Propagation of error for multivariable function
Now consider a multivariable function f(u, v, w,…). If measurements of u, v, w,. All have uncertainty δu, δv, δw, …., how will this affect the uncertainty of the function?
+∂∂
+∂∂
+∂∂
= L text)of (3.48)(Equation wwf v
wf u
u f f δδδδ
⎟⎟⎠
⎞⎜⎜⎝
⎛+
∂∂
+∂∂
+∂∂
±=∴ L w wf v
wf u
u f ) .... , w,v,f(u f 000 δδδ
This works for cases like systematic errors, when the errors of most of the variables have the same sign. For cases like random errors, this overestimate and give an upper bound of the actual error:bound of the actual error:
W ill t d th f d l t i th
w wf v
wf u
u f f L+
∂∂
+∂∂
+∂∂
≥ δδδδ
We will study the case of random error later in the course.
Example
If f = x – y, calculate δf in terms of δx and δy.
y y f x
xf f δδδ
∂∂
+∂∂
=
y1x1
y y
y)-(x x x
y)-(x
δδ
δδ
−+=
∂∂
+∂
∂=
y x y1 x 1
δδδδ
+=
+
Add not subtractAdd, not subtract
Example
If f = xyz, calculate the fraction error in f.
fff ∂∂∂
(xyz)(xyz)(xyz)
z z f y
y f x
xf f δδδδ
∂∂∂
∂∂
+∂∂
+∂∂
=
z xy y xz x yz
zz
(xyz) y y
(xyz) x x
(xyz)
δδδ
δδδ
++=
∂∂
+∂
∂+
∂∂
=
xyzz xy y xz x yz
ff
δδδ
δδδδ ++=∴
zz
yy
x x δδδ
++=
For product of variables the resultant fraction error is the sum of the fractionFor product of variables, the resultant fraction error is the sum of the fraction errors of the variables.
Example (Problem 3.7(d) of text)A t d t k th f ll i tA student makes the following measurement:a = 5 ± 1 cm, b = 18 ± 2 cm, c = 12 ± 1 cm, t= 3.0 ± 0.5 s, m= 18 ± 1 gramCompute the quantity mb/t with its uncertainties and percentage uncertainties
mb
t tf b
b f m
m f f
t
mb f
∂∂
+∂∂
+∂∂
=
=
δδδδ
g/s-cm 5.03
181823
1813
18
tt
mbb tm m
tb
2
2
⎟⎠⎞
⎜⎝⎛ ×
×+×+×=
−++= δδδ
/110/1081818mbf
g/s-40cm g/s-36cm
333
×≈=
⎠⎝
g/s-cm 40 110 f
g/s-cm110g/s-cm1083
t
f
±=∴
≈===
40% 36% %10011040 y uncertaint Percentage ≈=×=
Topics Part 1 – Single measurement
1. Basic stuff (Chapter 1 and 2)2. Propagation of uncertainties (Chapter 3)
Part 2 – Multiple measurements as independent results
1. Mean and standard deviation (Chapter 4)2. Basic on probability distribution function (not in text explicitly)3. The Binomial distribution (Chapter 10)4. The Poisson distribution (Chapter 11)5. Normal distribution (first half of Chapter 5)6. χ2 test – how well does the data fit the distribution model? (Chapter 12)
Part 3 – Multiple measurements as one sample
1. Central limit theorem (not in text explicitly)2. Normal distribution (second half of Chapter 5)3. Propagation of error (Chapter 3)4. Rejection of data (Chapter 6)5. Merging two sets of data together (Chapter 7)
Part 4 ‐ Dependent variablesPart 4 Dependent variables
1. Curve fitting (Chapter 8)2. Covariance and correlation (Chapter 9)
Mean and standard deviationMean and standard deviationThe mean and variance of a collection of data of sample size N are defined as:size N are defined as:
x ondistributisampleofmean x
i
N
1i∑===
tmeasuremen theof variance N
p
N
2 =σ
N
)x-(x
2i
1i∑==
deviation standard thecalled is N
)x-(x 2i
N
1i∑==σ
N
ExampleExample
3.74.34.23.94.54.03.7}4.3,4.2, 3.9,4.5,{4.0,dataofcollection aConsider
+++++
4.1 6
24.6
63.74.34.23.94.54.0 Mean
==
+++++=
4.1)-(3.74.1)-(4.34.1)-(4.24.1)-(3.94.1)-(4.54.1)-(4.0
deviationStandard
6
222
222
+++
++
0.07 6
0.42
6deviation Standard
==
=
of range in the fallprobably t willmeasuremennext expect the We
0.26 6
∴
=
0.264.1 ±
Two Definitions of Standard DeviationTwo Definitions of Standard Deviation
2N
∑
N
)x-(x 2i
1i∑==σ is called the population standard deviation
1N
)x-(x 2i
N
1i∑==σ is called the sample standard deviation
1-N
For large N, population standard deviation ≈ sample standard deviation. At this point, we will use population standard deviation, but later sample standard deviation will become more important.p
Population Standard DeviationPopulation Standard Deviation
)x-(x 2i
N
∑
N
)x(xi1i∑==σPopulation standard deviation is a measure
of the actual spread of the dataof the actual spread of the data.
