assessing and presenting experimental datadragon.ccut.edu.tw/~mejwc1/e-mea/pdf/em_3.pdf · what...

Assessing and Presenting Experimental DataCommon Types of error

Uncertainty and Precision Uncertainty

Theory Based on the PopulationTheory Based on the Sample

Introduction

• Actual data• Error

– The difference between the measured value and the true physical value

How good are the data?

Error in Measuring

• Error : the difference between the measured value and the true value

• bound (ε) ; uncertainty (u)

truem xxError −≡= ε

1):(n uu +≤≤− ε

1):(n xm uxxu mtrue +≤≤−

Common Types of Error

• Bias errors : systematic errors– The same way each time a measurement is

made– Example: the scale on an instrument

• Precision errors : random errors– Different for each successive measurement

but have an average value of zero– Example: mechanical friction or vibration

Bias and Precision errors

• Bias errors > Precision errors

Measured value, xm

Freq

uenc

y of

occ

urre

nce

xmxtrue

Total error

Bias error

Precision error

Bias and Precision errors

• Bias errors < Precision errors

Measured value, xm

Freq

uenc

y of

occ

urre

nce

xmxtrue

Total error

Bias error Precision error

Classification of Errors1) Bias or systematic error

a. Calibration errorsb. Certain consistently recurring human errorsc. Certain errors caused by defective equipmentd. Loading errorse. Limitations of system resolution

2) Precision or random errora. Certain human errorsb. Errors caused by disturbances to th equipmentc. Errors caused by fluctuating experimental

conditionsd. Errors derived from insufficient measuring-system

sensitivity

Classification of Errors3) Illegitimate error

a. Blunders and mistakes during an experimentb. Computational errors after an experiment

4) Errors that are sometimes bias error and sometimes precision error

a. From instrument backlash, friction, and hysteresisb. From calibration drift and variation in test or

environmental conditionsc. Resulting from variations procedure or definition

among experimental

Elements of instrument error

• Hysteresis error


• Linearity error


• Sensitivity error


• Zero shift(null) error


• Repeatability error

Calibration errors

• Ideal response : xmeasured=xtrue

• Actual response : xmeasured=βxtrue +xoffset

1

1

Ideal response1

β≠1

Input, xtrue

Out

put,

x mea

sure

d

xoffset

Actual response

hysteresis error

• Backlash and mechanical friction

Ideal response

Input, xtrue

Out

put,

x mea

sure

d

Actual response

In Rating Instrument Performance

• Accuracy – The difference between the measured and

true values– Maximum error as the accuracy– The extent to which a reading might be wrong,

and is often quoted as a percentage of the full-scale reading of an instrument

• For example: ±1% of full-scale reading• Accuracy:

100

1 ×

−−=

valuetruevalueindicatedvaluetrue

A


• Precision– The difference between the instrument’s

reported values during repeated measurements of the same quantity

– Determined by statistical analysis– A term which describes an instrument’s

degree of freedom from random errors


• Precision


• Accuracy & Precision

low accuracy ; low precision high accuracy ; low precision

high accuracy ; high precisionlow accuracy ; high precision


• Resolution – The smallest increment of change in the

measured value that can be determined from the instrument’s readout scale

– Same (or smaller) order as the precision• Sometimes specified as an absolute value and

sometimes as a percentage of full-scale deflection


• Sensitivity– The change of an instrument or transducer’s

output per unit change in the measured quantity

– Higher sensitivity will also have finer resolution, better precision, and higher accuracy

• The sensitivity of measurement is therefore the slope of the straight line in measured quantity v.s. output reading characteristic chart


• Sensitivity

Measured quantity

Outputreading

Gradient=Sensitivity of Measurement


• Sensitivity to disturbance

Measured quantity

Scalereading

Nominal characteristic

Characteristicwith zero drift

Zerodrift



Measured quantity

Scalereading


Characteristicwith sensitivity drift

Sensitivitydrift



Measured quantity

Scalereading


Characteristic withzero and sensitivity drift

Zero drift plus Sensitivity drift

Precision error and accuracy


• Hysteresis– The non-coincidence between these loading

and unloading curves


• Dead space– As the rang of different input values over

which there is no change in output value

-

Measuredvariable

Outputreading

+-

+

Dead space


• Range or span– An instrument defines the minimum and

maximum values of a quantity that the instrument is designed to measure

• Input span: Rinput=xmax-xmin

• Output span: Routput=ymax-ymin


• Threshold– If the input to an instrument is gradually from

zero, the input will have to reach a certain minimum level before the change in the instrument output reading is of a large enough magnitude to be detectable. This minimum level of input is known as threshold for instrument.

