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Atomic LabAtomic Lab
Introduction to Error AnalysisIntroduction to Error Analysis
Significant FiguresSignificant Figures For any quantity, x, the best measurement of x is For any quantity, x, the best measurement of x is
defined as xdefined as xbestbest ±±xx In an introductory lab, In an introductory lab, x is rounded to 1 x is rounded to 1
significant figuresignificant figure Example: Example: x=0.0235 -> x=0.0235 -> x=0.02 x=0.02
g= 9.82 ± 0.02g= 9.82 ± 0.02 Right and WrongRight and Wrong
Wrong: speed of sound= 332.8 ± 10 m/sWrong: speed of sound= 332.8 ± 10 m/s Right: speed of sound= 330 ± 10 m/sRight: speed of sound= 330 ± 10 m/s
Always keep significant figures throughout Always keep significant figures throughout calculation, otherwise rounding errors introducedcalculation, otherwise rounding errors introduced
Statistically the SameStatistically the Same
Student A = 30 Student A = 30 ± 2± 2Student B = 34 ± 5Student B = 34 ± 5Since the uncertainties for A & B overlap, Since the uncertainties for A & B overlap,
these numbers are statistically the samethese numbers are statistically the same
PrecisionPrecision
Mathematical DefinitionMathematical Definition
Precision of speed of sound= 10/330 ~ 0.33 or Precision of speed of sound= 10/330 ~ 0.33 or 33%33%
So often we write: speed of sound= 330 So often we write: speed of sound= 330 ± 33%± 33%
bestx
xprecision
Propagation of Uncertainties– Propagation of Uncertainties– Sums & DifferencesSums & Differences
Suppose that x, …, w are independent Suppose that x, …, w are independent measurements with uncertainties measurements with uncertainties x, …, x, …, w w and you need to calculate and you need to calculate
q= x+…+z-(u+….+w)q= x+…+z-(u+….+w)
If the uncertainties are independent i.e. If the uncertainties are independent i.e. w is not w is not sum function of sum function of x etc then x etc then
Note: Note: q < q < x+…+ x+…+ z+ z+ u+…+ u+…+ ww
2222 wuzxq
Propagation of Uncertainties– Propagation of Uncertainties– Products and QuotientsProducts and Quotients
Suppose that x, …, w are independent Suppose that x, …, w are independent measurements with uncertainties measurements with uncertainties x, …, x, …, w and you w and you need to calculate need to calculate
If the uncertainties are independent i.e. If the uncertainties are independent i.e. w is not sum w is not sum function of function of x etc then x etc then
wu
zxq
2222
2222
w
w
u
u
z
z
x
xqq
or
w
w
u
u
z
z
x
x
q
q
Functions of 1 VariableFunctions of 1 Variable
xdx
dqq
Suppose Suppose = 20 = 20 ± 3 deg and want to find cos ± 3 deg and want to find cos 3 deg is 0.05 rad3 deg is 0.05 rad |(d(cos|(d(cos)/d)/d|=| -sin|=| -sin|= sin|= sin (cos(cos)= sin)= sin* * = sin(20 = sin(20oo)*(0.05))*(0.05) (cos(cos 20 20oo) = 0.02 rad and cos 20) = 0.02 rad and cos 20oo= 0.94= 0.94
So So coscos= 0.94 = 0.94 ± 0.02 ± 0.02
Power LawPower Law
Suppose q= xSuppose q= xnn and x and x ± ± xx
x
xn
q
q
Types of ErrorsTypes of Errors
Measure the period of a revolution of a Measure the period of a revolution of a wheelwheel
As we repeat measurements some will be As we repeat measurements some will be more or some lessmore or some less
These are called “random errors” These are called “random errors” In this case, caused by reaction timeIn this case, caused by reaction time
What if the clock is slow?What if the clock is slow?
