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3.6 Some Important Distributions

77

0.4

 

 

 

 

 

normal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f=5

 

 

 

 

 

 

 

 

f=2

 

 

0.3

 

 

 

 

 

f=1

 

 

0.2

 

 

 

 

 

 

 

 

0.1

 

 

 

 

 

 

 

 

0.0-8

-6

-4

-2

0

2

4

6

8

 

 

 

 

X

 

 

 

 

Fig. 3.23. Student’s distributions for 1, 2, 5 degrees of freedom and normal distribution.

The only parameter is f, the number of degrees of freedom. For f = 1 we recover the Cauchy distribution. For large f it approaches the normal distribution N(0, 1) with variance equal to one. The distribution is symmetric, centered at zero, and bell shaped, but with longer tails than N(0, 1). The even moments are

i1 · 3 · · · (i − 1)

µi = f 2 (f − 2)(f − 4) · · · (f − i) .

They exist only for i ≤ f − 1. The variance for f ≥ 3 is σ2 = f/(f − 2), the excess for f ≥ 5 is γ2 = 6/(f − 4), disappearing for large f, in agreement with the fact that the distribution approaches the normal distribution.

The typical field of application for the t distribution is the derivation of tests or confidence intervals in cases where a sample is supposed to be taken from a normal distribution of unknown variance but known mean µ. Qualitatively, very large absolute values of t indicate that the sample mean is incompatible with µ. Sometimes the t distribution is used to approximate experimental distributions which di er from Gaussians because they have longer tails. In a way, the t distribution interpolates between the Cauchy (for f = 1) and the Gauss distribution (for f → ∞).

3.6.12 The Extreme Value Distributions

The family of extreme value distributions is relevant for the following type of problem: Given a sample taken from a certain distribution, what can be said about the distribution of its maximal or minimal value? It is found that these distributions converge with increasing sample size to distributions of the types given below.

78 3 Probability Distributions and their Properties

The Weibull Distribution

This distribution has been studied in connection with the lifetime of complex aggregates. It is a limiting distribution for the minimal member of a sample taken from a distribution limited from below. The p.d.f. is

 

p

 

x

p

 

1

 

x

p

 

f(x|a, p) =

 

 

 

 

 

exp −

 

, x > 0

(3.58)

a

a

 

 

a

with the positive scale and shape parameters a and p. The mode is

m

 

p

 

 

 

x

= a

p − 1

1/p

for p

 

1 ,

 

 

 

mean value and variance are

µ = aΓ (1 + 1/p) ,

σ2 = a2 Γ (1 + 2/p) − Γ 2(1 + 1/p) .

The moments are

µi = aiΓ (1 + i/p) .

For p = 1 we get an exponential distribution with decay constant 1/a.

The Fisher–Tippett Distribution

Also this distribution with the p.d.f.

f

 

(x x , s) =

1

exp

±

x − x0

e±(xx0)/s

±

 

s

 

| 0

s

 

belongs to the family of extreme value distributions. It is sometimes called extreme value distribution (without further specification) or log-Weibull distribution.

If y is Weibull-distributed (3.58) with parameters a, p, the transformation to x = − ln y leads for x to a log-Weibull distribution with parameters x0 = − ln a and s = 1/p. The first of these, the location parameter x0, gives the position of the maximum, i.e. xmod = x0, and the parameter s > 0 is a scale parameter. Mean value

µand variance σ2 depend on these parameters through

µ= x0 Cs , with Euler’s constant C = 0.5772 . . . ,

σ2 = s2 π2 .

6

Mostly, the negative sign in the exponent is realized. Its normal form f(x|0, 1) = exp −x − ex

is also known as Gumbel’s distribution and shown in Fig. 3.24.

Using mathematical properties of Eulers Γ function [22] one can derive the characteristic function in closed form:

φ(t) = Γ (1 ± ist)eix0t ,

whose logarithmic derivatives give in turn the cumulants for this distribution:

κ1 = x0 Cs , κi≥2 = ( 1)i(i − 1)!siζ(i) ,

with Riemann’s zeta function ζ(z) = Σn=11/nz. (see [22]). Skewness and excess are given by γ1 ≈ 1.14 and γ2 = 12/5.

 

 

 

3.6 Some Important Distributions

79

0.4

 

 

 

 

 

 

0.3

 

 

 

 

 

 

f(x)

 

 

 

 

 

 

0.2

 

 

 

 

 

 

0.1

 

 

 

 

 

 

0.0

-2

0

2

4

6

8

 

 

 

 

 

x

 

 

Fig. 3.24. Log-Weibull distribution.

