-
ExampleOneFacility
ExampleTwoFacility
1. Experimental standarddeviation, kg~/sec (s)
2. Degrees of freedom (v)
3. Systematic error,
kg~/sec (B)
4. Uncertainty, kg~/sec
0.078 7
>30
0.245 7
0.40
0.076 2
>30
0
0.22
Annex B Examples on estimating uncertainty in
open channel flow measurement
B.1 General
Evaluation of the overall uncertainty of a flow in anopen
channel will be demonstrated by considering (1)the velocity-area
method and (2) the weirs method.
The method of measuring the flow is such that it isimpractical
to eliminate interdependent variablesfrom the equation before
estimating flow uncertainty.Therefore, it involves evaluation of
the interdepen-dent uncertainties specified in 7.4. In addition,
mea-surement conditions often make it impossible toobtain the
replicate measurements needed forevaluation of experimental
standard deviations.Thus, it is desirable to express the random
errors aswell as the systematic errors as error limits. Underthese
conditions, it also is appropriate to assume thatall the random
error limits are equivalent to twoexperimental standard deviations.
Under this as-sumption, the random error limits can be
propagatedwith each other by means of the same root-sum-square
formulas as the systematic error limits (seeequations 19-22).
B.2 Example one velocity area method
B.2. 1 The equation for discharge in an open
channel velocity areaThe channel cross-section under
consideration isdivided into segments by m verticals. The
breadth,depth and mean velocity associated withany vertical iare
denotedby b1, d~and V, respectively. (see figure18) The product Q1
b,d~represents an approxima-tion to the discharge (volumetric flow
rate) in the i-thsegment. The sum over all segments,
39
i-i (78)
Table 12 Error comparisons of examples oneand two Q. = bd.
__________________________________________ ________________
___________ 1=1 iI (76)
represents an estimated or observed value of the
totaldischarge.
If x and y are respectively horizontal and verticalcoordinates
of all the points in the cross-section, andA is its total area,
then the precise mathematicalexpression for Q~,the true volumetric
flowrate (dis-charge) across the area, can be written as
ffA v(x,y) dx dy (77)The true discharge and the observed
discharge arerelated by a proportionality factor representing
theapproximation of the integral equation (77) by thefinite sum
equation (76), thus:
Q~= Fm Q~0= Fm Z
where
F~= [hA v(x,y) dx dy ] / L~b~d~~]In practice, Fm can be
evaluated from analysis ofmeasurements in which m is sufficiently
large for theeffects on Q~0of omitting verticals, in stages, to
bedetermined. Fm is subject to a random uncertainty.
It may be convenient in practice to take an Fmvariation with m
that is a mean value of values forsections of several different
rivers, taken together.Then the actual variations of Fm from river
to river,as compared with the meaned variation, will involveboth
systematic and random errors.
Fm is dependent on the number of verticals m, andtends to unity
as m increases without limit. Thus,equation 78 can be written
approximately as
= ~ (b~d~)t=1 (i9)
with increasing accuracy as mincreases.
This last form is the one that is given in Iso 748.
0260,
-
B.2.2 The overall uncertainty of the flowdetermination
It is plausible to assume that, at a given m, F and Q~canbe
treated as independent variables.
However, the Q1 in principle are not independent ofone another,
since the value corresponding to any onevertical will be related to
the values of adjacentverticals. Furthermore, there is an
interdependencebetween the d~and V, corresponding to any
particu-lar vertical. Thus, applying the principles for combin-ing
random errors (see clause 5) and denoting randomerror by S, the
following expression for SQ. theuncertainty of Q, can be derived
from equation 78.
F SQ ~2 f S~ 12L Q~J L Fm JZn / Q. \2
iI ~tVO
I Sb. 12 1 Sd~ 12 1 S~12Lb~i~La~i ~L~]~Q2 ~ s~+~
[(-~)sd~~]~(80)
where S~arise from the interdependence between Q1and and S5~from
the interdependence between d1and ~.
It is convenient to introduce the notation S forrelative random
error.
