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Measurement Errors
Measurement Errors: Gross error, systematic error,
absolute error and relative error, accuracy, precision,
resolution and significant figures, Measurement error
combination, basics of statistical analysis.
2/3/2017 1NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Measurement ErrorsA measurable quantity is a property of phenomena, bodies, orsubstances that can be defined qualitatively and expressedquantitatively. Measurable quantities are also called physicalquantities
True value of a measurand is the value of the measured physicalquantity, which, would ideally reflect, both qualitatively andquantitatively, the corresponding property of the object
Measurement Error is the deviation of the result of measurement from the true value of the measurable quantity, expressed in absolute or relative form
Error = Measured or Observed – True Value
2/3/2017 2NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Measurement Errors: TypesGross Errors: Errors due to human carelessness. Ex. misreading ofInstrument or using wrong range. It can be avoided by two means
–Great care in reading and recording of data.
–More observations of measurement to avoid same error.
Systematic Error: Errors from measurement system/ instrument ordue to wrong use of instrument. Ex. Offset error, Zero setting error
–These are predictable, is typically proportional to true value,
–It can be generally measured/ eliminated if the cause is known.
2/3/2017 3NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Measurement Errors: TypesSystematic errors are of three types
•Instrumental Error : errors due to inherent shortcomings, loading effects
•Environmental Error: errors due to factors external to the instruments
•Observational Error : Errors due to observations; ex. parallax error,
Random Error : errors of unexplainable origin (unknown sources oferror) are referred as random errors. These error results due tochanges in environmental variables.
–Random Errors and generally of Gaussian nature
–It can be estimated by multiple measurements and its effects can bereduced by averaging.
2/3/2017 4NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Absolute error and Relative error,
If Am is Observed or measured value of a physical quantity with At
is its true value, then
Error or Absolute Error A= Am- At
Relative Errors are Error expressed as fraction of true value.
Relative Error = A/ At
and can be expressed in % by multiplying with hundred.
2/3/2017 5NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Absolute error and Relative error,
Limiting Error (Guarantee Errors):– Manufacturers guarantee component value to lie in certain % of its rated value.
– In Instruments, accuracy is provided and is generally, a certain % of Full Scale.
The manufacturers has to specify deviation from nominal value ofquantity. The limit of these deviations is called Limiting Errors.
Thus a nominal value A with limiting error A is referred as A±A.
Relative limiting error r = A/A = 0 /A
Or absolute limiting error 0 = A = r A
2/3/2017 6NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Absolute error and Relative error, Ex. A resistance R of 600 is known to have possible absolute error as± 60 . Express the value of resistor in relative error.
R= 600 ± 60
Relative error= ± 60/600 = ± 0.1 = ± 10 %
Thus R = 600 ± 10%
Percentages are usually employed to express errors in resistances andelectrical quantities. The terms Accuracy & Tolerance are also used. Aresistor with ± 10% error is said to be accurate to ± 10 % or havingtolerance of ± 10%.
2/3/2017 7NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Accuracy
A measure of how close a measurement is to the true value of thequantity being measured.
2/3/2017 8NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
More accurate Less accurate
Ex. A voltmeter with 1% accuracyindicates a value as 200 V.
Possible error = ± 1% of 100 V
= ± 2V
Thus the true value lies between198 V to 202 V.
•Generally full scale accuracy isreferred by manufacturers
PrecisionThe term Precise means clearly and sharply defined.
Precision is measure of reproducibility of measurement, i.e. given afixed value of a quantity, precision is measure of degree ofagreement within a group of measurements.
2/3/2017 9NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
More Accurate, More Precise
Less Accurate More Precise
Less Accurate Less Precise
More Accurate Less Precise
PrecisionPrecision are composed of two characteristics
•Conformity
•Significant figures
High precision means conformity of repeated readings in a tightcluster
While low precision means conformity of repeated readings in abroad scattered cluster
The number of significant figure indicates the precision ofmeasurement.
