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 1
NEC 403 Unit I by Dr Naim R Kidwai,
Professor & Dean, JIT Jahangirabad

Measurement Errors
A measurable quantity is a property of phenomena, bodies, or
substances that can be deﬁned qualitatively and expressed
quantitatively. Measurable quantities are also called physical
quantities
True value of a measurand is the value of the measured physical
quantity, which, would ideally reﬂect, both qualitatively and
quantitatively, 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 2

Measurement Errors: Types
Gross Errors: Errors due to human carelessness. Ex. misreading of
Instrument 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 or
due 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.
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Measurement Errors: Types
Systematic 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 of
error) are referred as random errors. These error results due to
changes in environmental variables.
–Random Errors and generally of Gaussian nature
–It can be estimated by multiple measurements and its effects can be
reduced by averaging.
2/3/2017 4

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 5

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 of
quantity. 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 6

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 and
electrical quantities. The terms Accuracy & Tolerance are also used. A
resistor with ± 10% error is said to be accurate to ± 10 % or having
tolerance of ± 10%.
2/3/2017 7

Accuracy
A measure of how close a measurement is to the true value of the
quantity being measured.
2/3/2017 8
More accurate Less accurate
Ex. A voltmeter with 1% accuracy
indicates a value as 200 V.
Possible error = ± 1% of 100 V
= ± 2V
Thus the true value lies between
198 V to 202 V.
•Generally full scale accuracy is
referred by manufacturers

Precision
The term Precise means clearly and sharply defined.
Precision is measure of reproducibility of measurement, i.e. given a
fixed value of a quantity, precision is measure of degree of
agreement within a group of measurements.
2/3/2017 9
More Accurate,
More Precise
Less Accurate
More Precise
Less Accurate
Less Precise
More Accurate
Less Precise

Precision
Precision are composed of two characteristics
•Conformity
•Significant figures
High precision means conformity of repeated readings in a tight
cluster
While low precision means conformity of repeated readings in a
broad scattered cluster
The number of significant figure indicates the precision of
measurement.
2/3/2017 10

Significant Figures
The number of significant figure indicates the precision of
measurement.
Let the true value of a register is 1.35786 K, but observer reads
from the scale to 1.4 K. This precision error is caused by limitation
of 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 could
be 6.494V or 6.496 V, as the measurement has four significant digits.
2/3/2017 11

Significant Figures
•Let a measured resistance (R) is close to 872.4 , than 872.3  or
872.5 . The four significant figure show that measurement
precision 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  or
51.9 . Here ‘0’ is also a significant digit so the number of significant
figure of measurement is three.
•Let R is 5.00x106  (500000000 ). If the resistance is closer to the
value than 4.99x106  or 5.01x106 , then there are three
significant figures and not nine.
2/3/2017 12

Significant Figures
•Let measurement of Resistor is done by measuring the voltage
across the resister (by Voltmeter) and current through resistor (by
Ammeter) respectively. Let the measured Voltage is 50.31 V and
2.33 A, then
Resistance R = V/I = 5.31/2.33 = 2.2789699571 
Clearly the answer in 10 fraction digits is irrelevant. Answer should
be in same number of significant digits as the original quantities i.e
R = 2.28  (three significant digits)
2/3/2017 13

Resolution
If input is changed from some initial arbitrary value (may be zero),
then output will not change until a minimum increment of input is
observed.
Thus the smallest increment in input (quantity being measured) that
can be detected by the instrument is called resolution
(discrimination)
Smallest value of input which can be detected by instrument is
called Threshold.
2/3/2017 14

Measurement Error Combination
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 of
quantity. 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 15

2/3/2017 16
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 Quantities
Let y = ±u ±v ±w
Relative increment of the function
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2/3/2017 17
Ex. Three resistances have following ratings
R1= 50 ± 10%, R2= 100 ± 5%, R3= 200 ± 10%
What is the result, when three resistances are combined in series
%57.835030350Thus
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2/3/2017 18
Product of Quantities
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2/3/2017 19
Quotient of Quantities
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2/3/2017 20
Power of a factor/ Composite factor
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Basics of Statistical Analysis
A number of measurements of a quantity have data scattered
around a central value. Statistical tools can be employed to reach
best 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 value
can be obtained using arithmetic mean.
2/3/2017 21

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21 1
or 

Dispersion (Spread or Scatter)
The property which denotes the extent to which samples are
dispersed around a central value (mean) .
2/3/2017 22
Curve 1: Data spreads in the
range x1 to x2. The observations
are less disperse (more precise)
Curve 2: Data spreads in the
range x3 to x4. The observations
are more disperse (less precise)
Curve 1: More precision
Curve 2: Less precision
x3 x1  x2 x4 x
Curves showing different ranges and precision
Probability or
frequency of
occurrence

Range: Range is difference of largest and lowest value of the
samples. It is simplest measure of dispersion.
Deviation: deviation is departure from arithmetic mean of the
sample.
2/3/2017 23
  
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As average deviation will be zero, therefore deviation is not suitable
measure of dispersion
Variance: Variance is average of squared deviation of samples from
mean

Variance: Variance is average of squared deviation of samples from
mean
2/3/2017 24
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Variance is a finite value and is occasionally used for representing
dispersion
Standard deviation: is square root of averaged squared deviation of
samples from mean.
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Normal or Gaussian Curve
2/3/2017 25
-3 -2 -1 1 2 3 x
y
Probability or frequency
of occurrence
•Gaussian curve is symmetric about arithmetic mean and area under
the curve is zero. Therefore, data is normalized to be zero mean
•Most of natural events, measurements having some amount of
randomness follows Gaussian curve
For Normal curve, probability y is given as
0i.e)(deviationvaluenormalizedx
indexprecisionhwhere
22

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Normal or Gaussian Curve
2/3/2017 26

Normal or Gaussian Curve:
Specification
2/3/2017 27
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 28
Let X is combination of several independent variables each of which
is subject to random effects
       
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Uncertainty & Confidence Level
For single sample data, statistical analysis can not be applied as
its 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 upon
odds (confidence level)
2/3/2017
29
levelconfidenceasexpressedalsomaychance,orOdds1tob
intervalyuncertaintw
1)to(b


 wXX
Treatment of uncertainty is done in same way as to the erro
analysis

Thanks
2/3/2017
30

Measurement errors, Statistical Analysis, Uncertainty

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