Measurement Uncertainity

8/18/2019 Measurement Uncertainity

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Since there is always a margin of doubt about anymeasurement, we need to ask ‘How big is the margin?’

and ‘How bad is the doubt?’ hus, two numbers are really needed in order to !uantifyan uncertainty.

"ne is the width of the margin, or interval. he other is acon#dence level, and states how sure we are that the

‘true value’ is within that margin.

$e might say that the length of a certain stick measures%& centimetres 'lus or minus ( centimetre, at the )*'ercent con#dence level.

his result could be written+%& cm ( cm, at a level of con#dence of )*.

he statement says that we are )* 'ercent sure that thestick is between () centimetres and %( centimetres long.


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rror versus uncertainty

/t is im'ortant not to confuse the terms ‘error’ and

‘uncertainty’.rror is the di0erence between the measured valueand the ‘true value’ of the thing being measured.

1ncertainty is a !uanti#cation of the doubt about

the measurement result.$henever 'ossible we try to correct for any knownerrors+ for e2am'le, by a''lying corrections fromcalibration certi#cates. But any error whose value

we do not know is a sourceof uncertainty


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3easurement

‘3easure thrice, cut once’ ... operator error

here is a saying among craftsmen, ‘3easure thrice,cut once’. his means that you can reduce the risk ofmaking a mistake in the work by checking themeasurement a second or third time

before you 'roceed./n fact it is wise to make any measurement at leastthree times. 3aking only one measurement meansthat a mistake could go com'letely unnoticed.

/f you make two measurements and they do notagree, you still may not know which is ‘wrong’. But ifyou make three measurements, and two agree witheach other while the third is very di0erent, then youcould be sus'icious about the third.


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Basic statistical calculations

4ou can increase the amount of information you get from your

measurements by taking a number of readings and carrying out some

basic statistical calculations.

he two most im'ortant statistical calculations are to #nd the averageor arithmetic mean, and the standard deviation

for a set of numbers.

5etting the best estimate - taking the average of a number of readings

/f re'eated measurements give di0erent answers, you may not bedoing anything wrong. /t may be due to natural variations in what isgoing on. 67or e2am'le, if you measure a wind s'eed outdoors, it willnot often have a steady value.8 "r it may be because your measuringinstrument does not behave in a com'letely stable way. 67or e2am'le,ta'e measures may stretch and give di0erent results.8

/f there is variation in readings when they are re'eated, it is best totake many readings and take an average. 9n average gives you anestimate of the ‘true’ value. 9n average or arithmetic mean is usuallyshown by a symbol with a bar above it, e.g. 2 6‘2-bar’8 is the meanvalue of 2.


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Average / Standard deviation


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Where do errors and uncertainties come from

3any things can undermine a measurement. 7laws in the measurement maybe visible or

invisible. Because real measurements are never made under 'erfectconditions, errors and

uncertainties can come from+

; !he measuring instrument " instruments can su#er from errorsincluding bias, changes

due to ageing, wear, or other kinds of drift, 'oor readability, noise 6for electrical

instruments8 and many other 'roblems.; !he item being measured " which may not be stable. $Imagine tryingto measure the si%e of

an ice cube in a warm room.8

; !he measurement process " the measurement itself may be di&cultto ma'e. (or e)ample

measuring the weight of small but lively animals 'resents 'articular di


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perator s'ill " some measurements depend on the s'ill and judgement of the operator.

"ne 'erson may be better than another at the delicate work of setting u' a

measurement, orat reading #ne detail by eye. he use of an instrument such as a sto'watchde'ends on the

reaction time of the o'erator. 6But gross mistakes are a di0erent matter andare not to be

accounted for as uncertainties.8

; Sampling issues " the measurements you ma'e must be properlyrepresentative of the

'rocess you are trying to assess. /f you want to know the tem'erature at thework-bench,

don’t measure it with a thermometer 'laced on the wall near an air

conditioning outlet. /f you are choosing sam'les from a 'roduction line for measurement, don’talways take the

#rst ten made on a 3onday morning.

; !he environment " temperature, air pressure, humidity and many

other conditions cana0ect the measuring instrument or the item being measured.


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-andom or systematic

random " where repeating the measurement gives a randomlydi#erent result. If so, the

more measurements you make, and then average, the better estimate yougenerally can

e2'ect to get

systematic " where the same inuence a#ects the result for each ofthe repeated

measurements 6but you may not be able to tell8. /n this case, you learn nothinge2tra =ust

by re'eating measurements. "ther methods are needed to estimateuncertainties due to

Systema

!he two ways to estimate uncertainties tic e0ects, e.g. di0erentmeasurements, or calculations.

Type A evaluations - uncertainty estimates using statistics (usuallyfrom repeated readings)

Type B evaluations - uncertainty estimates from any other

information. This could beinformation from 'ast e2'erience of the measurements, from calibration


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ombining standard uncertainties

/ndividual standard uncertainties calculated by y'e 9 or y'e B evaluations

can be combinedvalidly by ‘summation in !uadrature’ 6also known as ‘root sum of the s!uares’8.

he result of

this is called the combined standard uncertainty, shown by uc or uc(y).

Summation in !uadrature is sim'lest where the result of a measurement isreached by addition

or subtraction. he more com'licated cases are also covered below for themulti'lication and

division of measurements, as well as for other functions.


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Having scaled the com'onents of uncertainty consistently, to #nd thecombined standard

uncertainty, we may then want to re-scale the result. he combined standard

uncertainty may bethought of as e!uivalent to ‘one standard deviation’, but we may wish to havean overall

uncertainty stated at another level of con#dence, e.g. )* 'ercent. his re-scaling can be done

using a coverage factor, k. Multiplying the combined standard uncertainty, uc ,

by a coveragefactor gives a result which is called the expanded uncertainty, usually shownby the symbol U,

i.e. U kuc

9 'articular value of coverage factor gives a 'articular con#dence level for thee2'anded

uncertainty.3ost commonly, we scale the overall uncertainty by using the coverage factork !, to give a

level of con#dence of a''ro2imately )* 'ercent. 6k ! is correct if thecombined standard

uncertainty is normally distributed.

Some other coverage factors 6for a normal distribution8 are+


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; he measurement result, together with the uncertainty#gure, e.g. ‘he length of the stick was %& cm ( cm.’

; he statement of the coverage factor and the level ofcon#dence. 9 recommended wording is+

‘he re'orted uncertainty is based on a standard uncertaintymulti'lied by a

coverage factor k !, providing a level of con#dence ofapproximately '&*.+


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; 1se of 'artial di0erentiation in

/f the calculated !uantity is a function of more than onevariable, then each variable needs to be considered for theircontribution.

-hen the calculated uantity is a function of the measuredvalues x,..., with uncertainties /x,...,/ then the uncertainty in

(x,...,) is obtained from


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et@s say, for e2am'le. that we are calculating by measuringthe

circumference and diameter of a disk. /f we measure both witha meter stick, and we assume that, we can measure to onehalf of the smallest marking on the meter stick.

3ost meter sticks go to (mm, so we@ll assume that ouruncertainty is &+* mm. So, for our uncertainty e!uation, f A ACD d, and 2( A 2% A &.&&*. aking this, we can calculate the

uncertainty in


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his is as far as we can go without using any numbers, so let@s saythat we measure the circumference to be E( cm and the diameterto be (& cm. Fow by using these values, we can get

his leads to a value of A&.&(G. So, the value of that E.( D-&.&(G mm

Measurement Uncertainity

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