worth
as an estimate of the true value of X which is for the case of zero
bias). C. G. Darwin has noted in this regard [4]:"It seethed to me
that there was a defect in the habit of thought of many in the
engineering profession, some sort of campaign was needed to
inculcate in people's minds the idea that every number has a fringe
that it is not to be regarded as exact but as so much a bit and
that the size of this bit is one of its really important
quantities."
Slide 18
Slide 19
This plus or minus fringe that accompanies every measurement is
called a confidence interval CI Thus a confidence interval for the
true value can be given by XCI(p), where X is the estimated best
value of X, CI is the confidence interval. And (p) is the
probability statement (and not a multiplier of CI). To form these
confidence intervals we need replicate data, and we note in this
regard that these intervals will differ according to the size and
number of sets available. Sets of Very Large N Many times in
engineering, a tabulation of how the, Various values of X occur in
replication is well approximated by the Gauss--Laplace normal
distribution relation [5]
Slide 20
(10.4) where the factor has the normalizing effect of making
the integral of f(X) over all values of X equal unity and where
represents the true standard deviation of X, which in turn is well
approximated by (10.5)
Slide 21
The standard deviation of a normal distribution o f X has the
following characteristics: 1. measure the scatter of X at a given
input, that is, it is a measure of the precision error. 2. has the
same units as X. 3. is the square root of the average of the sum of
the squares of the deviations of all possible observations from the
true arithmetic mean For any engineering applications this is not
good enough, and wider intervals must be expected to express
greater confidence. For example, 95.46% of the data can be expected
to fall within the +2delta interval and 99.73% within +3delta
(Figure 10.4).
Slide 22
We are assured that X is a very good estimate of by the large
size of the sample. We may ask, however, how typical a single
observation of X. is as we have just seen, one answer is X 3(at
99.73%) (10.6) Statement(10.6) indicates that the interval is
expected to include 99.7% of the time.
Slide 23
Slide 24
It further brings out the important point that, to be most
meaningful, a measurement should be given three parts [6], [7].
These are: 1. A magnitude (the indicate value of X) 2. A confidence
interval [which is your estimate of what the error might be: in
statement (10.6)] 3. A probability statement [an indication of your
confidence that the true value will be within the confidence
interval chosen; 99.73% in statement (10.6)].
Slide 25
Single Set of Small N We now face the unpleasant fact that, in
a practical experiment, we usually obtain only a relatively small
sample from all possible values of X. This means that we cannot
obtain the true arithmetic mean and hence we cannot form the true
standard deviation instead of the inaccessible deviations( ) we can
determine only the residuals. We note in this regard that the sum
of the squares of the residuals being always a minimum according to
least-squares of principle previously mentioned, is always less
than the sum of the squares of the deviations.
Slide 26
The precision index of the single sample is defined in terms of
the residuals and is patterned after equation (10.5) as (10.7)
where the factor (N-1) is used in place of the usual N in an
attempt to compensate for the negative bias that results from using
X in place of in forming the differences. However, a negative bias
unfortunately still remains in the small estimate of the standard
deviation and S, the obtainable does not equal delta the
desired.
Slide 27
Student's Distribution Recognizing this deficiency a method was
developed by the English chemist W.S.Gosset (writing in 1907 under
the pseudonym "Student"), by which confidence intervals could be
based on the precision index S of a single small sample. He
introduced the "Student's statistic whose values have been
tabulated in terms of degree of freedom miu and the desired degree
of confidence (quantified by the probability pi) (Table 10.1 and
Figure10.5) Careful perusal of these values will show that the t
statistic inflates the confidence interval (i.e.the uncertainty
margin) so as to reduce the effect of understand deviation delta
when a small sample is used to calculate S(Figure 10.6)
Slide 28
Degrees of freedom can be defined in general as the number of
observations minus the number of constants calculate from the data.
According to equation(10.3), X has N degree of freedom, whereas by
equation (10.7), S has N-1 degree of freedom because one constant,
X, is used to calculate S. The answer to the question how typical
is a single observation of X, is, in terms of S and t, (to a given
probability p) X= T v,p S (10.8)
Slide 29
Statement(10.8) indicates that will be included in the
interval, to the probability p. The counterpart of statement (10.8)
was given in statement 10.6 in terms of. the interval of statement
10.6 can be generalized in terms of the normal deviate z as XZp (to
a given probability P) The plus and minus quantities in statements
10.8 and 10.9, that is, tS and z,should be recognized as the
confidence intervals on the individual measurements of X.
