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: -Analyze data for Exp #2
Taylor: -Read chapters 6-7, HW 4
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
Lab 2
Due!!
3rd Quiz
4th Quiz
1
Gaussian Distribution, Rejection
of Data, Maximum Likelihood,
Weighted Mean
Lecture # 4
Physics 2BL
Summer Session I 2012
Physics 2BL Summer I 2012
2
Lecture #4:
• Issues from experiment 2?
– Tuesday you will need to turn in your lab
notebook/report before the end of lab
– Start it at home! (may need to retake data)
• Recap:
–
–
–
–
Error Propagation
Gauss (Normal) distribution
Confidence level
Data Rejection
Knock on west-facing
doors on first floor of
MHA or call office
phone if you need to be
let in for office hours
Visit course web site to
see if your iClicker is
unregistered (or forfeit
your extra credit for
coming to lecture?)
• Principle of Maximum Likelihood
• Experiment 2 writeup
• Homework 4
Physics 2BL Summer I 2012
3
Issues From Lab 1
• Good job on error propagation!
– MAKE SURE to show work and point out the
dominant source(s) of error
– (see Lecture 2, slide 22)
• Be careful to use the correct quantities for
calculations!
– Height of cliff
hcliff = h person + L cos θ ≠ L cos θ
Physics 2BL Summer I 2012
θ
θ
4
General Formula for Error Propagation
For independent, random errors:
2
 ∂q   ∂q 
 ∂q 
δq =  δx  +  δy  + ... +  δz 
 ∂x   ∂y 
 ∂z 
2
2
Whenever you are unsure, use this!
(All special cases are derived using this)
Set of measurements x1, x2, …, xN
Error for single measurement: xk ± σ x
Best guess with error: x ± σ x
Physics 2BL Summer I 2012
5
Gaussian (Normal) Distribution
• Expression for bell curve
Peak position
(~ mean value)
1
− ( x − X )2 2σ 2
G X ,σ ( x ) =
e
σ
2
π
Width (& height) parameter
Probability
(~ standard deviation)
density!
Normalization factor
• Probability of measuring within a t-value of true value
erf (t ) = ∫
X + tσ
X −tσ
t=
x A − xB
σA +σB
2
2
G X ,σ ( x )dx
=
xk − x
σx
Physics 2BL Summer I 2012
6
Gauss Distributions - Applications
1
− ( x − X )2
G X ,σ ( x ) =
e
σ 2π
• Confidence levels
– Compatibility of independent measurements
– Acceptability of a single measured result
• Rejection of Data
– Chauvenet’s criterion
x1 ,..., x N
tsus =
xsus − x
σx
n = N * Prob(|t| ≥ tsus)
If n < 0.5, the reject xsus
• Principle of Maximum Likelihood
• Weighted Average
Physics 2BL Summer I 2012
8
2σ 2
The Principle of Maximum Likelihood
• Gaussian is one example of probability density
1
− ( x − X )2
PX ,σ (x ) =
e
σ 2π
2σ 2
• Make repeated measurements of x to reduce errors
x1 , x2 ,..., x N
• Define the likelihood as the product of the
individual probabilities
L = P ( x1 )P (x2 )...P ( x N )
• Larger L, the more likely a set of measurements is
The best estimate for the parameters of P(x) (peak value,
X, and width, σ) are those that maximize L
Physics 2BL Summer I 2012
10
Using the Principle of Maximum Likelihood
to Prove the mean is the best estimate of X
L = P ( x1 )P ( x2 )...P ( x N )
∂L ∂χ 2
1 − χ 2  ∂χ 2 


= − Ce
∂χ ∂X
2
 ∂X 
∂χ 2
=0
∂X
∂χ 2
1 N
= − 2 ∑ 2( xi − X ) = 0
∂X
σ i =1
2
Assume X is a parameter of P(x)
When L is a maximum, must have:
∂L
=0
∂X
Assume a normal error distribution
and find the formula for the best
value of X
N
N
1
− ( xi − X )2
e
L = ∏ P ( xi ) = ∏
i =1 σ 2π
i =1
L=
χ
2
(σ
1
2π
−
)
N
e
N
∑ ( xi − X )2
i =1
N
=
∑ ( xi − X )2
i =1
σ2
2σ 2
= Ce
2σ 2
−χ2 2
(e e
A B
= e A+ B )
⋅
2
N
N
N
∑ (x − X ) = 0 = ∑ x −∑ X
i =1
i
i =1
i
i =1
 N 
 ∑ xi  − NX = 0
 i =1 
1 N
X = ∑ xi = x
N i =1
Physics 2BL Summer I 2012
11
Aside: What is the Uncertainty of the Mean?
• Formula for the mean
1
x=
N
N
∑x
i =1
i
• Use propagation of errors to get uncertainty
2
2
 ∂x

 ∂x
  ∂x

δx =  δx1  + 
δx2  + ... + 
δxN 
 ∂x1   ∂x2

 ∂x N

∂x
1
=
∂xi N
1

δx = ∑  σ x 

i =1  N
N
δxi = σ x
2
σx
σx 
=
= N 
N
N
2
Physics 2BL Summer I 2012
2
Just as we knew, the
SDOM is the uncertainty
on the mean value
What if the xi had
different errors?
12
Weighted Averages
We can use the principle of
maximum likelihood (χ2) to get
the best value of measurements
with different uncertainties
χ
2
N
=
∑
i =1
 xi − X