4

Measurement errors

4.1 General Considerations

When we talk about measurement errors, we do not mean mistakes caused by the experimenter, but the unavoidable random dispersion of measurements. Therefore, a better name would be measurement uncertainties. We will use the terms uncertainty and error synonymously.

The correct determination and treatment of measurement errors is not always trivial. In principle, the evaluation of parameters and their uncertainties are part of the statistical problem of parameter inference, which we will treat in Chaps. 6, 7 and 8. There we will come back to this problem and look at is from a more general point of view. In the present chapter we will introduce certain, in practice often well justified approximations.

O cial recommendations are given in “Guide to the Expression of Uncertainty of Measurement”, published in 1993 and updated in 1995 in the name of many relevant organizations like ISO and BIMP (Guide to the Expression of Uncertainty of Measurement, International Organization for Standardization, Geneva, Switzerland) [23]. More recently, a task force of the European cooperation for Accreditation of Laboratories (EAL) with members of all western European countries has issued a document (EAL-R2) with the aim to harmonize the evaluation of measurement uncertainties. It follows the rules of the document mentioned above but is more specific in some fields, especially in calibration issues which are important when measurements are exchanged between di erent laboratories. The two reports essentially recommend to estimate the expected value and the standard deviation of the quantity to be measured. Our treatment of measurement uncertainty will basically be in agreement with the recommendations of the two cited documents which deal mainly with systematic uncertainties and follow the Bayesian philosophy, but we will extend their concept in Sect. 8.2 where we introduce asymmetric error limits.

4.1.1 Importance of Error Assignments

The natural sciences owe their success to the possibility to compare quantitative hypotheses to experimental facts. However, we are able to check theoretical predictions only if we have an idea about the accuracy of the measurements. If this is not the case, our measurements are completely useless.

82 4 Measurement errors

Of course, we also want to compare the results of di erent experiments to each other and to combine them. Measurement errors must be defined in such a way that this is possible without knowing details of the measurement procedure. Only then, important parameters, like constants of nature, can be determined more and more accurately and possible variations with time, like it was hypothesized for the gravitational constant, can be tested.

Finally, it is indispensable for the utilization of measured data in other scientific or engineering applications to know their accuracy and reliability. An overestimated error can lead to a waste of resources and, even worse, an underestimated error may lead to wrong conclusions.

4.1.2 Verification of Assigned Errors

In some situations a system of variables is overconstrained and thus allows us to check whether our measurements are consistent within their error limits. An example is the comparison of the sum of measured angles of a triangle with the value 1800 which is common in surveying. In the experiments of particle physics we can apply among other laws the constraints provided by energy and momentum conservation. When we adjust curves, e.g. a straight line to measured points, the deviations of the points from the line permit us to check the goodness of the fit, and if the fit is poor, we might reject the presumed parametrization or revise the error assignment. A systematic treatment of the corresponding goodness-of-fit tests will be presented in Chap. 10.

4.1.3 The Declaration of Errors

There are several ways to present measurements with their uncertainties. Some of the more frequent ones are given in the following examples:

t= (34.5 ± 0.7) 10−3 s

t= 34.5 10−3 s ± 2 %

x = 10.3+0.7

−0.3

me = (0.510 999 06 ± 0.000 000 15) MeV/c2 me = 0.510 999 06 (15) MeV/c2

me = 9.109 389 7 10−31 kg ± 0.3 ppm

The abbreviation ppm means parts per million. The treatment of asymmetric errors will be postponed to Chap. 8. The measurement and its error must have the same number of significant digits. Declarations like x = 3.2 ± 0.01 or x = 3.02 ± 0.1 are inconsistent.

A relatively crude declaration of the uncertainty is su cient, one or two significant digits are adequate in any case, keeping in mind that often we do not know all sources of errors or are unable to estimate their influence on the result with high accuracy1. This fact also justifies in most cases the approximations which we have to apply in the following.

We denote the error of x with δx or δx. Sometimes it is convenient, to quote dimensionless relative errors δx/x that are useful in error propagation – see below.

1There are exceptions to this rule in hypothesis testing (see Chap. 10).

4.1 General Considerations

83

4.1.4 Definition of Measurement and its Error

Measurements are either quantities read from a measurement device or simply an instrument – we call them input quantities – or derived quantities, like the average of two or more input quantities, the slope of a street, or a rate which are computed from several input quantities. Let us first restrict ourselves to input quantities. An input quantity can be regarded as an observation, i.e. a random variable x drawn from a distribution centered around the true value xt of the quantity which we want to determine. The measurement process, including the experimental setup, determines the type of this distribution (Gauss, Poisson, etc.) For the experimenter the true value is an unknown parameter of the distribution. The measurement and its error are estimates of the true value and of the standard deviation of the distribution2. This definition allows us to apply relations which we have derived in the previous chapter

for the standard deviation to calculations of the uncertainty, e.g. the error δ of a sum

P of independent measurements with individual errors δi is given by δ2 = δi2.

