Statistical Methods in Measurement

7/24/2019 Statistical Methods in Measurement

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Errors, Uncertainty and

Statistical Methods in

MeasurementEngineering measurements taken repeatedly will show

variations in measured values.


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Introduction

The experimentalist should always know the

validity of data.

Error will creep into all experiments regardless ofthe care which exerted.

Some of these errors are of a random nature, an

other part, will be due to gross blunders on the

part of the experimenter

The experiment experimentalist will strive to

maintain consistency in primary data analysis


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Sources that contribute to variations:

easurement systems

- !esolution

- !epeatability easurement "rocedure and Techni#ue

- !epeatability

easured $ariable- Temporal variation

- Spatial variation


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%or a given set of measurements, we

want to #uantify:

&. ' single representative value that bestcharacteri(es the average of the data set

). ' representative value that provides ameasure of the variation in the measureddata set

*. +e need to know how well this singleaverage value represents the true averageof the variable measured.


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ncertainty 'nalysis

' more precise method of estimating

uncertainty in experimental result has been

presented by -line and clintock. /-line, S. 0 and %. '. clintock, &12*,Describing Uncertainty in Single-

sample Experiments, ech.Eng3

This method is based on a careful

specification of the uncertainties in variousprimary experimental measurements.


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Error 'nalysis

Suppose a set of measurements is made and

the uncertainty in each may be expressed

with some odd. The result ! given by

R4R/x1,x2,x3,x4, ..,xi, ..,xn, 3 /&3

5et wRbe the uncertainty in the result

wR467/R8 x13w19)7/R8 x23w29

)..7/R8 xn3wn9);&8)/)3


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Examples

Example&

Example)

!emark: The utility of the uncertainty

analysis is that It afford the individual a

basis forselection of a measrement met!o"

to produce a result with less uncertainty.


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=% E'S!EE>T ET?=@. ' resistor has a nominal stated valueof &A B C l percent. ' voltage is impressed on the resistor, and the power dissipation is

to be calculated in two different ways: /&3 from " 4 E)8 ! and /)3 from " 4 E I. In /l3

only a voltage measurement will be made, while both current and voltage will be

measured in /)3. alculate the uncertainty in the power determination in each case

when the measured values of E and I are

E 4&AA$ C&D /for both cases3

I 4 lA' ClD


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Statistical measurements theory

+e can estimate the true valuex# as:x# $x x %&'(

x represents the most probable estimate ofx#based on the

available data and xrepresents the uncertainty intervalin that estimate at some probability level,&'.

The uncertainty interval combines the estimate of the random

error and of the systematic error in the measurement ofx.

Statistical methods are used to estimate the random error

only, therefore we neglect the systematic error.


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Simple "robability oncept

"robability is a mathematical #uantity that

link to the fre#uency which certain

phenomenon occur after a large number oftries.

If several independent events occur at the

same time such that ach event has aprobabilitypi) thusp$* pi.


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"robability density function

!egardless of the care taken random scatter in the data

values will occur F the measured data behaves as a ran"om

+ariable.

@uring repeated measurements of a variable under fixed

operating conditions data point tend to lie within some

interval F central ten"ency.

!efer to eg. G.& pg &&)


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The fre#uency with which the measured variable assumes a particular value or interval of

values is described by its probability density F example as shown the measured values are

plotted on a single axis.


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@evelop and compare the histograms of the three data sets presented

under columns &, ) and *. If these are taken from from the same process,

why might the histogram varyH @o they appear to show a central

tendencyH%/>3 set & %/>3 Set ) %/>3 set *

2&.1 2&.1 2&.&

2&.A GI.J 2A.&

2A.* 2&.& 2&.G

G1.K 2&.J 2A.2

2&.A G1.1 G1.J

2A.A GI.I 2&.K

GI.1 2).2 2&.A

2A.2 2&.J G1.2

2A.1 2&.* 2).G

2).G 2).K G1.2

2&.* G1.G 2&.K

2A.J 2A.* G1.G

2).A 2A.) 2A.IG1.G 2A.1 2A.I

G1.1 2).& 2A.)

G1.) G1.* 2A.&

G1.G 2A.J 2).*

G1.J 2A.2 GI.1

2A.2 G1.J 2A.G2A.J 2A.* 2&.2


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SOLUTION

' distribution is not uni#ue and we give one possible solution.

%or > 4 )A values, - 4 J is selected. The interval /bin3 values are

Bin #Intervalrange

1


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!egression 'nalysis

' measured variable is often a function of one or more independent

variables that are controlled during the measurement. ' regression analysis /!'3 can be used to establish a functional

relationship between the dependent variable and the independent

variable.

' !' assumes that the variation found in the independent measured

variable follows a normal distribution about each axis value ofindependent variable

=ne of the regression analysis is 5eastLs#uare. This method attempts

to minimised the sum of the s#uare of the deviation between the actual

data the polynomial fit of a state of order by adMusting the value of the

coefficient


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When do we need to perform the

regression analysis?

+hen the data

collected are

obtained over therange of input

values /e.g. during

calibration3


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!egression 'nalysis


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Least Squares Regression Analysis

urve fitting is passing a function curve between thedata points, trying to obtain a Nbest fitO.

ommon choice of the function is a polynomial:

In the least s#uares procedure, we minimise the sum

of s#uares of the differences between the exp. @ata

pointsyiand the curve values yci at the same

locations,xi The goal of this method is to reduce @ to a minimum

for given order of polynomial

find min/@3 where


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ow to select polynomial order?

nfortunately, there are no

clear rules for the best

polynomial orderP

Qenerally, choose between)ndto 2thorder. Too high an

order creates curves that are

too wavy.

In the extreme, a polynomial

of >L& order would pass

through each of the >data

points, which is clearly not

the intent of curve fitting

procedure.


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!an we define Sample Mean and

Standard "e#iation in !ur#e $itting? Res. The curve itself represents now the sample mean

within the whole range.

The Sample Standard @eviation is now called the Stan"ar"

Error of t!e ,it) Syx

where m is the polynomial order, > is the total number of data,yiare the

raw data points and yciare the curve values at locationsxi and+$ D,


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Student%t"

istri&ution


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Example G.G

onsider the data of Table G.&. /a3 ompute

the sample statistics for this data set

/b3Estimate the interval of values overwhich 12D of the measurements should be

expected to lie. /c3 Estimate the true mean

value of the measurand at 12D probabilitybased on this finite data set.


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Example G., G.1


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hoices of the graph format

h i f h h f


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hoices of the graph format


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& ).AGK)K*) &.KJ2AA& &2

* &.*J)A)1A1

G &.&)**))G&

2 A.1l1K1KA*

K A.J2)122*

J A.K&KG1)G&

A.2AGJG&)121 A.G&*)GJ))l

&A A.****)A

&* A&2K*1GK

&J A.A*G**lJ2

)A AAG2J1A1J

x y


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Statistical Methods in Measurement

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