Iso Dis 5168part4
description
Transcript of Iso Dis 5168part4
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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,
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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,
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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
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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)
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022)r
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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
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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%)
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(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.
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)26~r
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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%
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(~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
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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
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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
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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,
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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
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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,
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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.
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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,
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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
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MeasurementLargest Negative Error.
(B t95 S)
Uncertainty Interval
(The True Value Should Fall Within This Interval)
Figure 24 Measurement uncertainty; non-symmetrical systematic error
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