Thus Sbjbj is written S~.,SF /Fm is written SFand,
ne~lectingS~,and Sd~,~uation (80) becom~s
S~= St.I- ~ (S~,~S~,+S~)
If the relative errors Sb are all nearlyenough equal, ofvalue
5b~and similarly for the S~and Sd~, then
S~= S~( S~+ S~+ S~) ~ (Q1/Q~0)2
If the verticals are so located that Q1 Q~,0/m,then
~
In multi-point velocity-area methods, velocity ismeasured at
several points on a vertical, and themean value is obtained by
graphical integration or asa weighted average. The latter treatment
can beexpressed mathematically for a particular value as
(81)
= ~
where the w~are constant weighting factors. Thesuffix i that
identifies the particular vertical isomitted to simplify the
symbolism. The points usuallyare chosen so that Z w~= 1. This
equation can alsorepresent the single-point method, by taking k =
1.
In all cases, the estimates ~ so computed are subjectto errors.
These errors are due to improper placementof the meter at depth and
to deviations of the actualvelocity profile from the presumed
profile. The effectof these errors can be expressed by means of
amultiplicative coefficient P analogous to the coeffi-cient F~used
for similar purposes in equation (78).The same analysis that led to
equation (80) thenyields the following expression for relative
randomerror of the average velocity ~:
S~= S2 + S2 I (w~v~V p v ~Ik pVp)
in which S denotes relative random error in thesubscript
variable, v is measured point velocity, andthe ratio of wv-sums
expresses the variability ofweighted ~~elocityover the depth of the
vertical, For auniform k-point velocity profile, this ratio
wouldequal 1/k. For an extremely non-uniform profile, inwhich a
single term dominated all the others, theratio would equal 1. The
latter value is adopted, atleast for small k values, for the sake
of conservatism,with the result
Si = S~S~
This choice also helps to represent the effect of
anyunaccounted-for correlations among point-velocityerrors in the
same vertical.
In practice, the random error in the velocity measure-ment at a
point is assumed to be due to a meter-calibration random relative
error, S~,together with astream pulsation random error Se. Then the
random
(82) relative error for point velocities is
sc~= s,~+ s~
The corresponding random relative error for averagevelocity in
the vertical is
40
-
Si = S~+ S~+
B.2.3 Calculation of uncertainty
It is required to calculate the uncertainty in acurrent-meter
gauging from the followingparticulars:
Number of verticals used 20
Exposure time of currentmeter at each point inthe vertical 3
mm
Number of points taken inthe vertical (singlepoint, two points,
etc.) 2
Type of current meter rating(individual or group) individual
Average velocity in measuringsection above 0.3 rn/s
Details of procedure are described in ISO 748.
The random and systematic errors are combined bythe
root-sum-square method as stated in 8.3, i.e., ifSQ and BQ are the
percentage overall random andsystematic relative errors
respectively, then UQ, thepercentage uncertainty in the current
meter gauging,is
UQ = \i( 2S~)2 + B~and UQ~= BQ + 2SQ
B.2.3. 1 The error equation used for evaluating theoverall
random error is (see equation (82).)
SQ I 1= S;+-(S~S~~2~L~~)where
SQ is the overall percentage random error
Sm is the percentage random error due to thelimited number of
verticals used;
5b is the percentage random error in measuringwidth of
segments;
Sd is the percentage random error in measuringdepth of
segments;
S~ is the percentage random error in estimatingthe average
velocity in each vertical
Zn + 5~+ SiI.,
(see equation (85))
where
S~, is the percentage error due to limited numberof points taken
in the vertical (in the presentexample the two-point method was
used, i.e.,at 0.2 and 0.8 from the surface respectively);
S~ is the percentage error of the current meterrating (in the
present example an individualrating was used at velocities of the
order of0.30 m/s);
5e is the percentage error due to pulsations(error due to the
random fluctuation of
velocity with time; the time of exposure in thepresent example
was three one-minute read-ings of velocity.)
The percentage values of the above partial errors atthe 95%
confidence level are tabulated in B.2.3.2.