In an ideal situation the following conditions are fulfilled:

1.The mean value of infinitely often repeated measurements coincides with the true value, i.e. the true value is equal to the expectation value hxi of the measurement distribution, see Sect. 3.2. The measurement is then called unbiased.

2.The assigned measurement error is independent of the measured value.

These properties can not always be realized exactly but often they are valid to a su ciently good approximation. The following two examples refer to asymmetric errors where in the first but not in the second the asymmetry can be neglected.

Example 43. Scaling error

A tape measure is slightly elastic. The absolute measurement error increases with the measured length. Assuming a scaling error of 1% also the estimate of the error of a measured length would in average be wrong by 1% and asymmetric by the same proportion. This, however, is completely unimportant.

Example 44. Low decay rate

We want to measure the decay rate of a radioactive compound. After one hour we have recorded one decay. Given such small rates, it is not correct to compute the error from a Poisson distribution (see Sect. 3.6.3) in which we replace the mean value by the observed measurement. The declaration R = 1 ± 1 does not reflect the result correctly because R = 0 is excluded by the observation while R = 2.5 on the other hand is consistent with it.

In Sect. 8.2 we will, as mentioned above, also discuss more complex cases, including asymmetric errors due to low event rates or other sources.

Apart from the definition of a measurement and its error by the estimated mean and standard deviation of the related distribution there exist other conventions: Distribution median, maximal errors, width at half maximum and confidence intervals. They are useful in specific situations but su er from the crucial disadvantage that

2Remark that we do not need to know the full error distribution but only its standard deviation.

84 4 Measurement errors

they are not suited for the combination of measurement or the determination of the errors of depending variables, i.e. error propagation.

4.2 Di erent Types of Measurement Uncertainty

There are uncertainties of di erent nature: statistical or random errors and systematic errors. Their definitions are not unambiguous, disagree from author to author and depend somewhat on the scientific discipline in which they are treated. Also the authors of this book have di erent opinions regarding the meaning of systematic errors. Hence, we have decided to present two parallel subsections for this topic.

4.2.1 Statistical Errors

Errors Following a Known Statistical Distribution

Relatively simple is the interpretation of measurements if the distributions of the errors follow known statistical laws3. The corresponding uncertainties are called statistical errors. Examples are the measurement of counting rates (Poisson distribution), counter e ciency (binomial distribution) or of the lifetime of unstable particles (exponential distribution). Characteristic for statistical errors is that sequential measurements are uncorrelated and thus the precision of the combined results is improved by the repetition of the measurement. In these cases the distribution is known up to a parameter – its expected value. We then associate the actually observed value to that parameter and declare as measurement error the standard deviation belonging to that distribution.

Example 45. Poisson distributed rate

Recorded have been N = 150 decays. We set the rate and its error equal to

 

≈ 150 ± 12.

Z = N ± N = (150 ±

 

150)

Example 46. Digital measurement (uniform distribution)

With a digital clock the time t = 237 s has been recorded. The error is

δt = 1/ 12 s ≈ 0.3 s, thus t = (237.0 ± 0.3) s.

Example 47. E ciency of a detector (binomial distribution)

From N0 = 60 particles which traverse a detector, 45 are registered. The e - ciency is ε = N/N0 = 0.75. The error derived from the binomial distribution

is p p

δε = δN/N0 = ε(1 − ε)/N0 = 0.75 · 0.25/60 = 0.06 .

Example 48. Calorimetric energy measurement (normal distribution)

The energy of an high energy electron is measured by a scintillating fiber calorimeter by collecting light produced by the electromagnetic cascade in

3G. Bohm’s definition is slightly di erent, i.e. more restrictive, see examples at the end of this section.

4.2 Di erent Types of Measurement Uncertainty

85

the scintillator of the device. From the calibration of the calorimeter with electrons of known energies E we know that the calorimeter response is well

described by a Gaussian with mean proportional to E and variance propor-

tional to E.

Many experimental signals follow to a very good approximation a normal distribution. This is due to the fact that they consist of the sum of many contributions and a consequence of the central limit theorem.