Slide 30
Slide 31
Slide 32
Slide 33
Appropriate values of z are given in Table 10.2. The z and
statistics are compared in Figure 10.6. Several Sets of Small N. A
useful measure of scatter in multiple set experiments is called the
precision index of the mean. This is patterned after equation 10.5
and 10.7 and is defined in general as (10.10) where M is the number
of sets involved and of the ith set, is the mean and is the number
of observations. The factor is the grand average of the M sets,
defined in general as
Slide 34
(10.11) which is naturally to be used in all multiple set
experiments as the best estimate of the true value of the
population mean at a given input. When a common number of
observations to all M sets equations(10.10) and (10.11) reduce to
(10.12)
Slide 35
And (10.13) where the subscript N signifies a common set size.
It is an observable feet that the means to of different sets of
measurements from the same population are always much closer to
each other than values of a single set. Equation (10.12) defines
the precision index of a set of M values of, whereas equation
(10.7) defines the precision index of a set of N values of X.
Statistical theory gives us an important relation between these two
statistics, namely,
Slide 36
(10.14) where N is consistently the number of observations
common to all M sets. Equation (10.14) says in effect: the average
value of a set has more precision than any of its parts by the
factor. Patterned directly after equation (10.14) is the precision
index of the grand average of M sets of observations, which can be
given as (10.15)
Slide 37
Equation (10.15) presents one apparent problem: which should be
used in a multiple-set experiment? One answer is to choose any one,
at random, but this leads to wide variation in. A more satisfying
answer, and the one recommended here, is to define a weighted
average of the weights being the appropriate degrees of freedom,
that is, replace the of equation (10.15) with defined as
(10.16)
Slide 38
On the basis of the t statistic, best estimates of the interval
with contains the true average can be given in (10.18) Since it is
often common practice to keep all sets of the same size, that is,
to keep the same for all sets, and since this practice assures us
that is a minimum equation 10.6 can be rewritten for the cast of an
N common to all M sets as (10.8)
Slide 39
The subscripts of t indicate degrees of freedom and probability
p. and the quantity indicates the confidence interval CI of the
best value estimate. Note that degrees of freedom is now given by
since not one but M were obtained from the data. Several examples
are given here to illustrate the ideas embodied thus far in these
statistical relations. And in terms of the mean of M sets of
measurements as (10.19)
Slide 40
which is the best estimate of. By equation 10.7 which is the
precision index of the mean. By Table 10.1 Example 1 For the
observations 7,8,7,6,5,6,7,8,6,9,8.find the best estimate of the
mean. The precision index of the mean the precision index of the
mean and the 95% confidence interval statement for [8]. Solution.
By equation 10.3,
Slide 41
which is the t statistic for the 95% confidence interval
statement. Hence according to equation 10.18 should be in the
interval Example 2 Express the best value and its 95% confidence
for a single sample experiment of eight observation. Solution
According to equation 10.18 and Table 10.1. at N=8 and v=7
Slide 42
Hence, At 95% Example 3 If five sets of the type of Example 2
were taken, how much more confidence could be placed in the best
value? Solution According to equation 10.19 and Table 10.1, at
MN=40, v=MN-M=35 and
Slide 43
Hence at 95% When the results of Examples 2 and 3 are rationed,
there results which indicates that the confidence interval could be
tightened by a factor of about 3 for the multiple-set case.
Slide 44
Example 4 From these sets of five measurements each, the
following table derived: Find the best estimate of and its 95%
confidence interval. Solution By equation 10.13 By equation
10.17
Slide 45
By equation 10.19, Hence is within the interval of the time
Range In addition to the CI statements based on, and tS, a third
type of CI statement can be based on the range estimate of S. The
range R is defined as the difference between the largest and small
measurements in a set, and can be used to estimate the precision
index S, that is, (10.20)
Slide 46
the average range of M sets, defined in turn by (10.21) For N
common to all M sets, and where d 2 is tabulated as a function of N
in Table 10.3. There is a loss of degrees of freedom with this
technique, and the estimate of S is less precise than those given
above, but the range estimate of S as given by equation ( 10.20) is
often convenient. Sometimes the range of a single set is used, via
equation 10.20, to estimate S[10]-[12], and occasionally d 2 is
used in place of in equation 10.20, where the degree of
approximation can be determined via Table 10.3.