 σi



2
i =1
Ex: How to find the best value
(with uncertainty) for two
groups who independently
determined the moment of
inertia (with uncertainty)
Uncertainty on weighted mean:
δx wav =
N
∂χ 2
xi − X
= −2 ∑
=0
2
∂X
σi
i =1
N
N
xi
1
∑ 2 − X∑ 2 =0
wi ≡
i =1
1
σi
2
σi
“weight”
N
N
∑w x − X ∑w
i =1
1
i i
wi
i =1
∑w x
i =1
N
Physics 2BL Summer I 2012
i i
∑w
i =1
i =1
i
=0
N
X=
N
∑
σi
= xwav
i
13
Example: Weighted Averages
Two sets of lab partners in 2BL
independently measure the acceleration
due to gravity in the lab. The results
were gA = 9.9 ± 0.6 m/s2 and gB = 10 ±
3 m/s2. What is the best estimate of the
acceleration due to gravity?
wA =
wB =
1
σ A2
1
σB
2
=
1
= 2.78
2
0.6
g wav =
=
1
= 0.111
2
3
δg wav =
N
X=
∑w x
i =1
N
i i
= xwav
∑w
i
i =1
δx wav =
1
N
∑ wi
i =1
wA g A + wB g B (2.78)9.9 + (0.111)10
=
= 9.90 m s 2
wA + wB
2.78 + 0.111
1
=
wA + wB
1
= 0.58 m s 2
2.78 + 0.111
g = 9.9 ± 0.6 m/s2
Physics 2BL Summer I 2012
14
Experiment 2
Make sure to label all measurements
with uncertainties and units!
Briefly mention how you estimated
the value of each uncertainty
d
Ex: Radius, R, of Racquetball?
Actually measure diameter, d, with calipers
d
δd
R=
δR =
2
2
δd = 0.1mm (instrument precision)
0.1mm
δR =
= 0.05mm
2
Physics 2BL Summer I 2012
Uncertainties
in Measured
Value and
exact constant:
q = Ax
δq = A δx
15
Experiment 2: Plots
How to plot (for 1 ball N times)
standard deviation and SDOM as
function of N? (using EXCEL)
(only valid for N ≥ 2 trials)
Create a column for st. dev., SDOM
Standard deviation (EXCEL):
σt(N) “=stdev([t1]:[tN])”
Ex: “=stdev(B$3:B4)”
SDOM (EXCEL):
“=[σt(N)]/sqrt([N])”
Ex: “=C4/sqrt(A4)”
Physics 2BL Summer I 2012
16
Exp 2 - Plots
Title???
0.025
0.025
Y--Axis Title?
0.02
Std Deviation
constant
0.015
0.015
stddev
dev
std
0.01
SDOM
SDOM
0.005
0.005
SDOM falls off
as 1/sqrt(N)
0
1
3
5
7
9
15 17
17 19
19 21
21 23
23 25
25 27
27 29
29
11 13 15
X-Axis Title?
Physics 2BL Summer I 2012
17
Exp 2: Plots (Histogram)
• Determine the range of your data
(largest value - smallest value)
• Choose number of bins ≥ 3 ≈ N
• Width of bins, ∆k, is range divided
by #
• List bin boundaries, count number
of data points, nk, in each bin
• Draw histogram, y-scale, fk, may be
the # measurements in each bin
(
8
7
6
5
4
3
2
1
0
)
Series1
Physics 2BL Summer I 2012
• (sort>custom sort)
• sqrt(30) = 5.48 ~ 6!
• =(1.3283 s – 1.2836 s)/6
=0.0075 s
• countif([range],[criteria])
• CHECK that total data
points equals N
• Don’t forget labels, units,
titles!
• Maybe label mean, std
dev on histogram (to
justify if your data can or
can not be represented
with a normal
distribution)
18
Exp 2 - Calculations
• Calculate rolling radius, moment of inertia
– Don’t forget error propagation (show work) & units
– Check how dimensionless I compares to lecture notes
• Variation (in time) due to manufacturing (st dev)
σ t (man ) = σ t (total )2 − σ t (meas )2
N ball 1
Check that σt(man) is real!
1 ball N
• Estimate fractional error in thickness
• Calculate error in thickness variation
1 σd
σ 
δ d  =
N d
 d 
Assumes manufacturing data
is normally distributed
• Determine N such that
σd 
 ≤ 1%
 d 
σd
d
= 27
σ t (man )
t
Mean value
for 1 ball N
δ
Physics 2BL Summer I 2012
19
Exp 2 Conclusion - Chauvenet
• Use outlier with the largest deviation from
the mean
x1 ,..., x N
tsus =
xsus − x
σx
n = N * Prob(|t| ≥ tsus)
If n < 0.5, the reject xsus
• Redo analysis if outlier gives n < 50%
Physics 2BL Summer I 2012
20
Exp 2 Conclusion – Normal Distribution
• For N measurements, how many are
expected to be within t0 std deviations of the
mean:
– # = N * Prob(|t| > t0) = N
∫
X + tσ
X −tσ
G X ,σ ( x )dx
• How many are expected to be beyond t0
standard deviations from the mean?
Physics 2BL Summer I 2012
21
Homework
Check σmanufacturing is REAL!
If you need to retake data, visit Chris’s
office hours (M 10am-12pm)
Read Taylor chapter 6-7
Start analysis so you finish Lab 2 on time!
HW4: Taylor problems 6.4, 7.2
Physics 2BL Summer I 2012
23