The equation for calculating the overall systematicerror is
BQ = ~jBi + B~+ B~
where
BQ is the overall percentage systematic uncer-tainty in
discharge;
Bb is the percentage systematic error in theinstrument measuring
width;
B~ is the percentage systematic error in theinstrument measuring
depth; and
Bd is the percentage systematic error in thecurrent meter rating
tank.
The systematic errors in the current meter gaugingare confined
to the instruments measuring width,depth and velocity and should be
restricted to 1% asshown in B.2.3.2.
(83)
41
022)r
-
Discharge(Combined) uncertainty, UQ95(Combined) uncertainty,
UQ99Random error (2SQ)Systematic error (BQ)
Gauged head, hBreadth of weir, bCrest height, PCoefficient of
discharge, CdCoefficient of velocity, Cv
(Q) me/s5.9%7.4%5.7%1.7%
0.67mlOrnim1.1631.054
= 1.7%
The combination of both random and systematicerrors then gives
the overall percentage uncertaintyin discharge, UQ.
Taylor series analysis of the discharge equation yieldsthe
following uncertainty equations, which can beused for both random
and systematic errors:
0,2 ~t2 1) c~\2cw2~0~c,+0b~~/h) 0h
B.2.3.2 The values of the error elements affectinguncertainty in
discharge are tabluated below aspercentage errors at the 95%
confidence level. Thenumerical values are taken from ISO 748. It
isrecommended, however, that each user determineindependently the
values of the errors for any partic-ular measurement.
Table 13 Error elements affecting uncertainty indischarge
UQ Zn ~/(2SQ)2+ B~ U~~= BQ + 2SQ
= ~j572+ 1.72 Zn 1.7 + 57
Zn 5~9% Zn 7,4%
B.2.3.3 The discharge measurement may be ex-pressed in the
following form:
Error source Units
(2S)random
errorlimit
(2S95%)
(B)percentagesystematic
errorlimit
Fm, number of verticals
b, segment width
d, segment depth
number of profilepoints
v~,meter calibration
Ve, meter exposure time
m
m
rn/s
rn/s
rn/s
5.0
0.5
0.5
7.0
2.0
10.0
1.0
1.0
1.0
Then, the overall random error in discharge is givenby
Uncertainties calculated in accordance with ISO5168.
B.3 Example two weir measurement
&3. 1 Weir data
It is required to calculate the discharge and theuncertainty in
discharge for a triangular profile weirgiven the following details:
(see figure 19)
Zn 2~]~
Zn 4~i~1~(0.25+0.25494+100)
Zn 5~7%
The overall systematic error is
BQ Zn ~12 + 12 + 12
The discharge equation is
Q Zn (2/3)3/2 CdC~.,j~b h312 (84)
Details of the procedure are described in ISO 4360.
B.3.2 Uncertainty equations
42
-
and
8.3.3 Evaluation of discharge and uncertainties
The values of the error elements affecting thisproblem are
tabulated below as error limits at the95% confidence level. The
numerical values are basedon information given in ISO 4360. It is
recommended,however, that each user determine independently
thevalues of the errors for any particular measurement.(See table
14) -
B Zn B2 + B~+ (3/2)~B~Q ~ (85)
in which S and B denote percentage errors of thesubscript
variables.
Hejd gauging section - -
3 tO4h,,,,~
Slope 1 5
Figure 19 Triangular profile weir
Table 14 Error element values
Variable UnitsNominal
value
(2S)randomerror
limit(2S:95%)
-
(B)systematic
errorlimit
h
b.
CdC~
g
m
m
m/s2
0.67
10.00
1.226
9.81
0.0030.45%
0.
0.5%
0.
0.0030.45%
0.010.1%
1.5%
0.
