Schedule
Lecture Exp
Date
A
July 5
3
B
July 10
4
A
July 12
5
B
July 17
6
A
July 19
B
July 24
A
July 26
B
July 31
Lecture Topics
Course Overview
Discussion of Exp 1 – Goals, setup
(Deduce mean density of the
earth)
Measurements, uncertainties.
Statistical Analysis
Intro to error propagation
Discussion of Exp 2 – goals, setup
(Deduction of mass distribution)
Histograms & distributions
The Gaussian Distribution,
Maximum likelihood,
Rejected data, Weighted mean
Discussion of Exp 3 – goals, setup
(Tune a shock absorber)
Fitting
Chi-squared test of distribution
Discussion of Exp 4 – goals, setup
(Calibrate a voltmeter)
Chi-squared
Covariance and correlation
Final Exam Review
August 2
Final Exam
1
July 3
2
1
2
3
7
8
9
10
4
Assignment
Lab: -Prepare for Quiz #1
Taylor: -Read chapters 1-3, HW 1
Lab: -Analyze data for Exp #1
Taylor: -Read chapter 4, HW 2
Lab: -Prepare for quiz #2
Taylor: -Read chapter 5, HW 3
Lab 1
Due!!
Lab: -Analyze data for Exp #2
Taylor: -Read chapters 6-7, HW 4
2nd Quiz
Lab:
Taylor:
Lab:
Taylor:
Lab:
Taylor:
Lab:
Taylor:
Lab:
8PM, York 2722
Physics 2BL Summer I 2012
-Prepare for quiz #3
-Read chapter 8, HW 5
-Analyze data for Exp #3
-Read chapters 9 & 12
-Prepare for quiz #4
-HW 6
-Analyze data for Exp #4
-Prepare for final exam
-Pick up graded work from
TAs
-Pick up final from LTAC
3rd Quiz
4th Quiz
1
Measurement, Uncertainties,
Statistical Analysis, Intro to Error
Propagation
Lecture # 2
Physics 2BL
Summer Session I 2012
Physics 2BL Summer I 2012
2
Lecture #2:
• Issues from experiment 1?
– Tuesday you will need to turn in your lab
notebook/report before the end of lab
– Start it at home!
• Errors – random & systematic
• Statistical analysis (uncertainties for
datasets)
• Error propagation – general formula and
special cases
Consult the course web site
regularly! Our main way to
• Homework
communicate with you…
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Measurement and Observation
• Measurement: deciding the amount of a
given property by observation
• Empirical
• Not logical deduction
• Not all measurements are created equal…
Physics 2BL Summer I 2012
4
Reproducibility
• Same results under similar circumstances
– Reliable/precise
• ‘Similar’ - a slippery thing
– Measure resistance of metal
•
•
•
•
need same sample purity for repeatable measurement
same size and shape
need same people in room?
same potential difference?
– Measure outcome of treatment on patients
• Can’t repeat on same patient
• Patients not the same
Physics 2BL Summer I 2012
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Precision and Accuracy
Not the same thing!
• Precise - reproducible
• Accurate - close to true value
• Example - temperature measurement
– thermometer with
• fine divisions (high precision)
• or with coarse divisions (low precision)
– and that reads
• 0° C in ice water (highly accurate)
• or 5 ° C in ice water (not so accurate)
Physics 2BL Summer I 2012
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Accuracy
Accuracy
vs. Precision
Distance from true value
Spread between trials
P
r
e
c
i
s
i
o
n
Accuracy
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What do we mean when we say “error(s)”?
Errors
Uncertainties
(not mistakes)
Inevitable and intrinsic part of any
experiment
How do we estimate uncertainties?
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8
How to estimate uncertainty on a single
measurement – Ex: Length
• For lengths with well-defined (easy to reach) edges:
– use precision of measurement tool
– Ex: measuring distance L from beach to cliff in Exp 1
• For things with unclear edges:
– Estimate the range within which the length is likely to fall and
use the center of that range as the best guess
– Ex: measuring l from pivot point to the center of mass of the
bob
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Types of Measurement Errors:
• Random Errors:
– Unavoidable
– Can be estimated by repeating measurements
– Can also be reduced by repeating measurements
• Systematic Errors:
– Hard to estimate, hard to reduce
– Can be due to calibration errors, neglecting
small corrections, or procedural mistakes
Get those clickers ready if you want extra credit…
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Statistical Analysis
• How to arrive at the the best estimate and
uncertainty for a collection of repeated
measurements as opposed to a single measurement
– We use repeated measurements to
• Improve accuracy
• Estimate uncertainty
xbest ± δx
How to determine xbest?
How to estimate δx??
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The mean
x1 , x2 ,..., xN
xbest = x
N measurements of the quantity x
Best estimate
mean
the average or
x1 + x2 + ... + xN
1
x=
=
N
N
N
∑x
i =1
i
 N

