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Lectures for chemists on statistics II. The normal distribution: a briefer on the univariate case

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Abstract

Motivated by the introduction of a supplement to the GUM suggesting Monte Carlo simulation as a method for establishing a complete measurement uncertainty budget, some properties of the normal distribution are reviewed. The normal distribution is the central distribution of parametric statistics and sampling from normal distributions is a regularly occurring activity in statistical simulation by Monte Carlo methods. Algorithms for computer generation of normal deviates, areas under the normal curve, and the error function are given to encourage practical simulation. Some critical issues, e.g. coverage and influential observations, are presented. The discussion is restricted to the univariate situation. There is no intention to provide an exhaustive presentation. Statistics is a specialised field requiring sophisticated competences. The text may serve as a golden thread to acquire a quick survey on a subject of general interest.

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Notes

  1. The term “observation” is used here, and at other locations in this manuscript, in its statistical sense: a result of an experiment or trial, where either numerical or categorical (e.g. head or tail of a tossed coin) may be obtained.

  2. The term “statistic” refers to a quantity characterising a set of observations and which is not dependent on unknown parameters.

  3. Those familiar with these definition will notice that the distinction between weakly and strongly consistent estimators is not elaborated because this text is of an introductory nature.

Abbreviations

N :

Normal distribution

μ :

Location parameter of normal distribution

σ :

Dispersion parameter of normal distribution

x :

arbitrary value, observation

π:

pi (3.1415...)

v :

Variance (v = s 2)

n :

Sample size

x i :

ith sample out of n samples

\(\bar{X}\) :

Mean value

P :

Error function (cumulative normal distribution)

e:

Eulerian number

ɛ i :

Disturbance

e i :

Residual

dz :

d operator; z variable

z :

Variable

Φ :

Cumulative normal distribution

χ 2 :

Chi-square distribution

t :

Student t distribution

m′:

Mid-range

m :

Integer number(s)

E:

Expectation operator

θ, θ′:

Statistic and its estimator

Pr:

Operator (probability)

ε :

Lower bound

M :

Median

g :

A function

F :

Empirical distribution function

y :

Threshold

p :

Probability

erf:

Error function

d max :

Kolmogorov parameter

CI:

Confidence interval

α :

Confidence level

df :

Degrees of freedom

k :

Coverage factor

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Authors and Affiliations

  1. RER Consultants Passau, Schiessstattweg 3a, 94032, Passau, Germany

    Günther Meinrath

  2. Institute of Inorganic Chemistry, Technische Universität Bergakademie Freiberg, Leipziger Str. 27, 09596, Freiberg, Germany

    Günther Meinrath

  3. Boojum Research Technology Ltd, 20 Mississauga Valley Blvd, Mississauga, ON, Canada, L5A351

    Günther Meinrath

Authors
  1. Günther Meinrath
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Correspondence to Günther Meinrath.

Appendix

Appendix

The programs for calculating a standardized cumulative normal distribution and the error function are written in QBasic and run in a DOS box. QBasic is part of the Windows 95 operating system and is included on most Windows 98 installation disks. It is also available on many Internet sites. There are also compilers available for QBasic. The code given here is intended for demonstration purposes only. All arithmetic operations are made in single precision, even though the introduction of appropriate statements would allow to run them in double precision without requiring complicated modifications. The algorithms are useful for those interested in implementing their own stochastic simulations.

The algorithm is an adaption of Adam’s Algorithm 39 [17]. The code has been tested against other code and tabulated values of the error function.

REM ****** Algorithm 39: Areas under the normal curve *******************

figure a

The second piece of code returns the error function erf(x). It uses the relationship between the cumulative normal distribution and the error function. An alternative Chebyshev polynomial approximation to the area under the normal curve is evaluated. From this value, erf(x) is obtained via Eq. 10.

REM ****** evaluates the error function erf(x) **************

  • CONST a1 = 1.26551223#

  • CONST a2 = 1.00002368#

  • CONST a3 = .37409196#

  • CONST a4 = .09678418#

  • CONST a5 = −.18628806#

  • CONST a6 = .27886807#

  • CONST a7 = −1.13520398#

  • CONST a8 = 1.48851587#

  • CONST a9 = -.82215223#

  • CONST a10 = .17087277#

  • CONST Pi2 = 6.283

  • sqr2 = SQR(2)

  • LOCATE 12, 12

  • INPUT "value, for which erf(x) should be evaluated"; x

  • erfin = x

  • GOSUB 100

  • CLS

  • LOCATE 12, 12

  • PRINT "erf:"; erf

  • END

REM ******* evaluates erf(x) ***********************

  • 100 z = ABS(x)

  •   t = 1! / (1! + .5 * z)

  •   t1 = a5 + t * (a6 + t * (a7 + t * (a8 + t * (a9 + t * a10))))

  •   ans = t * EXP(-z * z - a1 + t * (a2 + t * (a3 + t * (a4 + t * t1))))

  •   IF erfin >= 0 THEN erf = 1! - .5 * ans ELSE erf = .5 * ans

  •   erf = 2 * (erf - .5)

  •   RETURN

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Meinrath, G. Lectures for chemists on statistics II. The normal distribution: a briefer on the univariate case. Accred Qual Assur 13, 179–192 (2008). https://doi.org/10.1007/s00769-008-0359-9

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