Slide 47
Avoiding the determination of S entirely, confidence interval
statements for the case when the range is being used can be given
in terms of a substitute t statistic as the counterparts of
equation 10.8 and 10.19 as (10.22) For the single set, and (10.23)
In terms of the mean of M sets of measurements. Example 5 Estimate
the number of range by which can depart from by reference to Table
10.4 and equation 10.22 for N=2,3 and 4.
Slide 48
Solution For two observation in the sample, Ranges at 95%
Ranges at 99% For three observation in the sample, Ranges at 95%
Ranges at 99% Here we note how dramatically the confidence interval
tightens with one additional measurement. For four observation in
the sample, Ranges at 95% Ranges at 99%
Slide 49
Slide 50
Thus we conclude that the use of three readings over two
greatly improves our understanding of the dispersion of X, and the
worth of additional measurements becomes primarily an economic
question. Example 6 For two sets of five measurements each, the
following table is derived. Find the best estimate of and its
confidence interval. Solution By equation 10.13,
Slide 51
By equation 10.21, Hence is within the interval 0.24920.4178
95% of the time. Sample of small N with known Often can be
considered known in the sense of being established by experience.
In such favorable situations, single small sample can yield
reliable estimates.
Slide 52
For example, in terms of, the counterparts of equations (10.18)
(10.19) and of equations (10,22),(10.23) are (10.24) for the single
set and (10.25) for multiple sets. All of the confidence intervals
developed thus far are summarized in Table 10.5
Slide 53
Example 7 A certain temperature measurement yields an average
value of 150.75, with 95% assurance. What confidence interval
statement can be made concerning the true temperature? Solution
Based on a single measurement, according to equation(10.9) and
Table 10.2, should be in the interval. T1.96 95% of the time
Slide 54
Based on the mean of four measurements, according to equation
10.24, should be in the interval 95% of the time or with 1.96 =0.5,
we can write the interval as Thus can be given as (95%)` In words,
the most believable value of T is 150.75. furthermore, 95% of the
time the true temperature is believed to lie between 150.5 and
151.0.
Slide 55
10.3 UNCERTAINTY OF A SINGLE PARAMETER In section 10.1 both
systematic and random errors are considered. In Section 10.2 we
dwell at length on the confidence interval method. While systematic
errors must be estimated by no statistical methods. It follows that
these two types of errors should not be joined together lightly.
Indeed the best procedure to follow in describing the uncertainty
in a measured parameter is to quote the random and systematic
errors separately, and let it go at that..
Slide 56
However, there are often times in engineering when a single
number is required to describe the uncertainty of a set of
measurements. That is the random and systematic error accompanying
repeated measurements of a given parameter (like temperature or
pressure) are often combined to yield a single number for the
uncertainty. the number formed by combining the bias error B and
the precision error is called the uncertainty U. Since there can be
no rigorous basis in statistics for the required relation defining
uncertainty, its formulation must remain arbitrary. So it is not
surprising that two definitions for uncertainty are in common
usage[13].
Slide 57
The most conservative model for uncertainty is (at 95%) (10.26)
where the subscript ADD signifies that the bias and precision
errors are simply added and the 99% figure in parentheses indicates
the percent coverage of the true value expected of this model. A
more realistic estimate for uncertainty, presuming that there will
be some beneficial canceling of the errors or,to say it another
way, assuming that all errors will not be in the same direction, is
based on the familiar root-sum-square model as (at 95%)
(10.27)
Slide 58
where the subscript RSS signifies that the bias and precision
errors are combined by taking the square root of the sum of the
squares of these errors and the 95% figure in parentheses indicates
the percent coverage expected by this model. Either model indicates
the expected error limit of a measured parameter for a given
coverage, that is the uncertainty U is our best estimate of the
total error of equation 10.2. The coverage indicates the expected
probability that the interval defined by the best value of the
parameter plus ormolus the uncertainty, U, will include the true
value. The uncertainty interval is shown in figure 10.7.