43
)26~r
-
Substitution of the nominal values into the discharge
equation yields
Q = (3/2)3/2 x (1.226) x ~J~Ix 10 x (0.67)3~2
Zn 11.46 m31/s
Evaluation of the random errors yields
2SQ = 1(05)2 + (3/2)2 (0.45)~
= 1.65%
- Combining the random and systematic errors by
theroot-sum-square (RSS) method yields
UQ95 = ~J(2S~+ B~ UQw Zn BQ +
= ~(0.84)2+ (1.65)2 = 1.65 + 0.84
= 1.85% Zn 2.49%
Uncertainties calculated in - accordance with ISO5168.
Annex C Small sample methods
C.1 Students t.
When the experimental standard deviation is basedon small
samples (N 30), uncertainty is defined as:
U~D Zn B + t95S
U~8= ~JB2+ (t95S)
2
For these small samples, the interval t95S/~N,X + t95S/~N] will
contain the true
unknown average, ~t, 95% ofthe time. If the systemat-ic error is
negligible, this statistical confidence inter-val is the
uncertainty interval. t95 is the 95th percen-tile point for the
two-tailed Students t-distribution.For small samples, t will be
large, and for largersamples t will be smaller, approaching 1.96 as
a lowerlimit. The t-value is a function of the number ofdegrees of
freedom (v) used in calculating S. Since 30degrees of freedom (v)
yield a t of 2.05 and infinitedegrees of freedom yield a t of 1.96,
an arbitraryselection of ~ Zn 2 is used for simplicity for values
of vfrom 30 to infinity. See table 15.
C.2 Degrees of freedom for small samples
In a sample, the number of degrees of freedom (v) isthe sample
size, N. When a statistic is calculated fromthe sample, the degrees
of freedom associated withthe statistic is reduced by one for every
estimatedparameter used in calculating the statistic. For exam-ple,
from a sample of size N, X is calculated and hasN degrees of
freedom, and the experimental standarddeviation, 5, is calculated
using equation (1), and hasN-i degrees of freedom because X is used
tocalculate S. In calculating other statistics, more thanone degree
of freedom may be lost. For example, incalculating the standard
error of a curve fit, thenumber of degrees of freedom which are
lost is equalto the number of estimated coefficients for the
curve,N 2.
When all random error sources have large samplesizes (i.e.,
v~> 30) the calculation of is unnecessaryand 2 is substituted
for t95. However, for smallsamples, when combining experimental
standard de-viations by the root-sum-square method (see
equation(20) for example), the degrees of freedom (v) associ-ated
with the combined experimental standard devia-tions is calculated
using the Welch-Satterthwaiteformula (88).
(86)
(87)
Zn 0.84%
Evaluation of the systematic errors yields
BQ Zn ~~(1.5)~+ (0.lf + (3/2)z (0.45)~
B.3.4 Presentation of results
The discharge Q may be reported as follows:Discharge
6m3/s(Combined) uncertainty, UQ95 %(Combined) uncertainty, UQ~
2.5%Random error (2SQ) 0.8%Systematic error (BQ) 1.6%
44
-
(~S2)2j1 ii
VZn
3 K
j~1 i-i ~ii
Degrees of Degrees offreedom t9~ freedom
1 12.706 172 4.303 183 3.182 194 2.776 205 2.571 216 2.447 227
2.365 238 2.306 249 2.252 25
10 2.228 2511 2.201 2712 2.179 2813 2.160 2914 2.14515 2.13118
2.120
C.3 Propagating the degrees of freedom
The Students t value of table 16 to be used incalculating the
uncertainty of the test result (equa-tions (86) or (87)) is based
on Vr, the degrees offreedom of Sr. If the degrees of freedom of
anymeasurement standard deviation is less than 30, thedegrees of
freedom of the result also may be less than30. In such cases, the
following small sample method
may be used to determine Vr This is defined for theabsolute
experimental standard deviation accordingto the Welch-Satterthwaite
formula by:
Vr =
(9, S1, )4i-I Vp~~ (92)
For example: the degrees of freedom for the calibra-tion
experimental standard deviation (S1) given byequation (20),
is:(~)~
~ S~i-I ~il
________ (S~~S~S~1S~1)2 (89)S4 54 S4 54
_~~!!.+ + + If the test result is an average, X, based on a
sample
(88) of size N,
where v~1is the degrees of freedom of each elemental 5-
=experimental standard deviation in the calibration X
(90)process.