Uncertainty

• Bias uncertainty, Bx

• precision uncertainty, Px

• Total uncertainty, Ux

)( 22xxx PBU +=

Uncertainty exampleA brass rod axial strain, yielding an average strain of ε=520 µ-strain(520 ppm). A precision uncertainty Pε=21 µ-strain with 95% confidence. The bias uncertainty is estimated to be Bε=29 µ-strain with odds of 19:1 (95% confidence). Whatis the total uncertainty of the strain ?

Solution.The total uncertainty for 95% coverage is

Uε=(Bε2+Pε2 ) 1/2 =36 µ-strain (95%)In other word, with odds of 19:1 , the true strain liesin the interval 520 ± 36 µ-strain:

484 µ-strain ≤ ε ≥ 556 µ-strain.

Sample versus Population

• Sample • population

Sample versus Population

群體(population)(所製造的所有品目)

x1 x2 xn

樣本(sample)由母體取出之樣本

Probability Distributions

• Probability is an expression of the likelihood of a particular event taking place, measured eithreference to all possible events.

Probability Distributions

• The Gaussian, or normal, probability distribution– Z-distribution

• Student’s t-distribution– Only a small sample of data is available

• The x2-distribution– in predicting the width of a population’s distribution, in

comparing the uniformity of samples, and in checking the goodness of fit for assumed distributions

Theory Based on the population

• Normal distribution curve


• probability density function, (PDF)

• Gaussian probability density function

∫=→2

121

)(yProbabilit )(x

x

xx dxxf

−−= 2

2

2)(exp

21)(

σµ

πσxxf

x= the magnitude of a particular measurementµ= the mean value of the entire populationσ= the standard deviation of the entire population


• the arithmetic average

• the deviation d = x - µ

• the standard deviation

∑=

=+⋅⋅⋅++

=n

i

in

nx

nxxxx

1

21

µ : The most probable single value for the quantity

nddd n

222

21 +⋅⋅⋅++

≈σ

Standard normal distribution curve

Example a.What is the area under the curve between z=-1.43

and z=1.43 ?b.What is the significance of this area ?

Solution.a. From Table 3.2, read 0.4236. This represent half

the area sought. Therefore,the total area is 2×0.4236=0.8472.

b. The significance is that for data following thenormal distribution, 84.72% of the population lies within the range –1.43 < z < 1.43.

Example What range x will contain 90% of the data ?

Solution.We need to find z such that 90%/2=45% of the

data lie between zero and +z; the other 45% willlie between –z and zero. Entering Table 3.2, we find z0.45≈1.645 (by interpolation). Hence, sincez=(x-µ)/σ, 90% of the population should fall within the range

(µ- z0.45) < x < (µ+ z0.45)or

(µ- 1.645) < x < (µ+1.645)

Theorey Based on the Sample

• We deal with samples from a population and not the population itself– to use average values from the sample to