We would never know if our clock is slow; we We would never know if our clock is slow; we would have to compare to another clockwould have to compare to another clock
This is a “systematic error”This is a “systematic error”
In some cases, there is not a clear difference In some cases, there is not a clear difference between random and systematic errorsbetween random and systematic errors Consider parallax:Consider parallax:
Move head around: random errorMove head around: random error Keep head in 1 place: systematicKeep head in 1 place: systematic
Mean (or average)Mean (or average)
N
xx
N
xxxx
N
ii
N
1
21
DeviationDeviation
ii xxd Need to calculate an average or “standard” deviation
To eliminate the possibility of a zero deviation, we square di
11
2
N
dN
ii
x
When you divide by N-1, it is called the population standard deviation
If dividing by N, the sample standard deviation
Standard Deviation of the MeanStandard Deviation of the Mean
The uncertainty in the best measurement is given by the standard deviation of the mean (SDOM)
N
N
d
xx
N
ii
x
11
2If the xbest = the mean, then best =mean
HistogramsHistograms
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
1 2 3 4 5 6
Value
Number of times that value has occurred
Distribution of a Craps GameDistribution of a Craps Game
Histogram
05
101520253035404550
2 3 4 5 6 7 8 9
10
11
12
Mo
re
Bin
Fre
qu
en
cy
Bell CurveOr Normal Distribution
Bell CurveBell Curve
0
50
100
150
200
250
300
350
400
0 2 4 6 8 10 12 14
Dice Value
Fre
qu
ency
Dice
Gaussian
Centroid or Mean
x- x+
68 %
Between x-2 to x+2, 95% of population2 is usually defined as Error
GaussianGaussian
0
50
100
150
200
250
300
350
400
0 2 4 6 8 10 12 14
Dice Value
Freq
uenc
y
Dice
Gaussian
X0
2
20
2* x
xx
eAGaussian
In the Gaussian, x0 is the mean and x is the standard deviation.
They are mathematically equivalent to formulae shown earlier
Error and UncertaintyError and Uncertainty
While definitions vary between scientists, While definitions vary between scientists, most would agree to the following most would agree to the following definitionsdefinitions
Uncertainty of measurement is the value Uncertainty of measurement is the value of the standard deviation (1 of the standard deviation (1 ))
Error of the measurement is the value of Error of the measurement is the value of two times the standard deviation (2 two times the standard deviation (2 ))
Full Width at Half MaximumFull Width at Half Maximum
A special quantity is the full width at half A special quantity is the full width at half maximum (FWHM)maximum (FWHM) The FWHM is measured by taking ½ of the maximum The FWHM is measured by taking ½ of the maximum
value (usually at the centroid)value (usually at the centroid) The width of distribution is measured from the left side The width of distribution is measured from the left side
of the centroid at the point where the frequency is this of the centroid at the point where the frequency is this half valuehalf value
It is measured to the corresponding value on the right It is measured to the corresponding value on the right side of the centroid. side of the centroid.
Mathematically, the FWHM is related to the Mathematically, the FWHM is related to the standard deviation by FWHM=2.354*standard deviation by FWHM=2.354*xx
Weighted AverageWeighted Average
Suppose each measurement has a unique Suppose each measurement has a unique uncertainty such as uncertainty such as
xx11 ± ± 11
xx22 ± ± 22
……
xxNN ± ± NN
What is the best value of x?What is the best value of x?
We need to construct statistical We need to construct statistical weightsweights
We desire that measurements with small errors We desire that measurements with small errors have the largest influence and the ones with the have the largest influence and the ones with the largest errors have very little influencelargest errors have very little influence
Let w=weight= 1/Let w=weight= 1/ii22
N
ii
N
iii
best
w
xwx
1
1
*
This formula can be used to determine the centroid of a Gaussian where the weights are the values of the frequency for each measurement
N
ii
x
wbest
1
1
Least Squares FittingLeast Squares Fitting
What if you want to fit a straight line What if you want to fit a straight line through your data?through your data?
In other words, yIn other words, yii = A*x = A*xii + B + BFirst, you need to calculate residualsFirst, you need to calculate residualsResidual= Data – Fit orResidual= Data – Fit orResidual= yResidual= yii – (A*x – (A*xii+B)+B)When as the Fit approaches the Data, the When as the Fit approaches the Data, the
residuals should be very small (or zero). residuals should be very small (or zero).