As ..,/F.~ is a known constant, the degrees of freedomThe
degrees of freedom for the measurement experi- of S~is the same as
S, i.e.mental standard deviation (S), as given by equation -(21)
is: =
(91)
Table 15 Two-tailed students t table
SMALL SAMPLE METHODSDegre~esof freedom
-
and for the relative experimental standard deviationby:
Vr Zn
where
(Sr/r)4
(9~S1,.. /F1)4Vpr
Sr Zn \/E (0~S~)2
NOTE: The degrees of freedom for the relative and -absolute
experimental standard deviations are identi-cal.
Welch-Satterthwaite degrees of freedom may containfractional,
decimal parts. The fractions should bedropped or truncated as
rounding down is conserva-
(93) tive with Students t, i.e. v = 13.6 should be treated asVZn
13.0.
Annex D Outlier treatment
D.1 General
Zn (N~ i)
-J
E
0.
(94)
All data should be inspected for spurious data pointsas a
continuing check on the measurement process.Points should be
rejected based on engineering analy-sis of instrumentation,
thermodynamics, flow profilesand past history with similar data. To
ease the burdenof scanning large masses of data, computerized
rou-tines are available to scan steady-state data and flagsuspected
outliers. The flagged points should then besubjected to an
engineering analysis.
The effect of these outliers is to increase the randomerror of
the system. A test is needed to determine if aparticular point from
a sample is an outlier. The testshould consider two types of errors
in detectingoutliers:
(1) Rejecting a good data point(2) Not rejecting a bad data
point
and the degrees of freedom of the experimentalstandard deviation
(Sr.) of the independent measure-ments isusually given l~y:
All measurement systems may produce spurious datapoints. These
points may be caused by temporary orintermittent malfunctions of
the measurement sys-tem or they may represent actual variations in
themeasurement. Errors of this type should not beincluded as part
of the uncertainty of the measure-ment. Such points are meaningless
as test data. Theyshould be discarded. Figure 20 shows a spurious
datapoint calledan outlier.
Spurious Data Point
x S S S a
X - x ~ Thandom_X Error~ X ~ X X X X~X X - Limits
x ~ __S a a S S S S S a a
Figure 20 Outlier outside the range of acceptable data
46
-
The probability for rejecting a good point is usually Table 16
Rejection values for Grubbs methodset at 5%. This means that the
odds of rejecting agood point are 20 to 1 (or less). The odds will
beincreased by setting the probability of (1) lower.However, this
practice decreases the probability ofrejectingbad data points. The
probability of rejectinga good point will require that the rejected
points befurther from the calculated mean and fewer bad datapoints
will be - identified. For large sample sizes,several hundred
measurements, almost all bad datapoints can b~identified. For small
samples (five orten), bad data points are hard to identify.
One test in common usage for determining whetherspurious data
are outliers is Grubbs Method.
D.2 Grubbs method
Consider a sample (X1) of N measurements. Themean (X) and an
experimental standard deviation(S) are calculated by equation (1).
Suppose that (X~)~the j-th observation, is the suspected outlier;
then, theabsolute statistic calculated is:
I X.-XTflL ~s
Using table 16, a value of T~is obtained for thesample size (N)
and the 5% significance level (P).This limits the probability of
rejecting a good point to5%. (The probability of not rejecting a
bad data pointis not fixed. It will vary as a function of sample
size.
The test for the outlier is to compare the calculatedT0 with the
table T0.
IfT~calculated is larger than or equal to T0 table, wecall X3 an
outlier.
If T0 calculated is smaller than T~table, we say X~isnot an
outlier.