estimate the mean or standard deviation of the population

• the sample mean

nxxx

nxx n

n

i

i +⋅⋅⋅++==∑

=

21

1

Theorey Based on the Sample

• the sample standard deviation

• Difference between population and sample

1)x(

1)()()(

n

1i

22i

222

21

−

−=

−−+⋅⋅⋅+−+−

=

∑ =

nxn

nxxxxxxs n

x

Sxσ or σxStandard Deviation

µ or µxMean

For sampleFor population

x

An Example of Sampling Results of a 12-hour pressure test

14.05024.04064.030174.020334.010254.000123.99033.98013.970

Number of results, mPressure p, in Mpa

Solution

• Histogram of the pressure data

Solution

• Sample mean and standard deviation

0.0420.0320.0220.0120.002-0.008-0.018-0.028-0.038

Deviationd

1261733251231

Number of results

176.4×10-5

102.4×10-5

48.4×10-5

14.4×10-5

0.4×10-5

6.4×10-5

32.4×10-5

78.4×10-5

144.4×10-5

d2

4.0504.0404.0304.0204.0104.0003.9903.9803.970

Pressurep

Solution

• Sample mean and standard deviation 77.400=∑ p

52 101858 −×=∑d

100n ==∑m

Mpa 008.4100/77.400 ==p

Mpa 014.099/101858 5 =×= −pS

Goodness of Fit

• A given set of data may or may not abide by the assumed distribution and since, at best, the degree of adherence can be only approximate, some estimate of goodness of fit should be made before critical decisions are based on statistical error calculations.

Goodness of Fit

• Normal probability plot

Goodness of Fit

• Graphical effects

Propagation of Uncertainty

• What is that uncertainty?– Finding the uncertainty in a result due to

uncertainties in the independent variables is called finding the propagation of uncertainty.

A linear function y of several independent variablesxi with standard deviations σi; The standard deviation of y is

22

22

2

11

∂∂

+⋅⋅⋅+

∂∂

+

∂∂

= nn

y xy

xy

xy σσσσ

Propagation of UncertaintyWe assume that each uncertainty is small enough that a first-order Taylor expansion of y(x1,x2,…,x3) provides a reasonable approximation:

nn

n

nn

uxyu

xyu

xyxxxy

uxuxuxy

∂∂

+⋅⋅⋅+∂∂

+∂∂

+≈

+++

22

11

21

2211

) ..., , , (

) ..., , , (

Propagation of UncertaintyUnder this approximation, y is linearfunction of the independent variable.

The uncertainties are:22

22

2

11

∂∂

+⋅⋅⋅+

∂∂

+

∂∂

= nn

y uxyu

xyu

xyu

Uncertainty exampleSuppose that y has the form

y=Ax1+Bx2

and that the uncertainties in x1 and x2 are knownwith odds of n:1. What is the uncertainty in y?

Solution.

1).:(n )()(

Eq., above Using

;

22

21

21

BuAuu

BxyA

xy

y +=

=∂∂

=∂∂

Uncertainty example例：兩個電阻值為100Ω之電阻，每個電阻值之公差(不確定度)為 5% 。試求將兩電阻串聯後之總電阻值及其電阻不確定度為何？

Solution.

( ) ( )Ω±Ω

Ω=×+×=

∂∂

+

∂∂

=

Ω=×

Ω=+=

07.7 200 07.75151

RR

RRu

55%100 200RR

22

2

22

2

11

y

21

串聯總電阻為

總共不確定度為

每個電阻之不確定度為

一般串聯總電阻為

uu

R

Uncertainty exampleA cylindrical body of circular section has a normal length of 5000 ± 0.5mm, an outside diameter of 200 ± 0.05mm. Determine the uncertainty in calculated volume.

Solution.

%101.12about or ,1076.1u is volume theofy uncertaint The

1076.1%1012.11057.1

%1012.1 %)01.0(%)0025.02( )()2(v

u

Eq., above Using

0.0025% 2000.05 ; 1%0.0

50000.5

1057.150002004

v; 4

234v

428

22222v

3822

−

−

−

×±×=

×≈×××=

×=+×=+=

====

×=×==

mmuor

lu

du

du

lu

mmldV

v

ld

dl

ππ

Graphical Presentation of Data

• When used to present facts, interpretations of facts, or theoretical relationships, a graph usually serves to communicate knowledge from the author to his readers, and to help them visualize the features that he considers important.

• A graph should be used when it will convey information and portray significant features more efficiently than words or tabulations.

