Uncertainties of measurement in EXCEL

History and basic terms

RNDr. Ctibor Henzl, Ph.D.

VŠB – Technical University Ostrava

RNDr. Ctibor Henzl, Ph.D.

VŠB – Technical University Ostrava

Example

References

Conclusions

Uncertainties of measurement represent a statistical approach to the

evaluation of measured data. Alas, approximately since150 years we hear

that there are three forms of falsehoods:

falsehoods

punishable falsehoods

statistics

This often highly cited proposition about statistics is ascribed on Benjamin Disraeli, other times Lord Palmerston is regarded as

the authorBenjamin Disraeli(1804-1881) Lord Palmerston(1784-1865)

In his beautiful book „Moderní statistika, “Dr. Helmut Swoboda indicates two reasons of this abusive statement: 1.Insufficient knowledge of

methods, aims and facilities of the statistics 2.Many people consider statistics

as something which is actually pseudostatistics

The International Committee for The International Committee for Weights and Measures (CIPM) hasWeights and Measures (CIPM) has committed itself since 70 years of committed itself since 70 years of

the last century to create an unique the last century to create an unique methodology for processing and methodology for processing and

evaluating results of measurement evaluating results of measurement

1977 The International Committee for Weights and Measures (CIPM) asked the International Bureau of Weights and Measures (BIPM) to collaborate with national metrology institutes and elaborate recommendations for the solution of the situation, i.e. recommendation of an uniform approach 1980 recommendation is published as

Recommendation INC-1 (1980), see [5].

1990 West European Calibration Committee issued document WECC Doc. 19-1990 see [6] which is a source for a lot of recommendations and norms, see References.

Uncertainties of „A“ type originate from random errors. Their evaluation is based on a statistical analysis of series of observations.

Estimation of uncertainties of „A“ type

The arithmetic mean (the estimate of the quantity) is calculated

mjn

xx

n

i

jij ,...,1,

1

m is number of values, n is number of measurements

The sample variance of is calculated

n

ijjixAx xx

nnsu

jj1

222

1

1

Experimental standard deviation of the mean is used as a standard uncertainty of „A“ type

n

ijjixAx xx

nnsu

jj1

2

1

1

jx

jxs

Uncertainties of „B“ type originate from systematic errors. The evaluation of this uncertainty can not be based on statistical analysis of series of observations. Relevant information are

Technical documentation

Experience with relevant materials and instruments

Knowledge of previous data etc.

Guess zmax – maximal deviation of value, appropriate to source z and look up relevant probability distribution in the following table

Estimation of uncertainties of „B“ type

Determine uncertainty of „B“ type

2

2max2

z

uBz If deviation cannot be practically

exceeded = 3, if deviaton can be exceeded = 2

Gaussian law of propagation of uncertainties

Let us assume that y is a function of values x1, x2, … , xm

),,,( 21 mxxxfy

),,,( 21 mxxxfy

The mean of y is

Uncertainty of the every value x1, x2, … ,xm contributes to the resulting uncertainty

m

xxx

xxx

xxx

mA

x

f

x

fx

f

sss

sss

sss

x

f

x

f

x

fu

mmm

m

m

y

2

1

2

2

2

21

2

2,1,

,221,2

,12,11

,,,

Uncertainty of , symbolized by uAy is calculated by means of Gaussian law of propagation of uncertainties. The “Matrix form“ of this law is very suitable for computer programming.

y

mm xXxX ,,11

The partial derivatives f /xj (referred to as sensitivity coefficients) are evaluated at

There are sample variances of mean of values x1,…,xn in the main diagonal of the matrix

n

ijjiAxx xx

nnus

jj1

222

1

1

m

xxx

xxx

xxx

mA

x

f

x

fx

f

sss

sss

sss

x

f

x

f

x

fu

mmm

m

m

y

2

1

2

2

2

21

2

2,1,

,221,2

,12,11

,,,

kki

n

ijjix xxxx

nns

jk

11

1

There are sample covariance of means in the adjacent diagonal of the matrix

kj xx ,

Off-diagonal elements are zero, if variables x1,…,xn are not correlated. The result is then more simple

m

x

x

x

mA

x

f

x

fx

f

s

s

s

x

f

x

f

x

fu

m

y

2

1

2

2

2

21

2

00

00

00

,,, 2

1

2

2

,,,1

2

1211

j

mm

x

xXxXxX

m

j jAy s

X

fu

Gaussian law of propagation of uncertainties must again be applied when calculating uncertainties of type „B“.