Samsize
pieN
5%(1-sided)
Samplesize
5%(1-sided)
3 - 1.150 20 2.564 1.46 21 2.585 1.67 22 2.606 1.82 23 2.627
1.94 24 2.648 2.03 25 2.669 2.11 30 2.75
10 2.18 35 2.8211 2.23 40 2.8712 2.29 45 2.9213 2.33 50 2.9614
2.37 60 3.0315 2.41 70 3.0916 2.44 80 3.1417 2.47 90 3.1818 2.50
100 3.2119 2.53
26 79 58 24 1 103 121 22011 137 120 124 129 38 25 60148 52
216
12 56 89 8 29107 20 9 40 40 2 10 166126 72 179 - 41 127 35
334
555
suspected outliers are 334 and -555 (underlined).
To illustrate the calculations for determining whether-555 is an
outlier from figure 21.
555 1.125= 140.813 6 Zn 3~95
from table 16 using Grubbs Method for N Zn 40 5%level of
significance (one-sided),
T Zn 2.87
Therefore, since 3.95 > 2.87
(T~,~)> (T,~bl)
555 is an outlier according to Grubbs test.
D.3 Example
In the followingsample of 40 values,
Mean (X)Exp. Std. Dcv.
Sample Size
Zn 1.125= 140.813 6Zn 40
47
-
Suspectedoutlier
CalculatedTn
Table TnP5
Samplesize(N)
Experimentalstandard
deviation(s)Mean
X555 3.95 2.87 40 140.8 1.125
334 2.91 (stop) 2.86 39 109.6 15.385220 2.33 2.85 38 97.5
7.000
aa.
Figure 21 is a normal probability plot of this datawith the
suspected outliers indicated. In this case, theengineering analysis
indicated that the 555 and 334readings were outliers, agreeing with
the Grubbs testresults.
Figure 21 Results of outlier tests
Annex E Statistical uncertainty intervals
It is usually impossible to determine the
statisticaldistribution of the systematic errors (~)because theyare
usually subjective judgments, i.e. not based ondata. However, if
there is information to justify adistribution assumption, it is
possible to use rigorousstatistical methods to calculate the
uncertainty inter-val. The validity of this assumption must be left
tothe judgement of the reader. The purpose of thisannex is to
describe the methods, given the assump-tion.
E.1 Assumed systematic error distribution
If it is assumed that the systematic errors (B) areactually the
maximum possible upper and lower limitof the true, unknown
systematic error (13), and that 13is equally probable anywhere
within the limits, thenthe standard deviation of the systematic
error may bedetermined by
B=
As depicted in figure 22.
(95)
600
~
..3~Re1ot~.Mean 1.125000
- Std. 0ev. 140.83b- N - 40
Data a Not Normal
at 90 PCI. Confidence
-400
-600
1~
G
,
.~
-800
iFl~l~ti [I~.01 0.1- 1 lii
Cumulative Frequency . Percent99.99
F,
48
-
The validity of this assumption cannot be proved or Students t
and the Welch-Satterthwaite approxima-disproved. It is a matter of
judgement. tion will be needed as described in annex C.
E.2 URSS E.3 UADD
The systematic error limit of the measurement result With the
additive model of uncertainty, the assumedmay be calculated as
before distribution does not affect the answer. The system-
atic error, B, is still determined as equation (~6)andB Zn ~
(9.B.)2 there is no advantage to calculating a standard
V (96) deviation of systematic error.
The experimental standard deviation ofthe systemat- U~D Zn B +
t85sic error is estimated as: (99)
s B E.4 Monte Carlo exampleB (97)To illustrate the Central Limit
Theorem, the sum of arandom sample from each of the ten
rectangulardistributions with means zero was repeated 1000times. In
sets of three, the distributions had a Zn 0.5,
(98) 1.0, 2.0 respectively, and the tenth, a Zn 4.0. If
thetendency toward normality and the Monte Carlosimualtion were
both perfect
Figure 22 Thethe limit B.
>~
z
assumed frequency rectangular distribution of the systematic
error (13) as a function of
49
The uncertainty is
URSS Zn ~J(1.645S8)2
+
for large samples, where S is the experimentalstandard deviation
of the random error.
Assuming there are many sources of systematic andrandom errors,
~ay ten or more, the Central LimitTheorem states that sums of
samples taken from anydistribution(s) will tend toward normality.
