Summary of Coleman and Steele uncertainty methods

7/25/2019 Summary of Coleman and Steele uncertainty methods

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Summary of Coleman and Steele and how we might use what they present in ourcase

CHPT1 summary:Two main type of errors in experiment depending on their values (P.9):

rrors that stay constant during the experimental period (systematic or!ias ) rrors that vary with the experimental period (random)

The uncertainty that we are interested in and what we report is an estimateregion in which the actual error in the experimental result might lie.

"mportant concept is the standard uncertainty which is: an estimate of thestandard deviation of the parent population from which a particular elementalerror originates.(P.#$)

rror sources that vary !etween measurements are random and they are all

included in the calculation of Sx. rror sources that do not vary !etweenmeasurements are not included in Sx. (P.#$)

Standard uncertainties are either type % or &: (P.#')Type %: has the sym!ol s and is evaluated !ased on statistical analysis fromseries of measurementsType &: has the sym!ol u and is evaluated !y means other than statisticalanalysis.

xpanded uncertainty has the upper case sym!ol and is associated withcondence level

xpansion from measurement uncertainty to *experimental uncertainty+ (P.#,)Sometimes the uncertainty specication must correspond to more than an estimate ofthe *goodness+ with which we can measure something. This is true for cases in which the-uantity of interest has a varia!ility unrelated to the errors inherent in the measurementsystem.

ven if the operating condition is steady there will !e time variations in themeasured -uantity that will appear as random errors. %lso the ina!ility to resetthe system to the exact operating condition from test to test will cause test to testvariations and hence more data scattered.

/eplication level (0ery "mportant)(P.#9)

"n terms of uncertainty analysis replication levels are '

1eroth2order: happens when one test is collected (li3e one 456 pointscollection) and detailed uncertainty analysis is done. The uncertaintyo!tained from this results give rise directly to the errors and accuracy inthe measuring system. 7ence only the measuring system errors is shownin this analysis. This is the !est uncertainty that we can get

8irst2order: this is when multiple test is compared (three tests at the same

pro!e angle). The uncertainty given !y the three tests will include themeasuring system errors plus any other inuencing factors that varies


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!etween tests to tests. "t is expected that the uncertainty at this level ishigher than the eroth2order since it contains more error sources. ;otehere a!out systematic uncertainty(mentioned in C7PT,): sometimes thesystematic uncertainty doesn%? system is used large num!er of sample can !e gathered in short time. "nthis short time it might !e possi!le that not all the random factors that inuencethe measured varia!le !e present at the time of collecting (due to long timescale of that error source). Coleman and Steele suggest that the mean of onecollection should !e regarded as one measurement of the varia!le.

Pooled standard deviation e-uating (-. $.$6) (P.55)

/eview section $25 (P.5@) for outlier reAection criterion

Systematic standard uncertainty can !e from diBerent parent distri!ution (suchas aussian rectangular and triangular). >epending on the parent distri!utionproper value of the standard deviation should !e used(P.,$ and P.,').

;ote that the standard systematic uncertainty has the same values regardless ofwhether the measured varia!le is a single reading or a mean value. The averagingprocess does not aBect the systematic uncertainties(P.,5).

The way to com!ine the systematic and random standard uncertainty is given in -. $.$,(P.,5)

To associate condence interval level the !oo3 recommends using the coverage factoras -. $.$4 (P.,5)

The value of the coverage factor depends on the error source parent distri!utions !ut itis argued that the central limit thermo permits the use of the t student value as the valueof the coverage factor. The degrees of freedom for the coverage factor can !e o!tainedusing -. $.$=. please revise %ppendix C in which the authors show that it is very goodassumption and very applica!le in many engineering applications that the large sampleapproximation is used and no need to worry a!out #. $.$=. so if 9,D condence intervalis needed the coverage factor can !e ta3en as $. (P.,5)

CHPT3 summary:


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rror propagation (P.4'):Ehen propagate error through a >/ correlated error !etween severalmeasured varia!les most li3ely to !e present. This hold true for !oth systematicand random error although random error are usually uncorrelated with diBerentvaria!les !ut they can !e due to unsteadiness in the measured process.

rror propagation of a result can !e done using -. '.#' instead of the usualerror propagation formula. The !enet of using -.'.#' is that any correlatedeBect due to unsteadiness for instance is already captured in -. '.#'. it isrecommend that results from -. '.#$ and '.#' !e compared. >iBerences!etween the two e-uations suggest correlated errors (P.45).

