1 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: The types of errors include: systematic errors and random errors.

2 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: [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 Systematic errors 4. MEASUREMENT ERRORS Systematic errors

3 Example: Measurement of the voltage source value VSVS Temperature sensor RsRs R in V in Measurement system V S  V in V S  ·V in  R in + R S R in 4. MEASUREMENT ERRORS Systematic errors

4 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 Random errors Uncertainty and inaccuracy 4.2. Random errors Uncertainty and inaccuracy

5 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 Random errors Uncertainty and inaccuracy

6 Measurements t True value Example: Random and systematic errors 4. MEASUREMENT ERRORS Random errors Uncertainty and inaccuracy (0.14%) 66 Maximum random error 22 Bending point Amplitude, 0  p rms Inaccuracy UncertaintySystematic error f (x)f (x) Measurements Mean measurement result

Crest factor One 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 (k  0). This inequality accretes that the probability deviations that exceed k   is not greater than one over the square of the crest factor. *Crest stands here for ‘peak’. 1 P{  x  x   k  }    k2k2 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 Random errors Crest factor

8 22  (x  x) 2 f(x)dx +  x  k    (x  x) 2 f(x)dx x  k     (x  x) 2 f(x)dx x  k  x  k   1 k22k22 P{  x  x   k  }   f(x)dx  x  k   f(x)dx   x  k  Proof:  k 2   2 f(x)dx  x  k    k 2   2 f(x)dx x  k    1 k22k22  1 k2k2 k22k22 k22k22 4. MEASUREMENT ERRORS Random errors Crest factor  (x  x) 2 f(x)dx  x  k    (x  x) 2 f(x)dx x  k    1 k22k22 x  x  k  (x  x) 2  k 2   2 x  x  k  (x  x) 2  k 2   2

9 4. MEASUREMENT ERRORS Random errors Crest factor Note that the Chebyshev-Bienaymé inequality can be derived from the Chebyshev inequality which can be derived from the Markov inequality 22 P{  x  x   a  }    a2a2 x P{ x  a  }  a x 

10 Tchebyshev (most pessimistic) limit any pdf Probability of excess deviations Crest factor, k Normal pdf Illustration: Probability of excess deviations 4. MEASUREMENT ERRORS Random errors Crest factor

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 Systematic errors Let us define the absolute error as the difference between the measured and true values of a physical quantity,  a  a  a 0, 4. MEASUREMENT ERRORS Error propagation

12 Reference: [1] and the relative error as:  a   a  a 0 a0a0 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:  x = f(a,b,c,…)  f(a 0,b 0,c 0,…). aa a0a0 4. MEASUREMENT ERRORS Error propagation Systematic errors

13 Reference: [1] 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. f(a,b,c,…)f(a,b,c,…)  x =  a aa f(a,b,c,…)f(a,b,c,…) +  b + …, bb 4. MEASUREMENT ERRORS Error propagation Systematic errors (a 0,b 0,c 0,…)

14 Reference: [1] One never knows the actual value of  a,  b,  c, …. Usually the individual measurements are given as a ±  a max, b ±  b max, …  in which  a max,  b max are the maximum possible errors. In this case  f(a,b,c,…)  x max =  a max aa  f(a,b,c,…) +  b max + …. bb 4. MEASUREMENT ERRORS Error propagation Systematic errors (a 0,b 0,c 0,…)

15 Reference: [1] 4. MEASUREMENT ERRORS Error propagation Systematic errors ff S x a , …, aa Defining the sensitivity factors: this becomes:  x max   S x a   a max  S x b   b max  + …. (a 0,b 0,c 0,…)

16 Reference: [1] This expression can be rewritten to obtain the maximal relative error:  f a  a max  x max  =  a f 0 a + + ….  f b  b max  b f 0 b  x max x0x0 this becomes:  x max   s x a   a max  s x b   b max  + …. f af a s x a , …,  a f 0  f aa Defining the relative sensitivity factors: 4. MEASUREMENT ERRORS Error propagation Systematic errors f/f0f/f0 a/aa/a

17 4. MEASUREMENT ERRORS Error propagation Systematic errors Illustration: The rules that simplify the error sensitivity analysis 2. s x a  s x b s b a b = a 2 x = b  aa 6  a2a2a 1. s x a  s x a m n mnmn 2a2a x = a 2 aa 3. s x 1 x 2 a  s x 1 a + s x 2 a x 1 = a 2 aa 2a2a x 2 =  a aa aa  a a +

18 Reference: [1] Random errors 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: Again, we have neglected the higher order terms of the Taylor expansion.   f aa (a,b,c,…)(a,b,c,…) bb (a,b,c,…)(a,b,c,…) cc (a,b,c,…)(a,b,c,…) d x = da + db + dc + …. 4. MEASUREMENT ERRORS Error propagation Random errors

19 Reference: [1] Since dx= x  x,  f aa    = (dx) 2 = da + db + dc + ….  f bb cc 2 = (da) 2 + (db) 2 + …+ da db + …  f aa 2 bb 2 aa bb squarescross products (=0) 4. MEASUREMENT ERRORS Error propagation Random errors = (da) 2 + (db) 2 + ….  f aa 2 bb 2 (a,b,c,…)(a,b,c,…)(a,b,c,…)(a,b,c,…)

20 Reference: [1] Considering that (da) 2  a 2 …, the expression for  x 2 can be written as (Gauss’ error propagation rule):  x 2 =  a 2 +  b 2 +  c 2 + …  f aa 2 bb 2 cc 2 (a,b,c,…)(a,b,c,…)(a,b,c,…)(a,b,c,…)(a,b,c,…)(a,b,c,…) x = f(a,b,c,…) NB: In the above derivation, the shape of the pdf of the individual measurements a, b, c, … does not matter. 4. MEASUREMENT ERRORS Error propagation Random errors

21 Example A:Let us apply Gauss’ error propagation rule to the case of averaging in which x =   a i : 1n1n i = 1 n or for the standard deviation of the end result:  x =  a. 1   n Thanks to averaging, the measurement uncertainty decreases with the square root of the number of measurements. x  a.  x 2  n  a    a   , 1 n 2 1n1n 4. MEASUREMENT ERRORS Error propagation Random errors

22 Example B:Let us apply Gauss’ error propagation rule to the case of integration in which x =   a i : i = 1 n or for the standard deviation of the end result: Due to integration, the measurement uncertainty increases with the square root of the number of measurements. x  a.  x 2  n  a    4. MEASUREMENT ERRORS Error propagation Random errors  x =  n  a.

23 4. MEASUREMENT ERRORS Error propagation Random errors Illustration: Noise averaging and integration Gaussian white noise Averaging (  10) and integration Averaging Integration  x =  a 1n1n  x =  n  a OutputInput

24 4. MEASUREMENT ERRORS Error propagation Random errors Illustration: LabView simulation

25 Next lecture Next lecture:

26 The repeatability is determined by the uncertainty of measurements (maximum random error). The repeatability of a measurement is a measure or the extent of agreement between consecutive measurements of the same physical quantity, using the same method and equipment, and under the same operating conditions over a short period of time. The reproducibility of a measurement is the closeness of agreement among repeated measurements of the same measurand performed in different locations under different operating conditions, or spread over a long period of time The repeatability is determined by systematic errors and long- term drift. Reference: [1] 4. MEASUREMENT ERRORS Random errors Uncertainty and inaccuracy