measurement errors, statistical analysis, uncertainty

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


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.


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.


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.


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


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%.


Accuracy

A measure of how close a measurement is to the true value of thequantity being measured.


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.


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.


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.


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.


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)


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.


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


Measurement Error Combination


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

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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.


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Basics of Statistical AnalysisDispersion (Spread or Scatter)

The property which denotes the extent to which samples aredispersed around a central value (mean) .


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.


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Variance: Variance is average of squared deviation of samples frommean

Basics of Statistical AnalysisVariance: Variance is average of squared deviation of samples frommean


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Standard deviation: is square root of averaged squared deviation ofsamples from mean.

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Normal or Gaussian Curve


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•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

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Normal or Gaussian Curve


Normal or Gaussian Curve: Specification


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


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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

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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

measurement errors, statistical analysis, uncertainty

Engineering