2/3/2017 10NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Significant FiguresThe number of significant figure indicates the precision ofmeasurement.
Let the true value of a register is 1.35786 K, but observer readsfrom the scale to 1.4 K. This precision error is caused by limitationof the instrument (less significant digits).
Let a measured voltage is 6.495 V.
– It indicates that instrument can read minimum value of 0.001 V (resolution).
– If the measurement is made with precision of 0.001 V, then the value read couldbe 6.494V or 6.496 V, as the measurement has four significant digits.
2/3/2017 11NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Significant Figures
•Let a measured resistance (R) is close to 872.4 , than 872.3 or872.5 . The four significant figure show that measurementprecision is 0.1 .
•Let R is close to 872.4 K, then precision is 0.1 K or 100 .
•If R is 52.0 then it implies that R is closer to 52 than 52.1 or51.9 . Here ‘0’ is also a significant digit so the number of significantfigure of measurement is three.
•Let R is 5.00x106 (500000000 ). If the resistance is closer to thevalue than 4.99x106 or 5.01x106 , then there are threesignificant figures and not nine.
2/3/2017 12NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Significant Figures
•Let measurement of Resistor is done by measuring the voltageacross the resister (by Voltmeter) and current through resistor (byAmmeter) respectively. Let the measured Voltage is 50.31 V and2.33 A, then
Resistance R = V/I = 5.31/2.33 = 2.2789699571
Clearly the answer in 10 fraction digits is irrelevant. Answer shouldbe in same number of significant digits as the original quantities i.e
R = 2.28 (three significant digits)
2/3/2017 13NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Resolution
If input is changed from some initial arbitrary value (may be zero),then output will not change until a minimum increment of input isobserved.
Thus the smallest increment in input (quantity being measured) thatcan be detected by the instrument is called resolution(discrimination)
Smallest value of input which can be detected by instrument iscalled Threshold.
2/3/2017 14NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Measurement Error CombinationLimiting Error (Guarantee Errors):– Manufacturers guarantee component value to lie in certain % of its rated value.
– In Instruments, accuracy is provided and is generally, a certain % of Full Scale.
The manufacturers has to specify deviation from nominal value ofquantity. The limit of these deviations is called Limiting Errors.
Thus a nominal value A with limiting error A is referred as A ± A.
Relative limiting error r = A/A = 0 /A
Or absolute limiting error 0 = A = r A
2/3/2017 15NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Measurement Error Combination
2/3/2017 16NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
When two or more quantities, (each having limiting errors) is combined,limiting error of result can be computed using algebraic relation .Sum or difference of QuantitiesLet y = ±u ±v ±wRelative increment of the function
)(
error limiting resulting Thus
then, and , , as drepresente are errors limiting if
)(
wvuwvuw
ww
v
vv
u
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Measurement Error Combination
2/3/2017 17NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Ex. Three resistances have following ratingsR1= 50 ± 10%, R2= 100 ± 5%, R3= 200 ± 10%What is the result, when three resistances are combined in series
%57.835030350 Thus
30)2055( error, limiting resulting
35020010050 ,resistance theof Magnitude
20200200100
10200
5100100100
5100
55050100
1050
321
3
2
1
R
wvuR
RRRR
R
R
R
Measurement Error Combination
2/3/2017 18NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Product of QuantitiesLet y = uvTaking logarithm
v
v
u
u
y
y
vu
v
dv
u
du
y
dy
dy
dv
vdy
du
uy
vuy eee
then, and , as drepresente are errors limiting if
or
111
y w.r.t.ationdifferenti Taking
logloglog
Measurement Error Combination
2/3/2017 19NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Quotient of QuantitiesLet y = u/v
v
v
u
u
y
y
hv
v
u
u
v
v
u
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y
y
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error, limiting relative so
signs) opposite ave and whenoccurserror (Max.
then, and , as drepresente are errors limiting if
or
111 y, w.r.t.ationdifferenti Taking
logloglog
Measurement Error Combination
2/3/2017 20NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Power of a factor/ Composite factorLet y = un vm
u
vm
u
un
y
y
vu
u
dvm
u
dun
y
dy
dy
dv
v
m
dy
du
u
n
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vmuny eee
then, and as drepresente are errors limiting if
or 1
y, w.r.t.ationdifferenti Taking
logloglog
Basics of Statistical AnalysisA number of measurements of a quantity have data scatteredaround a central value. Statistical tools can be employed to reachbest approximation to the true value of the quantity.
