ERRORS CW

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Electronic Measurements & Instrumentation.

Prof S.Lakshminarayana,M.Tech.,Ph.D.,

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FACT!S I" MA#I"$ MEAS%!EME"TS

The 'oo(ness of measurements in)ol)es

se)eral im*ortant conce*ts. Some of the moresi'ni+cant of these are

• Error,

• -ali(ity,

• !eliaility,

• !e*eataility,

• Accuracy,

• Precision,• !esolution.

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FIGURE 2-5: The accuracy of a

measurement is indicated by the size of ∆!

1. Error In all measurements there is a certain degree of error present.

The word error in this context refers to normal random variation

and in no way means "mistakes" or "blunders.”

o is the true value.

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The measured value may deviate from the true value

!o by a certain amount !# $

If measurements are made repeatedly on the same parameter !which is truly unchanging$

or if different instruments or instrument operators are

used to make successive measurements.

∆X = Absolute error.

X i = Measured Value

X o = True steady state value.

∆X = X i –X o If ∆X → 0, X i →X o

Relative Error

Percentage Error Percentage Error = E r X00

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Errors in Measurement

Systems

Error is the difference between measured value and the truevalue of a physical %uantity. The accuracy of measuring

system is measured in terms of error.

Errors may be positive !&eading higher value or negative

!reading lower value.

TYPES OF ERRORS.

1. Errors due to calibration

'. (uman Errors.

). *oading Error.

+. Environmental Error. !Temperature,. &andom Error.

-. Instrument Error.

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1. Accuracy: The maximum expecteddiference in ma'nitu(e et/een measure( an(

true )alues 0often e1*resse( as a *ercenta'e ofthe full2scale )alue3.2. Precision: The ability o the instrument tore*eat the measurement of a constantmeasuran(. More *recise measurements ha)e

less ran(om error.3. Resolution: The smallest possible increment(iscernile et/een measure( )alues. As the term isuse(, hi'her resolution means smaller increments.

Thus, an instrument /ith a +)e2(i'it (is*lay 0say,4.4444 to 5.55553 is sai( to ha)e hi'her resolutionthan an other/ise i(entical instrument /ith a three2(i'it (is*lay 0say, 4.44 to 5.553. 0Su6ect to thecon(ition that the source has the re7uire(resolution3

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Accurate but not precise.

The average is close

to the center but the

individual values are

not similar.

Precise but not accurate Accurate and Precise

Precision vs. Accuracy

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WHAT IS THE DIFFERENCE BETWEEN

ACCURACY AND PRECISION?

Accuracy is de!ined as "The ability o! a measurement to match theactual value o! the #uantity being measured". $! in reality it is %&.' F

outside and a temperature sensor reads %&.' F then than sensor is

accurate.

Precision is de!ined as "()* The ability o! a measurement to beconsistently reproduced" and "(+* The number o! signi!icant digits to

,hich a value has been reliably measured". $! on several tests the

temperature sensor matches the actual temperature ,hile the actual

temperature is held constant then the temperature sensor is precise. -y

the second de!inition the number %.)&) is more precise than the

number %.)&

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• An e/ample o! a sensor ,ith -A0 accuracy and -A0 precision1 Suppose

a lab re!rigerator holds a constant temperature o! %2.' F. A temperaturesensor is tested )' times in the re!rigerator. The temperatures !rom the

test yield the temperatures o!1 %3.& %2.) %3.% %4. %2.% %3.) %4.) %4.2

%2.2 %3.'. This distribution sho,s no tendency to,ard a particular

value (lac5 o! precision* and does not acceptably match the actual

temperature (lac5 o! accuracy*.

• An e/ample o! a sensor ,ith 6OO0 accuracy and -A0 precision1

Suppose a lab re!rigerator holds a constant temperature o! %2.' F. A

temperature sensor is tested )' times in the re!rigerator. Thetemperatures !rom the test yield the temperatures o!1 %4.2 %2.% %2.)

%2.' %4.7 %2.+ %2.' %2.' %4.& %2.%. This distribution sho,s no

impressive tendency to,ard a particular value (lac5 o! precision* but

each value does come close to the actual temperature (high accuracy*.

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• An e/ample o! a sensor ,ith -A0 accuracy and 6OO0 precision1

Suppose a lab re!rigerator holds a constant temperature o! %2.' F. A temperature

sensor is tested )' times in the re!rigerator. The temperatures !rom the test yield the

temperatures o! 1 %3.+ %3.% %3.) %3.' %3.) %3.% %3.+ %3.) %3.+ %3.+. This

distribution does sho, a tendency to,ard a particular value (high precision* but everymeasurement is ,ell o!! !rom the actual temperature (lo, accuracy*.

• An e/ample o! a sensor ,ith 6OO0 accuracy and 6OO0 precision1

Suppose a lab re!rigerator holds a constant temperature o! %2.' F. A temperature

sensor is tested )' times in the re!rigerator. The temperatures !rom the test yield thetemperatures o!1 %2.' %2.' %4.2 %2.) %2.' %4.3 %2.' %2.+ %2.' %4.3. This

distribution does sho, a tendency to,ard a particular value (high precision* and is

very near the actual temperature each time (high accuracy*.