Slide 59
In engineering work, the RSS model of equation (10.27) is used
most often. and is she approach recommended here for combining
systematic and random errors. Sometimes the mode is called the
probable uncertainty.
Slide 60
Example 8 A given parameter P is measured with an estimated
bias B of 2 units and a calculated precision index S of l unit when
the number of observations N is 10. Give the uncertainty interval
statements based on 99% and 95% coverage. When the number of
measurements of a given parameter is extreme small, or when no
statistical information is available, one must estimate the
uncertainty U in place of the calculated uncertainty of equation
10.26 and 10.27.
Slide 61
In such cases the uncertainty represents the experimenters best
estimate of the maximum error to be reasonably associated with the
parameter. Thus one could say, for example,that based on
experience, the uncertainty of a temperature measurement is (95% of
the tithe) without reference to any particular set of measurements,
without application of the statistics. Or one could that all flow
measurements made with uncalibrated nozzles can be counted on to
(95% of the time). and use this as the uncertainty
Slide 62
Although it is true that the uncertain thus conceived includes
both systematic and random errors the idea of separating these
errors and dealing with them separately is too arbitrary to be
practical, and we recommend considering such uncertainties as
describing systematic errors alone.
Slide 63
10.4 PROPAGATION OF MEASUREMENT ERRORS INTO A RESULT Often a
result r is derived by combining a number of independent parameters
according to some functional relationship (10.28) where (10.29) and
the subscript J indicates the number of parameters involved.
Slide 64
In some cases important question must be considered, namely,
how are the measurement of independent parameters propagated into
the result? To ask this in another way what are the precision and
bias errors of a derived result? The uncertainty interval of
interest is now (10.30) where patterned after
equations(10.26)and(10.27) the uncertainty of the result is defined
as either (10.31) or (10.32)
Slide 65
A recognized concept of statistics provides the answer to one
of our question The precision index of the result is given by
(10.33) On an absolute basis, and by (10.34) Note once again the
use of the root-sum-square principle. The so-called sensitivity
factors of equations (14.33) and (10.34), namely the absolute
Slide 66
(10.35) and the relative (10.36) Must be evaluate (analytically
or numerically) and used as multipliers of the precision indices Si
of each of the parameters. When all parameters have sample sixes
greater than 30, the number of degrees freedom of S, is -2, as seen
by reference to Table 10.1. For all usher cases the number of
degrees of freedom of S, is determined by the Welch-Satterthwaite
equation [15], [16]:
Slide 67
(10.37) where If equation(10.37) results in a non-integer
number it should be rounded downward to the next integer. By
analogy with equations (10.33) and (10.34), but with little basis
in statistics, the bias of a result [17] is given by (10.38) as an
absolute basis and by (10.39)
Slide 68
It follows from equation 10.35 and 10.36 that and The mean
values, the sensitivity factors, and the biases for each of the
parameters are tabulated below
Slide 69
It follows from equation 10.40 that determine the bias of the
result in absolute and relative terms. Solution By equation 10.38
By equation 10.39
Slide 70
As a check on consistency, B can be determined from b/r as
Example 10 Using the relationship (10.49) between result and
parameters, and the values tabulated in Example 9, estimate the
precision error of the result in absolute and relative terms if the
Following information applies:
Slide 71
Solution ] By equation 10.33 By equation 10.34 Check
Slide 72
To get the precision error of the result (i.e., t95S,), it is
necessary to determine first the number of degrees of freedom of S,
via equation (10.37)
Slide 73
And rounded downwards. By Table 10.1 we have for the result and
the precision error PE of the result on an absolute basis is On a
relative basis, Example 11 If the instrumentation used in
determining the measurements of the three parameters in equation
(10.40) was such that both the systemic and the random errors of
Examples 9 and 10 applied, what would be the maximum and probable
values of the uncertainty of the result on an absolute and relative
basis?
Slide 74
Solution On an absolute basis, by equation 10.31 By equation
10.32 On a relative basis