A typical property of statistical errors is that the relative error is proportional

to 1/ N, the inverse of the square root of the number of events. This is obvious in

the first and third example, but also in the second example the relative error would

show the 1/ N behavior if we were able to repeat the measurement N times. If the relative error is su ciently small, we usually can treat it as independent of the observed value.

Errors Determined from a Sample of Measurements

An often used method for the estimation of errors is to repeat a measurement several times and to estimate the error from the fluctuation of the results. The results presented below will be justified in subsequent chapters but are also intuitively plausible.

In the simplest case, for instance in calibration procedures, the true value xt of the measured quantity x is known, and the measurement is just done to get information about the accuracy of the measurement. An estimate of the average error δx of x from N measurements is in this case

(δx)2 = N1 XN (xi − xt)2 .

i=1

We have to require that the fluctuations are purely statistical and that correlated

systematic variations are absent, i.e. the data have to be independent from each

other. The relative uncertainty of the error estimate follows the 1/ N law. It will be studied below. For example with 100 repetitions of the measurement, the uncertainty of the error itself is reasonably small, i.e. about 10 % but depends on the distribution of x.

When the true value is unknown, we can approximate it by the sample mean

 

 

1

N

 

 

recipe:

 

 

=

 

Pi=1 xi

and use the following

 

 

 

 

 

x

 

 

 

 

 

 

N

N

 

 

 

 

 

 

 

Xi

 

 

 

 

 

(δx)2 =

 

1

(xi

 

)2 .

(4.1)

 

 

 

 

N 1

x

 

 

 

 

 

 

 

=1

 

 

 

In the denominator of the formula used to determine the mean quadratic deviation (δx)2 of a single measurement figures N − 1 instead of N. This is plausible because, when we compute the empirical mean value x, the measurements xi enter and thus they are expected to be in average nearer to their mean value than to the true value. In particular the division by N would produce the absurd value δx = 0 for N = 1, while the division by N − 1 yields an indefinite result. The derivation of (4.1) follows from (3.15). The quantity (δx)2 in (4.1) is sometimes called empirical variance. We have met it already in Sect. 3.2.3 of the previous chapter.

86 4 Measurement errors

Table 4.1. Error estimate for a mean value.

measurements

 

 

quadratic deviations

 

 

xi

 

 

 

(xi

 

 

)2

 

 

 

x

2.22

 

 

 

0.0009

 

2.25

 

 

 

0.0000

 

2.30

 

 

 

0.0025

 

2.21

 

 

 

0.0016

 

2.27

 

 

 

0.0004

 

 

xi = 11.25

 

 

 

(xi

 

)2

 

=2

0.0054

 

2

 

x

 

 

 

 

 

 

 

 

 

Px = 2.25

 

=

 

 

 

 

(δx)

 

P

P

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Frequently, we want to find the error for measurements xi which are constrained by physical or mathematical laws and where the true values are estimated by a parameter fit (to be explained in subsequent chapters). The expression (4.1) then is generalized to

 

N

 

 

Xi

 

(δx)2 =

1

(xi − xˆi)2 .

(4.2)

N Z

 

 

=1

 

where xˆi are the estimates of the true values corresponding to the measurements xi and Z is the number of parameters that have been adjusted using the data. When we compare the data of a sample to the sample mean we have Z = 1 parameter, namely x¯, when we compare coordinates to the values of a straight line fit then we have Z = 2 free parameters to be adjusted from the data, for instance, the slope and the intercept of the line with the ordinate axis. Again, the denominator N − Z is intuitively plausible, since for N = Z we have 2 points lying exactly on the straight line which is determined by them, so also the numerator is zero and the result then is indefinite.

Relation (4.2) is frequently used in particle physics to estimate momentum or coordinate errors from empirical distributions (of course, all errors are assumed to be the same). For example, the spatial resolution of tracking devices is estimated from the distribution of the residuals (xi − xˆi). The individual measurement error δx as computed from a M tracks and N points per track is then estimated quite reliably to

 

 

1

M×N

(δx)2 =

 

 

Xi

(N

 

Z)M

(xi − xˆi)2 .

 

 

 

 

=1

Not only the precision of the error estimate, but also the precision of a measurements can be increased by repetition. The error δx of a corresponding sample mean is, following the results of the previous section, given by

(δx)2 = (δx)2/N ,

 

 

N

 

1

Xi

(4.3)

=

 

 

(xi

x

)2 .

N(N 1)

 

 

 

=1

 

 

 

Example 49. Average from 5 measurements

In Table 4.1 five measurements are displayed. The resulting mean value is x = 2.25 ± 0.02. We have used that the error of the mean value is smaller by

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