• The goal o! any instrument is to have high accuracy (sensor matching reality as close

as possible* and to also have a high precision (being able to consistently replicateresults and to measure ,ith as many signi!icant digits as appropriately possible*.

$nstruments including radar need to be calibrated in order that they sustain high

accuracy and high precision.

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). Arithmetic mean 1

The arithmetic mean$ the average of number of readings taken. The best

approximation will be made when the number of readings of the same%uantity is very large. Theoretically an infinite number of readings

would give the best result$ although in practice only a finite number of

measurements can be made. The arithmetic mean is given by

Statistical Analysis1

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+. Mean 0eviation

eviation is the departure at given reading

from the arithmetic mean of the group of

readings. If the deviation of the first reading x1 is d1$ and that of the second reading x

' is called

d' and so on then the deviation from the mean

can be expressed as

The deviation from the mean may have a

positive or a negative value and the algebric

sum of all deviation must be /ero.

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%. Average 0eviation 0

0verage deviation is an indication of the precision of

the instrument used in making the measurements.

(ighly precise instrument will yield a low average

deviation between readings.

o average deviation is the sum of the absolute values

of the deviation divided by the number of readings. Theabsolute value of the deviation is the value without

respect to sign. 0verage deviation may be expressed as

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&. Standard 0eviation

The standard deviation of an infinite number of data is the %uare

root of the sum of all the individual deviations s%uared$ divided by

the number of readings.It is represented by and it is also known as root mean s%uare

deviation. 2athematically it is given by

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8ariance

3ariance or mean s%uare deviation is same as thestandard deviation except that the s%uare root is

not obtained.

Therefore3ariance !v 4 mean s%uare deviation 4 5'

variance are used in many computations because

variances are additive.

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PRO-A-$9$TY OF ERRORS

8O9TA6E MEAS:REME;TS

2ains voltage is measured with a digital voltmeter to the nearest to 6.1 V at short

time intervals. The data are presented in table 1.'. The nominal value of the

measured voltage is ++' V. The data are presented in the histogram of 7ig. 1.+.The ordinate represents the number of observed readings !fre%uency of

occurrence of a particular value. It is also called the fre%uency distribution curve.

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8O9TA6E MEAS:REME;TS

An e/amination o! the. histogram reveals the !ollo,ing points 1

!i The largest number of readings !'6 have their values at the central value of

''6.6 3. 8ther readings are situated symmetrically. mall errors are more

probable than large errors.

!ii The tendency of the curve indicates that if more readings were taken at smaller

interval$ the curve remains symmetrical. 8nly the contour of the histogramwould become smooth. There is an e%ual probability of positive and negative

errors.

!iii The bell shaped curve is called the 9aussian distribution curve.

!iv 0 sharp and narrow distribution curve enables the observer to state that the

most probable value of the true value is the central value.

T0:*E 1.'

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PRO-A-$E ERROR

The 9aussian distribution curve as a function of standard deviation is shown in

7ig. 1.,. ;hen a large number of readings are taken$ the relation is given by

;here x 4 2agnitude of deviation from the mean.

4 <umber of readings at any given deviation x,

!the probability of deviation.

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The error distribution curve sho,s symmetrical distribution o! errors. $t can be

treated as the limiting !orm o! the histogram o! Fig. ).&. The area under the error

distribution curve bet,een the limits < =and > = represents the entire number o!

observations. The area under the curve < and > represents the case that di!!er!rom the mean by no more than the standard deviation (a*. The area under the

curve bet,een the limits < = and > = a gives the total number o! measurements ,ith

in these limits. About 72 percent o! all the cases lie bet,een the limits < and >

!rom the mean. ?ence the more !robable error is ta"en as '.74& .

Fig. ). ERROR 0$STR$-:T$O; @:R8E

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To illustrate the meaning of the probable error$ consider

measurement of nominally 166 pf capacitors. ;hen a large no. of

measurements are taken$ the mean value is found to be 166.66 pF.This means that ≈ -= percent of capacitors have values which lie

between limits of > 6.'6 pf of the mean. This indicates that if a

capacitor is selected from a lot of capacitors at random$ there is

then approximated two to one chance that the value of the capacitor

selected lies between the limits of ? 6.'6pf .@robable error r ? 6.-A+,

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@E&78&20<BE @?ARA@TER$ST$@S 87 I<T&C2E<T

Stability D The ability of a measurement system to maintain its

performance over prolonged periods of time is called stability. If a

system is highly stable$ it need not be calibrated fre%uently.7re%uent calibration is re%uired for less stable instruments.

ero Stability D It indicates the ability of an instrument to restore to

/ero reading after the input is made /ero$ while other conditions

remain the same.Resolution D It is the smallest change of input to which the

measuring system responds. If a digital instrument has a read out of

$ its resolution is 1 in .

0ccuracy and resolution are not the same.

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PRO-9EMS

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ERRORS CW

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