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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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