1 4. MEASUREMENT ERRORS 4. MEASUREMENT ERRORS Practically all measurements of continuums involve errors. Understanding the nature and source of these errors can help in reducing their impact. In earlier times it was thought that errors in measurement could be eliminated by improvements in technique and equipment, however most scientists now accept this is not the case. Reference: www.capgo.com The types of errors include: systematic errors and random errors.
The types of errors include: systematic errors and random errors . 4. MEASUREMENT ERRORS. 4.MEASUREMENT ERRORS. Practically all measurements of continuums involve errors. Understanding the nature and source of these errors can help in reducing their impact. - PowerPoint PPT Presentation
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14. MEASUREMENT ERRORS
4. MEASUREMENT ERRORS
Practically all measurements of continuums involve errors. Understanding the nature and source of these errors can help in reducing their impact.
In earlier times it was thought that errors in measurement could be eliminated by improvements in technique and equipment, however most scientists now accept this is not the case.
Reference: www.capgo.com
The types of errors include: systematic errors and random errors.
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Systematic error are deterministic; they may be predicted and hence eventually removed from data.
Systematic errors may be traced by a careful examination of the measurement path: from measurement object, via the measurement system to the observer.
Another way to reveal a systematic error is to use the repetition method of measurements.
References: www.capgo.com, [1]
NB: Systematic errors may change with time, so it is important that sufficient reference data be collected to allow the systematic errors to be quantified.
4.1. Systematic errors
4. MEASUREMENT ERRORS. 4.1. Systematic errors
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Example: Measurement of the voltage source value
VS
Temperature sensor
Rs
RinVin
Measurement system
VSVin
VSVin
Rin+ RS
Rin
4. MEASUREMENT ERRORS. 4.1. Systematic errors
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Random error vary unpredictably for every successive measurement of the same physical quantity, made with the same equipment under the same conditions.
We cannot correct random errors, since we have no insight into their cause and since they result in random (non-predictable) variations in the measurement result.
When dealing with random errors we can only speak of the probability of an error of a given magnitude.
Reference: [1]
4. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.1. Uncertainty and inaccuracy
4.2. Random errors4.2.1. Uncertainty and inaccuracy
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NB: Random errors are described in probabilistic terms, while systematic errors are described in deterministic terms. Unfortunately, this deterministic character makes it more difficult to detect systematic errors.
Reference: [1]
4. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.1. Uncertainty and inaccuracy
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Measurements
t
True value
Example: Random and systematic errors
4. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.1. Uncertainty and inaccuracy
(0.14%)(0.14% )
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Maximum random error
2 Bending point
Amplitude, 0p rms
Inaccuracy
UncertaintySystematic error
f )x(
Measurements
Mean measurement result
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4.2.2. Crest factorOne can define the ‘maximum possible error’ for 100% of the measurements only for systematic errors.
Reference: [1]
For random errors, an maximum random error (error interval) is defined, which is a function of the ‘probability of excess deviations’.
where k is so-called crest* factor )k0(. This inequality accretes that the probability deviations that exceed kis not greater than one over the square of the crest factor.*Crest stands here for ‘peak’.
1P{x xk}
k2
The upper (most pessimistic) limit of the error interval for any shape of the probability density function is given by the inequality of Chebyshev-Bienaymé:
4. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.2. Crest factor
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2
(xx)2f)x(dx+
xk (xx)2f)x(dx
xk
(xx)2f)x(dx xk
xk1
k22
P{x xk} f)x(dx
xk
f)x(dx
xk
Proof:
k22f)x(dx
xk k22f)x(dx
xk
1k22
1k2
k22
k22
4. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.2. Crest factor
(xx)2f)x(dx
xk (xx)2f)x(dx
xk
1k22
x xk)xx(2k22
x xk)xx(2k22
94. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.2. Crest factor
Note that the Chebyshev-Bienaymé inequality can be derived from the Chebyshev inequality
which can be derived from the Markov inequality
2
P{x xa} a2
xP{ x a}a
x
10
Tchebyshev (most pessimistic) limit
any pdf
10 0
10-1
10-2
10-3
10-4
10-5
10-6
0 1 2 3 54
Pro
babi
lity
of e
xces
s de
viat
ions
Crest factor, k
Normal pdf
Illustration: Probability of excess deviations
4. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.2. Crest factor
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4.3. Error sensitivity analysis
The sensitivity of a function to the errors in arguments is called error sensitivity analysis or error propagation analysis.
Reference: [1]
We will discuss this analysis first for systematic errors and then for random errors.
4.3.1. Systematic errors
Let us define the absolute error as the difference between the measured and true values of a physical quantity,
a a a0,
4. MEASUREMENT ERRORS. 4.3. Error propagation
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Reference: [1]
and the relative error as:
a a a0
a0
If the final result x of a series of measurements is given by:
x = f)a,b,c,…( ,
where a, b, c ,… are independent, individually measured physical quantities, then the absolute error of x is:
With a Taylor expansion of the first term, this can also be written as:
in which all higher-order terms have been neglected. This is permitted provided that the absolute errors of the arguments are small and the curvature of f)a,b,c,…( at the point )a,b,c,…( is small.