Section '2#., suggests that FG increment is not necessarily the *uncertainty inthe varia!les+ as what 8ig suggests. "t seems also that it is a good chec3 tochange the increment and see if the results of the derivatives converged or ischanging (P.@6)

"n using HCH for error propagation correlated error eBects is straightforward toimplement (P.@5)

&e aware when using HCH in estimating the expanded uncertainty. after nishing thesimulation loo3 at histogram and ma3e sure that there is no s3ewness in the resultotherwise the I2 uncertainty won


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"n steady2state tests where the variations of the measured varia!les are not correlated(sr )TSH and (sr )directshould !e a!out e-ual (within the approximations inherent in theTSH e-uation). 7owever when the variations in diBerent measured varia!les are notindependent of one another the values of sr from the two e-uations can !e signicantlydiBerent. % signicant diBerence !etween the two estimates may !e ta3en as anindication of correlated variations among the measurements of the varia!les and the

direct estimate should !e considered the more *correct+ since it implicitly includes thecorrelation eBects.(P.#$@)

Data sets for determining estimates of standard deviations of measured variablesor of experimental results should be acquired over a time period that is large relative tothe time scales of the factors that have a signicant inuence on the data and thatcontribute to the random uncertainty. "n some cases thismeans random uncertaintyestimates must !e determined from auxiliary tests thatcover the appropriate time frame(P. #$= and #$9).

calculating and interpreting sr!ar (P.#$9)sing high2speed data ac-uisition systemsone could ta3e an increasingly large num!erof measurements during short time ma3ing sr!ar goes to ero. How useful theseestimates would be from an engineering viewoint would!ertainly be oen to"uestion#

8or a digital output the minimum 9,D random uncertainty in the reading resulting fromthe reada!ility (assuming no ic3er in the indicated digits) should !e ta3en as one2half ofthe least digit in the output. $f !ourse% therandom un!ertainty !ould besigni&!antly less than this value. 'hen there is no(i!)erin the output of a digital instrument (at a steady2state condition) the randomerrors are essentially damed by the digiti*ing ro!ess. (P.#'6)

review the example in section ,2'.# (P.#'6)

due to correlation !etween the two pressure stations random error in the dischargecoeMcient has correlation terms !etween the pressure. ;ot accounting for correlation inthe usual error propagation formula caused the error to !e very high compared to thedirect method of calculating sr. see ta!le ,.$ (P.#'$) for num!er comparison.

/eview example in section ,2'.' (P.#',)

>epending on the sign of the partial derivative of the correlated term uncertainty thedirect method of calculating sr can !e either smaller or larger from sr determined fromTSH.

"n the authors< experience in many timewise tests on complex engineering systems(such as one of the roc3et engine tests in Section ,2'.' or determination of the drag on a

ship model during one run down a towing tan3) the result of the test should !econsidered a single data point since it is ac-uired over a time period that is shortcompared to the variations that inuence the system !eing tested. "nsuch situations sG!ar and sr!ar have little engineering meaning or use particularly if in theshort period data are ta3en at a high rate and the #JN; factor is used to drive sG!ar andsr!ar to ar!itrarily small num!ers.(P.#'=)

systemati! errors:

"n some cases !ecause of time or cost the measurement system is not cali!rated in itstest conguration. "n these cases the systematic errors that are inherent in theinstallation process must !e included in the overall systematic uncertainty determination(P.#56).(Hay!e cali!rating the 8ive2hole !y applying the pressure to the actual pro!e instead ofthe current method would eliminate the installation errors)


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"n the a!sence of cali!ration the instrument systematic uncertainty should !e o!tainedfrom the manufacturerepending on the information availa!le sometimes we estimate !G to !e constant (i.e.not a function of G) and this value is often reported as a *D of full scale.+ (P.#5$). (somay!e what is given to us !y the manufacturer are the systematic error of theinstrumentOO)

