Download Lecture 4 - Error Propagation

Scientific Measurement
PHY-103
Dr. Eram Rizvi & Dr. Jeanne Wilson
Lecture 4 - Error Propagation
Measurement Uncertainty
Imagine you need to determine area of a table
You measure x and y the sides of the table
x = 95.0 ± 0.5 cm
y = 190.0 ± 0.5 cm
Area = 1.81 m2
What is the uncertainty on the area?
Any quantity determined from measurements will also have an uncertainty!
Dr Eram Rizvi
Scientific Measurement - Lecture 4
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Measurement Uncertainty
Consider i measurements of area A
Uncertainty of A is given by std deviation of all i measurements
ignore terms like ΔxΔy if x and y are independent
!
!
!
this is the covariance term
!
!
!
is correlation of errors in x and y
!
!
!
= 0 if x, y are random & indep.
Dr Eram Rizvi
Scientific Measurement - Lecture 4
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For F=function(x,y) then
σF2
=
�
∂F
∂x
�2
σx2
+
�
∂F
∂y
�2
σy2
provided that x and y are
independent measurements
This is the last formula you will need in this course - memorise it!
Consider the table area example:
!
Area = x.y
2
σA
=
�
∂A
∂x
�2
σx2
+
= y 2 σx2 + x2 σy2
� σ �2
A
A
Dr Eram Rizvi
=
� σ �2
x
x
+
�
σy
y
�2
�
∂A
∂y
�2
σy2
Fractional error on A is quadratic sum of fractional errors on x & y
Pythagorus' Theorem is an example of a quadratic sum: a2=b2+c2
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Why a quadratic sum?
Combined error can never be smaller than on measured quantities
x = 95.0 ± 0.5 cm means:
y
High probability true value lies between x-Δx and x+Δx
Note: you do not guarantee that this is true
(we will quantify this probabilistic statement later)
measurements
2.1 2.2
x
2.3 2.4 2.5
If errors are random and x, y measured independently then equally probable
to measure underestimate OR overestimate in y
Probability to measure overestimate in x AND y simultaneously is small
Independent measurements = subject to only random uncertainties
Dr Eram Rizvi
Scientific Measurement - Lecture 4
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Lets look at pendulum example...
T = 2π
�
L
g
σg2
σg2
=
�
�
g = 4π 2
=
4π 2
T2
σg
g
∂g
∂L
�2
�2
σg
=
g
Dr Eram Rizvi
�
�2
2
σL
2
σL
+
+
�
�
∂g
∂T
�2
σT2
�2
σT2
−8π 2 L
T3
L
T2
2
σL
σT2
= 2 +4 2
L
T
�
2
σT2
σL
+
4
L2
T2
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Consider several simple examples - F is a function of variables x,y,z... and k is a constant
F =x+k
σF = σx
F = xy
� σ �2
F = kx
σx
σF
=
F
x
F = xn
� σ �2
F
F
F
F
=
absolute error remains the same
� σ �2
= n2
x
x
+
�
σy
y
�2
fractional errors add in quadrature
fractional error remains unchanged
� σ �2
x
x
fractional error is scaled by n
All these results derived from general formula
Results always given in terms of absolute / fractional error on measured quantity
Total error is never smaller than measurement error!
Dr Eram Rizvi
Scientific Measurement - Lecture 4
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How about sums and differences?
F =x+y
σF2 = σx2 + σy2
absolute errors add in quadrature
F =x−y
σF2 = σx2 + σy2
absolute errors add in quadrature
In both cases error is the same, but F can be very different!
Consider F=x+y
!
!
!
!
!
!
!
!
!
Consider F=x-y
!
!
!
!
!
!
!
!
!
Dr Eram Rizvi
x = 10 ± 1
y=9±2
!
F = 19 ± √(12+22)
!
F = 19 ± √5
!
F = 19 ± 2
number of sig.figs not ±2.23
!
!
!
!
F = 1 ± √(12+22)
F = 1 ± √5
F=1±2
error is larger than the central value!
Scientific Measurement - Lecture 4
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Formula can be trivially extended to many independent variables, x,y,z...
σF2
=
�
∂F
∂x
�2
σx2
+
�
∂F
∂y
�2
σy2
+
�
∂F
∂z
�2
σz2 ...
Note: Largest error contribution comes from largest error source
Can sometimes neglect one error if it is much smaller than others
σg
=
g
�
2
σT2
σL
+
4
L2
T2
If
σL
= 1%
L
and
σT
= 5%
T
then we can neglect error on L
By inspection we see
Dr Eram Rizvi
σT
σg
=2
g
T
Scientific Measurement - Lecture 4
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Full example involving products, sums, powers & multiplicative factors!
Newton's equations for distance travelled by object
velocity v = 200 ± 10 m/s
acceleration a = 12 ± 2 m/s2
time t = 6.0 ± 0.2 s
s = 1416 m
s = 1416 ± 89 m
Dr Eram Rizvi
Scientific Measurement - Lecture 4
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Beware: some complex formulae can lead to long calculations
May not be worthwhile - use your judgement
In such cases use:
In other words calculate the value of F for x+!x and for x-!x and take half the difference
Dr Eram Rizvi
Scientific Measurement - Lecture 4
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