 ∑ xi = ∑ xi = ∑ xi 
i
 i =1

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Standard Deviation (RMS)
d i = xi − x
≈
σx =
1
N
∑ di
2
Deviation of ith measurement from
the mean:
Average sum of the squares of the
uncertainty
1
2 Standard deviation: uncertainty
(xi − x )
∑
in any single measurement of x
N −1
xi ± δx = xi ± σ x
68% of all measurements will fall within the range of xtrue ± σx
How to estimate the uncertainty of xbest?
Physics 2BL Summer I 2012
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The Standard deviation of the Mean
(SDOM)
σx =
σx
N
Uncertainty in mean (best guess) is
the standard deviation of the mean
Based on the N measured values x1, …, xN we
can state our final answer for the value of x:
(best value of x) = xbest ± δx
xbest = x
Physics 2BL Summer I 2012
δx = σ x = σ x
N
15
Example: lab 1
We measure the period of a pendulum 3 times and find the
results: T = 1.43 s, 1.48 s, 1.42 s. Give the best guess for T?
Best guess? T =
1.43 + 1.48 + 1.42
= 1.443s
3
Standard deviation (error for 1.43 s, 1.48 s, 1.42 s):
σT =
[
]
1
2
2
2
(
)
(
)
(
)
1
.
43
−
1
.
443
+
1
.
48
−
1
.
443
+
1
.
42
−
1
.
443
= 0.032 s
∑
3 −1
Error for the best guess (SDOM): σ T =
Best guess:
σT
0.032
=
= 0.018s
N
3
T = 1.443 ± 0.018s
Physics 2BL Summer I 2012
ALWAYS state values
with uncertainties (and
units), without which the
value is meaningless
16
Propagation of error: Ex Lab 1
g = 4π l T
2
2
• Several period measurements: T = 1.443 ± 0.018 s
• Suppose you estimate l = 52.0 ± 0.3 cm
• From l and T, get best estimate of g
4π 2 (0.520m )
2
g=
(1.443s )
2
= 9.8589 m s
• What about uncertainty in g: δg?
• Error analysis tells us how errors propagate
through mathematical functions
Suppose you measure x ± δx, what is the
uncertainty in q(x)
Physics 2BL Summer I 2012
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Error propagation
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Error Propagation
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Error Propagation
What if multiple dependent variables, i.e. g(l, T)?
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General Formula for Error Propagation
Single variable function q(x)
∂q
δq =
δx
∂x
For any function q(x, y,…, z)
2
Partial derivative:
Differentiate w.r.t
one variable
while treating the
others constant
2
 ∂q   ∂q 
 ∂q 
δq =  δx  +  δy  + ... +  δz 
 ∂x   ∂y 
 ∂z 
2
Add independent & random errors in
quadrature (similar to Pythagorean Theorem)
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Ex: Error Propagation for Experiment 1
Acceleration due to gravity: g = 4π 2l T 2
Lab data: T = 1.443 ± 0.018s l = 0.520 ± 0.003m g = 9.8589 m s
Error in g: δg =  ∂∂gl δl  +  ∂∂Tg δT 
2


2
± δg
2


∂g
4π 2
4π 2
2
(
)
δl = 2 δl =
0
.
003
m
=
0
.
0569
m
s
∂l
T
(1.443s )2
∂
 ∂l ( Al ) =