Big ProblemBig Problem
Some residuals >0Some residuals >0Some residuals <0Some residuals <0 If there is no bias, then rIf there is no bias, then rjj = -r = -rkk and then r and then rjj
+ r+ rkk =0 =0
The way to correct this is to square rThe way to correct this is to square r jj and and
rrk k and then the sum of the squares is and then the sum of the squares is
positive and greater than 0positive and greater than 0
Chi-square, Chi-square, 22
N
i i
ii BxAy
1
2
2
We need to minimize this function with respect to A and B soWe take the partial derivative of w.r.t. these variables and set the resulting derivatives equal to 0
0
0
1
22
1
22
N
i i
ii
N
i i
ii
BxAy
BB
BxAy
AA
Chi-square, Chi-square, 22
NBxAy
xBxAyx
NBB
BxAyBxAy
xBxAyxxBxAyx
BxAyBxAy
BB
xBxAyBxAy
AA
N
ii
N
ii
N
ii
N
ii
N
iii
N
i
N
i
N
ii
N
ii
N
iii
N
ii
N
ii
N
iii
N
iiiii
N
iii
i
N
i i
ii
N
iiii
i
N
i i
ii
11
11
2
1
1
1111
2
11
2
11
2
12
1
22
12
1
22
0
0
02
02
Using DeterminantsUsing Determinants
N
ii
N
ii
N
iii
N
ii
N
ii
N
ii
N
iii
N
ii
N
ii
N
ii
N
ii
N
ii
N
ii
N
ii
N
iii
yx
yxx
B
Ny
xyx
ANx
xx
NBxAy
xBxAyx
11
11
2
1
11
1
11
2
11
11
2
1
A PseudocodeA PseudocodeDim x(100), y(100)Dim x(100), y(100)xsum=0xsum=0x2sum=0x2sum=0Xysum=0Xysum=0N=100N=100Ysum=0Ysum=0For i=1 to 100For i=1 to 100
xsum=xsum+x(i)xsum=xsum+x(i) ysum=ysum+y(i)ysum=ysum+y(i)
xysum=xysum+x(i)*y(i)xysum=xysum+x(i)*y(i)x2sum=x2sum+x(i)*x(i)x2sum=x2sum+x(i)*x(i)
Next INext IDelta= N*x2sum-(xsum*xsum)Delta= N*x2sum-(xsum*xsum)A=(N*xysum-xsum*ysum)/DeltaA=(N*xysum-xsum*ysum)/DeltaB=(x2sum*ysum-xsum*xysum)/DeltaB=(x2sum*ysum-xsum*xysum)/Delta
2 2 ValuesValues
If calculated properly, If calculated properly, 2 2 start at large values andstart at large values and
approach 1approach 1 This is because the residual at a given point should This is because the residual at a given point should
approach the value of the uncertaintyapproach the value of the uncertainty Your best fit is the values of A and B which give Your best fit is the values of A and B which give
the lowest the lowest 22
What if What if 22 is less than 1?! is less than 1?! Your solution is over determined i.e. a larger number Your solution is over determined i.e. a larger number
of degrees of freedom than the number of data pointsof degrees of freedom than the number of data points Now you must change A and B until the Now you must change A and B until the 2 2 doesn’t doesn’t
vary too muchvary too much
Without ProofWithout Proof
N
iy
B
i
N
iy
A
i
i
N
x
1
2
2
1
2
2
Extending the MethodExtending the Method
Obviously, can be Obviously, can be expanded to larger expanded to larger polynomials i.e. polynomials i.e. Becomes a matrix Becomes a matrix
inversion probleminversion problem
Exponential FunctionsExponential Functions Linearize by taking Linearize by taking
logarithmlogarithm Solve as straight lineSolve as straight line
N
i i
iii CxBxAy
1
222
tAtA
eAtA t
0
0
ln)(ln
)(
Extending the MethodExtending the Method
Power LawPower Law
Multivariate Multivariate multiplicative multiplicative functionfunction
tAEfficiency
tAEfficiency
AxNI
AxI N
loglogloglog
logloglog
Uglier FunctionsUglier Functions
q=f(x,y,z)q=f(x,y,z) Use a gradient search Use a gradient search
methodmethod Gradient is a vector Gradient is a vector
which points in the which points in the direction of steepest direction of steepest ascentascent
f = a direction f = a direction So follow So follow f until it f until it
hits a minimumhits a minimum
zz
yy
xx
Correlation Coefficient, rCorrelation Coefficient, r22
2
11
2
2
11
2
1112
N
ii
N
ii
N
ii
N
ii
N
ii
N
ii
N
iii
yyNxxN
yxyxNr
rr22 starts at 0 and approaches 1 as fit gets better starts at 0 and approaches 1 as fit gets better rr22 shows the correlation of x and y … i.e. is shows the correlation of x and y … i.e. is
y=f(x)?y=f(x)? If rIf r22 <0.5 then there is no correlation. <0.5 then there is no correlation.