Systematic uncertainty in the %J> converter (P.#5$)The 9,D condence uncertainty resulting from the digitiing process is usually ta3en to!e QS& (least signicant !it). The resolution of an QS& is e-ual to the full2scale voltagerange divided !y $; where ; is the num!er of !its used in the %J> converter. Thesystematic uncertainty associated with the digitiing process is then ta3en as QS&. Thissystematic uncertainty will !e in addition to the other !iases inherent in theinstrumentation system +su!h as gain% *ero% and

nonlinearity errors,#

"t is highly recommended that the >%? system !eing used has full range voltage close towhat the >%? system is !eing as3ed to read. This reduces the systematic error due todigitiing. (P.#5$)

Correlated systemati! error +P#145,:$bviously% the systemati! errors in the variables measured with the sametransdu!er are not indeendentof one another. %nother common example occurswhen di-erent variables are measured using di-erent transdu!ers% all of whi!hhave been !alibrated against the same standard a situation typical of theelectronically scanned pressure (SP) measurement systems in wide use in aerospacetest facilities. "n such a case at least a part of the systematic error arising from thecali!ration procedure will !e the same for each transducer and thus some of theelemental error contri!utions in the measurements of the varia!les will !e correlated.

stimation of the covariance terms for the correlated systematic errors in Gi and in G3 isgiven !y -. ,.#@ (P.#5,).

/eview example ,25.$.# (P.#5@) it shows how to consider correlated systematic error inuncertainty analysis.

that the eBects of systematic uncertainties do not cancel out in comparative experimentsin which the result is the diBerence in the results of two tests. however the systematicuncertainty can !e signicantly less than the systematic uncertainty in the result of

either test a or !.(P.#,5)

Sections ,2, and ,24 have extremely useful examples and can !e used as a guide.

0ery "mportant and useful (P.#4, and #44)

The utility of a rst2order replication comparison in the de!ugging phase of anexperiment uses the following logic. "f all the factors that inuence the random error ofthe measured varia!les and the result have !een accounted for properly in determiningsr then the random standard un!ertainty obtained from the s!atter in theresults at a given set oint should be aro.imated by sr. 7ere sr is the randomuncertainty of the result determined from propagating the estimated random uncertaintyof the measured varia!les using the TSH or an sr directly calculated previously in similar

tests. /f the random un!ertainty in the results is greater than the anti!iatedvalues% this indi!ates that there are fa!tors in(uen!ing the random error of the


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e.eriment that are not roerly a!!ounted for. This should !e ta3en as anindication that additional de!ugging of the experimental apparatus andJor experimentalprocedures is needed.

/andom order of test point settings rather than a se-uential order is preferred. This isdone to reduce hysteresis eBects. (P.#@')

CHPT0 summary:

>iBerent use of the regression model (P. $$# and $$6):

#. Some or all (Gi Ri) data pairs from diBerent experiments$. %ll (Gi Ri) data pairs from the same experiment'. ;ew G from same apparatus5. ;ew G from diBerent apparatus,. ;ew G with no uncertainty

"f the systematic standard uncertainties for the (Gi Ri) data and the (GiI# RiI#) data are

o!tained from the same apparatus and thus share the same error sources theirsystematic errors will !e correlated.(P.$$6)

P m0 I CThe uncertainty in 0new includes the uncertainty in the cali!ration curve as well as theuncertainty in the voltage measurement 0new. "f the same voltmeter is used in theexperiment as was used in the cali!ration the systematic error from the new voltagemeasurement will !e correlated with the systematic error of each 0i used in nding theregression and appropriate correlation terms are needed. (P.$$6)

Ee use same >%? for !oth cali!ration and collecting actual data so correlatedsystematic error shows up.

ften this perfect intercept is R 6 at G 6. This perfect value of the intercept shouldnever !e forced on the data. (P.$$$)

-s. @.$@ and @.$= represent the full uncertainty of the cali!ration process. "t includeserrors in each standard pressure we use plus the errors in the voltage reading using theGducer and >%? (!oth systematic and random) (Gducer has random uncertainties in theoutput voltage and the >%? system has small errors in reading the output voltage).Correlation errors !etween the diBerent measured pressures do exist (!oth systematicand random) and correlation errors !etween diBerent measured voltages do exist as well.;ote that the rst summation terms in -s. @.$@ and @.$= can !e given !y -s. @.$# and@.$$ if we !elieve that the errors are the same for each data point used in the cali!ration.

(P.$$4).

Summary of Coleman and Steele uncertainty methods

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