According to the American Standards Association


• For example: atmospheric pressure– The data are tabled

Time of Day Pressure (mbar)

10:00 A.M.11:3001:00 P.M.02:1503:4004:4005:40

1009.0984.2999.8989.0977.1981.2990.0


• For example: atmospheric pressure– The data are graphed

General Rules for Making graphs

• For example: a temperature data


• Minimum effort in understanding• The axes should have clear labels• Use scientific notation• Use real logarithmic axes• The axes should usually include zero• The scales should be commensurate with the

relative importance of the variations• Use symbols for data points

Other rules see textbook pp.101~103


• A pool graph


• improved by graphing guidelines

Choosing Coordinates

• Linear coordinates


• Semi-logarithmic coordinates


• Full logarithmic coordinates


• Polar coordinates

Producing Straight Lines

• For example: cooling dataBy linear coordinates


• For example: cooling dataBy semi-logarithmic coordinates


• For example: plot of y=1.0 + (2.5/x)

As y versus x


• For example: plot of y=1.0 + (2.5/x)As y versus (1/x)

Straight-line Transformations

• y=f(x) ⇒ Y=A+BX

baxnyy=a+bxn

blog alog xlog yy=axb

b log clog axlog yy=acbx

log blog axlog yy=abx

baxX/yy=x/(a+bx)bax1/yy=1/(a+bx)ba1/xyy=a+b/xBAXYF(x)

Line Fitting

• The simplest approach is just to draw appears to be a good straight line through the data

• When this approach is used, the probable tendency is to draw a line that minimizes the total deviation of all points from the line

Bias and precision error in line fitting

Least Square for Line Fits

• y=a + bx

• Correlation coefficient, r

∑ ∑∑ ∑ ∑ ∑

−

−=

22

2

)( ii

iiiii

xxnyxxxy

a∑ ∑∑ ∑ ∑

−

−=

22 )( ii

iiii

xxnyxyxn

b

( )( )∑

∑−+

−= 22

22

)()(

mi

mi

yxySyxy

r[ ]

2

1

2 )(

deviations squared the

∑=

−=n

iii xyyS

Least Square for Line Fits Example

A cantilever beam deflects downword when a mass is attachedto its free end. T deflection, δ(m), is a function of the beam stiffness, k(N/m), the applied mass, M(kg), and the gravitational body force, g=9.807m/s : k δ=MgTo determine the stiffness of a small cantilevered steel beam, astudent place various masses on the end of the beam and measures the corresponding deflections. The deflections are measured using a scale (a ruler) marked in 1mm increments. Each mass is measured in a balance. His results are as follow:

7.56.26.04.83.63.01.80.60Deflection(mm)

401.00

350.05

299.95

250.20

200.05

150.05

099.90

050.15

0Mass(g)


Solution :Setting y=δ and x=M

n=9 ; Σx=1801g ; Σx2=5.109×105 g2 ; Σy=33.50mm ; Σy2=179.3mm2 ; Σxy=9959g·mm ;

The least squares results are theny = a + bx [or δ= a + (g/k)M ] ; a= -0.0755mm ; b=g/k= 0.0190 mm/g ; r= 0.995886 ;

The experimental stiffness of the beam isk= g/b = 9.807/0.0190 = 516 N/m


Beam deflection for various masses


Solution :From the figure, these data do appear to fall on a straight line.The correlation coefficient, r, is nearly unity, but a better test is to consider (1-r2) 1/2 = 0.0906 ≈ 9%. This value indicates that the vertical standard deviation of the data is only about 9% of the total vertical variation caused by the straight-line relationship between y and x.

Solution :

y= 0.645 + 3.6506 x r =1.00


For the following data, determine the equation for y=y(x) by graphical analysis.

x 0 0.43 0.76 1.21 2.60 3.5y 1.00 1.54 3.61 5.25 10.0 13.50

645.0)( 22

2

=−

−=

∑ ∑∑ ∑ ∑ ∑

ii

iiiii

xxnyxxxy

a 6506.3)( 22 =

−

−=

∑ ∑∑ ∑ ∑

ii

iiii

xxnyxyxn

b

( )( )

00.1 ; )(

)(22

22 =

−+

−=

∑∑ r

yxySyxy

rmi

mi


Solution :

0 1 2 3 4

5

10

15

x

y

assessing and presenting experimental datadragon.ccut.edu.tw/~mejwc1/e-mea/pdf/em_3.pdf · what...

Documents