A

∂g
8π 2l
8π 2 (0.520m )
−2
−3 
2  ∂
(
)
AT
=
−
2AT
(
)
0
.
018
s
=
0
.
246
m
s
δT = 3 δT =
 ∂T

∂T
T
(1.443s )3
Dominant uncertainty
δg =
(0.0569 m s ) + (0.246 m s )
2 2
2 2
= 0.25 m s 2 ≈ 0.3 m s 2
You should show
this much work
when you do error
propagation!
2
g
=
9
.
9
±
0
.
3
m
s
Acceleration due to gravity:
Physics 2BL Summer I 2012
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Error Propagation – Special cases
Sum and differences: q = x + y
q= x− y
Independent errors
Upper limit (just FYI)
δq = (δx ) + (δy )
δq ≤ δx + δy
2
2
Example: error for ∆t = t1 – t2
t1 ± δt1
t2 ± δt2
2
2
(
)
(
)
(
)
δ ∆t = δt1 + δt2
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Error Propagation – Special cases
Products and quotients: q = xy
q=x y
Independent errors
Ex: error for ρ = 3g/4πGRE
g ± δg
RE ± δRE
2
 δg   δRE 
δρ

=   + 
ρ
 g   RE 
(just FYI)
2
δq
 δx   δy 
=   +  
q
 x  y
δq δx δy
≤ +
q
x
y
2
Fractional errors: ratio
of error to the true value
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2
Error Propagation – Special cases
nth order polynomial:
q = Ax
δq
q
n
=n
δx
δq = A δx
x
Example: error for g = 4πl/T2
l ± δl
T ± δT
δg
 δl 
2
Measured value & exact
constant: q = Ax
 δT 
=   + 2 
g
 l   T 
2
Ex: error for single period, T,
from 10 periods, T = (10T)/10:
(10T) ± δ(10T)
δT =
Physics 2BL Summer I 2012
δ (10T )
10
25
Uncertainties
in Counting:
Error Propagation Summary
q = N (integer #)
δq = N
Uncertainties in Products
and Ratios:
q = xy
q = x/ y
δq = (δx )2 + (δy )2
δq ≤ δx + δy
Uncertainties
in Measured
Value and
exact constant:
q = xn
Uncertainties in Sums
and Differences:
q= x+ y
q= x− y
Uncertainties
in nth order
polynomial:
δq
 δx 
 δy 
=   +  
q
 x  y
δq δx δy
≤
+
q
x
y
2
q = Ax
2
δq
q
=n
δx
δq = A δx
x
 ∂q   ∂q 
δq =  δx  +  δy 
General Rule:
 ∂x   ∂y 
For independent random errors
(≤upper bound)
∂q
∂q
δq ≤ δx + δy
∂x
∂y
*always use radians when calculating the errors on trig functions
2
Physics 2BL Summer I 2012
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2
When all else fails, use general rule
for error propagation!
 ∂q   ∂q 
δq =  δx  +  δy 
 ∂x   ∂y 
2
2
Example: error for height of cliff
hcliff = hperson + Lcos θ
hperson ± δhperson
L ± δL
θ ± δθ
2
δhcliff
 ∂h
  ∂h
  ∂h

=  cliff δh person  +  cliff δL  +  cliff δθ 
 ∂h
  ∂L
  ∂θ

 person

∂hcliff
∂hcliff
∂hcliff
=1
= − L sin θ
= cos θ
∂h person
∂θ
∂L
δhcliff =
(δh
2
Use radians!!!!!
) + (cosθδL ) + (L sin θδθ )
2
person
2
2
Physics 2BL Summer I 2012
2
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How to combine random and
systematic error?
δxtot =
(δxrandom ) + (δxsystematic )
2
Physics 2BL Summer I 2012
2
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Homework
Measure time between sunsets!
If you didn’t finish pendulum, visit Chris’s
office hours (M 12-2pm)
Read Taylor chapter 4
Start analysis so you finish Lab 1 on time!
HW2: Taylor problems 4.6, 4.14, 4.18, 4.26
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