Arithmetic mean
Let x1, x2,--------, xn are n different measurements of same quantity(samples). Most probable value from various observations of valuecan be obtained using arithmetic mean.
2/3/2017 21NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
n
i
in x
nn
xxxX
1
21 1or
Basics of Statistical AnalysisDispersion (Spread or Scatter)
The property which denotes the extent to which samples aredispersed around a central value (mean) .
2/3/2017 22NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Curve 1: Data spreads in therange x1 to x2. The observationsare less disperse (more precise)
Curve 2: Data spreads in therange x3 to x4. The observationsare more disperse (less precise)
Curve 1: More precision
Curve 2: Less precision
x3 x1 x2 x4 xCurves showing different ranges and precision
Probability or frequency of occurrence
Basics of Statistical AnalysisRange: Range is difference of largest and lowest value of thesamples. It is simplest measure of dispersion.
Deviation: deviation is departure from arithmetic mean of thesample.
2/3/2017 23NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
n
i
i
n
i
i
ii
xn
dn
D
xd
11
011
As average deviation will be zero, therefore deviation is not suitablemeasure of dispersion
Variance: Variance is average of squared deviation of samples frommean
Basics of Statistical AnalysisVariance: Variance is average of squared deviation of samples frommean
2/3/2017 24NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
n
i
i
n
i
i xn
dn
V1
2
1
2 11
Variance is a finite value and is occasionally used for representingdispersion
Standard deviation: is square root of averaged squared deviation ofsamples from mean.
Vxn
dn
n
i
i
n
i
i 1
2
1
2 11
Normal or Gaussian Curve
2/3/2017 25NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
-3 -2 -1 1 2 3 x
y Probability or frequency of occurrence
•Gaussian curve is symmetric about arithmetic mean and area underthe curve is zero. Therefore, data is normalized to be zero mean
•Most of natural events, measurements having some amount ofrandomness follows Gaussian curve
For Normal curve, probability y is given as
0 i.e )(deviation valuenormalizedx
index precision h where
22
xheh
y
Normal or Gaussian Curve
2/3/2017 26NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Normal or Gaussian Curve: Specification
2/3/2017 27NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Parameter Data range probability Explanation
Standard Deviation
1 range ± 0.6828 68.28 % of data lie in the range
2 range ±2 0.9546 95.46 % of data lie in the range
3 range ±3 0.9974 99.74 % of data lie in the range
Probable Error
r range ±3r 0.5 50.00 % of data lie in the range
Combination of Components
2/3/2017 28NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
Let X is combination of several independent variables each of which is subject to random effects
nx
n
xxX
n
n
n
n
n
n
Vx
XV
x
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x
XV
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22
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21
Thus zero. to tendllproduct wi cross of sum t variableindependenfor as
X, in variationtheninput, in change small are ,,, if
),,,(
Uncertainty & Confidence LevelFor single sample data, statistical analysis can not be applied asits scatter can’t be observed.
Thus sample value (X) can be expressed in terms of mean value(value for single samples) and uncertainty interval based uponodds (confidence level)
2/3/2017NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
29
level confidence as expressed alsomay chance,or Odds 1 tob
intervaly uncertaint w
1) to(b
wXX
Treatment of uncertainty is done in same way as to the erroanalysis
Thanks
2/3/2017NEC 403 Unit I by Dr Naim R Kidwai, Professor & Dean, JIT Jahangirabad
30