Therefore,the true error () should be distributed as a
normaldistribution with standard deviation equal to
theroot-sum-square of the systematic and random errorexperimental
standard deviations. This will be illus-trated in E.4. If small
samples are used to estimatethe random error experimental standard
deviations,
a Zn V3(0.521.02+2.02)+42
= 5.585. -
The average S for 1000 trials was S Zn 5.671. Theresults are
shown in figure 23. The bell shape of thenormal distribution is
apparent. A goodness-of-fittest could not reject normality at the
90% level ofconfidence. .- -
B i~1EASLTRE~NTSCALE
0260,
-
30. 00.
Ls~)LUL)Z 20.00uJ
D
U-
z
10.00
LUa-
20.00 -15.00- OS/2a/83 15,30,12 BGR
Figure 23 Distribution of sum of 10 rectangular systematic
errors
Annex F: Uncertainty interval coverage
Introduction
A rigorous calculation of confidence level or thecoverage of the
true value by the interval is notpossible because the distributions
of systematic errorlimits, based on judgement, cannot be
rigorouslydefined. Monte Carlo simulation of the intervals
canprovide approximate coverage5 based on assuming
- various systematic error limits.
F. 1 Simulation results
As the actual systematic error and systematic errorlimit
distributions will probably never be known, thesimulation studies
were based on a range of assump-tions. The result of these studies
comparing the twointervals are:
* Coverage as used herein is the propOrtion of Monte Carlo
trialswhere the measurement uncertainty interval contains the
truevalue.
50
20.00 25.00 ~
a) U99 averages approximately 99.1% coveragewhile U95 provides
95.0% based on system-atic error limits assumed to be 95%.
For 99.7% systematic error limits, U99averages 99.7% coverage
and U95, 97.5%.
b) The ratio of the average U99 interval size toU95 interval
size is 1.35:1.
c) If the systematic error is negligible, bothintervals provide
a 95% statistical confi-dence (coverage).
d) If the random error is negligible, bothintervals provide 95%
or 99.7% dependingon the assumed systematic error limit size.
25.00.
aIDEAL = 5.585
BlB3 0.5 a84_86 1.0
aB7B9 2.0
c~BlO 4.0
1000 TRIALS
CALC = -0.0075= 5.67
5.00
0.0010.00 ~.00 0~00
SUM 5~00 10.00
0260,
-
Assumptions and Simulation Cases Considered
(1) From 3 to 10 error sources, both systemat-ic and random
(2) Systematic errors distributed both nor-mally and
rectangularly
(3) Random error distributed normally
(4) Systematic error limits at both 95% and99.7% for both the
normal and the rectan-gular distributions
(5) Sample standard deviations based on sam-ple sizes from 3 to
30
(6) Ratio of random to systematic errors at1/2, 1.0 and 2.0.
F.2 Non-symmetrical interval
If there is a non-symmetrical systematic error limit,the
uncertainty (U) is no longer symmetrical aboutthe measurement. The
interval is defined by theupper limit of the systematic error
interval (B)~.Thelower limit is defined by the lower limit of
thesystematic error interval (B). (see clause 7.3)
Figure 24 shows the uncertainty (U ) for non-sym-metrical
systematic error limits. (See table 17.)
Zn B~+ t95S
U=B_t95S
(100)
(101)
Table 17 Uncertainty intervals defined bynon-symmetrical
systematic error limits
B B~ t95s,
U~9(Lower limitfor U)
U,(Upper limit
for U)0 deg K +10 deg K 2 deg K 2 deg K +12 deg K
3 Kg +13 Kg 2 lb 7 Kg +17 Kgo 1~a +7 P3 2 P3 2 ~a8 deg K 0 deg K
2 deg K 10 deg K +2 deg K
51
-
MeasurementLargest Negative Error.
(B t95 S)
Uncertainty Interval
(The True Value Should Fall Within This Interval)
Figure 24 Measurement uncertainty; non-symmetrical systematic
error
52