Combined uncertainty

22ByAyCy uuu

Combined uncertainty uCy includes both types of uncertainties. It is computed as the geometrical mean of uAy and uBy

Expanded uncertainty

Expanded uncertainty u is the product of the combined uncertainty uCy and the coverage factor k.

Cyuku

Result

The result can be written in the form

If k = 2, estimated values are approximately normally distributed and expanded uncertainty is uCy, then the unknown value of y is believed to lie in the interval defined by u with a level of confidence of approximately 95%.

uyuyY ;

uyY

Let us compute the Body mass index (BMI) of a group of students.

Example

The personal weighting-machine LUXA and folding rule LOGAREX 38031 were used.

A group of 15 students 15 years old was chosen among the first year students of theTelecommunication school in Ostrava

2l

mBMI

Body mass index is defined as the ratio of mass in kg and the square of height in m.

Mass m and length l are the directly measured values. BMI is calculated for each student..

The arithmetic mean, complete with uncertainty of measurement is calculated for the group.

BMI and health risk are related (see Table).

Very HighObesity 3. level40,0 and more

HighObesity 2. level35,0 – 39,9

AverageObesity 1. level30,0 – 34,9

IncreasedOverweight25,0 – 29,9

MinimalNormal18,5 – 24,9

Health riskStatusBMI

Very HighObesity 3. level40,0 and more

HighObesity 2. level35,0 – 39,9

AverageObesity 1. level30,0 – 34,9

IncreasedOverweight25,0 – 29,9

MinimalNormal18,5 – 24,9

Health riskStatusBMI

n

ijjiAx xx

nnu

j1

22

1

1

2

2max2

z

ujBx

Gaussian law

Table ofresults

Uncertainties of „A“ type are computed using formulas

2

1

22 kg12,21

1

n

immiAm xx

nnu

Commentaries

2

1

22 m0001,01

1

n

illiAl xx

nnu

The formulas are based on statistical analysis of series of observations

CommentariesUncertainties of „B“ type are computed by

2

2max2

z

ujBx

zmax is estimated, is determined from distribution of probability, see table

Commentaries

2

2

2

2

2max2

1max kg3,03

1

3

1rectangle1

z

uon distributikg,z Bm

22

2

2max2

2max kg702,09

5,0 normal50

z

uondistributikg,,z Bm

26

2

2

2

2max2

max m1083

005,0rectangle0050

z

uondistributim,,z Bl

Estimation of „B“ type uncertainties in our example

The partial derivatives f/xi are calculated

from their analytical expression2

22,

21

m32,077,1

11

ll

m

mX

f

llmm

3334

,2

2

mkg2,2477,1

0,67222

l

m

l

lm

l

m

lX

f

llmm

422

2

2

2

2

1

2 mkg28,0

AlAmAy uX

fu

X

fu

Commentaries

We use the Gaussian law of propagation of uncertainties for calculating uncertainties of „A“ type

422

2

2

22

21

2

1

2 mkg04,0

BlBmBmBy uX

fuu

X

fu

The combined uncertainty uc is

222 kgm57,0 yByAc uuu

Commentaries

The Gaussian law of propagation of uncertainties is used for calculating uncertainties of „B“ type

The expanded uncertainty u is

2kgm1,12 cuu

Commentaries

The final result is given by

2-__________

kgm1,14,21 BMI

ConclusionsComputing of uncertainties of A and B

type is described in this paperComputation has been made according

to documents WECC doc. 19-1990 and EAL-4/02

A practical-oriented example has been solved

Spreadsheet EXCEL has been used for computations

References

[1] ISO, Guide to Expression of Uncertainty in Measurement (International Organisation for Standardization), Geneva, Switzerland, 1993

[2] NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, US Government Printing Office, Washington, 1994

[3]Publication Reference EAL-4/02 (European cooperation for Accreditation of Laboratories),1999

[4] WECC Doc. 19-1990 : "Guidelines for Expression of the Uncertainty in Calibrations

[5] http://physics.nist.gov/cuu/index.html

[6] http://www.european-accreditation.org

[7] http://www.bipm.org/CC/documents /JCGM/bibliography_on_uncertainty.html/