When the sample size is N=1, the population standard deviation equals to 0, because there is no spread in the data. However, this does not mean this is a good collection of data.data.
HistogramFrequency N
Variable xbin
Assuming all bins have the same size, if you do one more measurement, probability for itsAssuming all bins have the same size, if you do one more measurement, probability for its value to fall in bin i = Pi . This is probably the best you can guess from the data. N
N N
N i
jj
i ==∑
Normalized Histogram
Variable xAssuming all bins have the same size, you can just read Pi from the vertical axis of a normalized histogram.
Note that 1P =∑Note that 1 P j
j =∑
Probability Density Function
When N →∞, we can make the P(x)bin size → 0 (=dx). When this happens, the normalized histogram will become a probability density function:probability density function:
Pi → P(x)dx
∫∑ 1dx P(x) 1Pj
j =⇒= ∫∑
Random variable x
Unfortunately, we will never know P(x) because it is impossible to satisfy N →∞.
ExamplepFor a fair dice, the probability distribution function is discrete for integers from 1 to 6:
P(x)
1/6
xx
1 2 3 4 5 6
ExamplepConsider probability density function:
P(x)
11
xx0
1 2
0.51and0.5between occur for x toy Probabilit 0.005 0.01 0.5
y=×≈
Normalization condition of the probability density functionprobability density function
:case Discrete].x,[xx If ba∈
1 )x(p bi
i
xx
xx=∑
=
=
:case Continuousb
ai
x
xx =
1 dx )x(p b
a
x
x
=∫
ExamplepFor a fair dice, the probability distribution function is discrete for integers from 1 to 6:
P(x)
1/6
xx
1 2 3 4 5 6
1111116
∑ 161
61
61
61
61
61 )x(p i
1xi
=+++++=∑=
ExampleConsider probability density function:
P(x)
11
xx01 2
2xx2
2x dx x)(2 dx x dx P(x)
221221
⎥⎦
⎤⎢⎣
⎡−+⎥
⎦
⎤⎢⎣
⎡=−+= ∫∫∫
212
244 0
21
22 1010
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −−⎟
⎠⎞
⎜⎝⎛ −+⎥⎦
⎤⎢⎣⎡ −=
⎦⎣⎦⎣∫∫∫
123
24
21 =−+=
ExampleExampleA fair dice is thrown, if the out come is 1, I pay you $5.9. Otherwise, you pay me $1.00. What is the expectation of your earning?
x∈{1,2,3,4,5,6}. If f(x)= your earning,. f(1) = 5.9, f(2)=f(3)=f(4)=f(5)=f(6)=‐1.0
Since the dice is fair, p(1)=p(2)=p(3)=p(4)=p(5)=p(6)=1/6
f( ) $5 9/6 $1 0/6 $1 0/6 $1 0/6 $1 0/6 $1 0/6∴<f(x)>=$5.9/6‐$1.0/6 ‐$1.0/6 ‐$1.0/6 ‐$1.0/6 ‐$1.0/6
=$5.9/6‐$5.0/6
= $0.15
∴ I expect you to earn $0.15 per game.
MeanMeanThe mean of a distribution p(x) is just the expectation of x, <x> or x.
].,[x x If a∈ bx
)x(pxx
:caseDiscrete
∑=><= bi xx
:case Continuous
)x(px x ∑=><= ai
iixx
dx )x(xp x ∫=><bx
ax
ExamplepFor a fair dice, the probability distribution function is discrete for integers from 1 to 6:
P(x)( )
1/6
x
1 2 3 4 5 6
)x(px xMean ii
6
1x=>< ∑
=The mean is the average f l f
3.5621
66
65
64
63
62
61
1xi
==+++++=
= face value after many trials.
ExampleConsider probability density function:
P(x)
11
xx01 2
xxd)(dd( )23
2132
21 ⎤⎡⎤⎡
∫∫∫
1241118401
3xx
3x dxx)x(2dx xdx xP(x) x
1
2
01
2
0
=−+=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −−⎟
⎠⎞
⎜⎝⎛ −+⎥⎦
⎤⎢⎣⎡ −=
⎥⎦
⎤⎢⎣
⎡−+⎥
⎦
⎤⎢⎣
⎡=−+==>< ∫∫∫
333333 ⎥⎦
⎢⎣
⎟⎠
⎜⎝
⎟⎠
⎜⎝⎥⎦⎢⎣