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Propagation of errors
Compute error in q(x,y) from errors in x and y
_ δx
x=x +
_ δq
q=q
+
independent, random errors
_
y=y + δy
δq
σ
q
σ = (sigma) = RMS error = Standard Deviation
root mean square
2
⎛ ∂q ⎞ ⎛ ∂q ⎞
σq = ⎜ σx ⎟ + ⎜ σ y ⎟
⎝ ∂x ⎠ ⎝ ∂y ⎠
δq
δx
2
propagation of errors using
the RMS
δy
Physics 2BL
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Statistical analysis
l
T = 2π
g
period of pendulum
uncertainty
error propagation
statistical analysis
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The mean
N measurements of the quantity x
the best estimate for x
the average or mean
sigma notation
common abbreviations
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The standard deviation
_
deviation of x from xi
average uncertainty of the
measurements x1, …, xN
standard deviation of x
RMS (route mean square) deviation
uncertainty in any one measurement of x
δx = σx
_ σx
68% of measurements will fall in the range xtrue +
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The standard deviation of the mean
_
uncertainty in x 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:
_
_ δx
(value of x) = x +
_
xbest = x
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Systematic errors
random component
systematic component
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Histograms and Distributions
different values
number of occurrences
weighted sum – each value of xk is weighted
by the number of times it occurred, nk
total number of measurements
fraction of our N measurements that gave the result xk
_
x is sum of xk weighted by Fk
normalization condition
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Fk specify the
distribution of xk
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Histograms and Distributions
bar histogram
bin histogram
26.4, 23.9, 25.1, 24.6, 22.7, 23.8, 25.1, 23.9, 25.3, 25.4
fk∆k = fraction of measurements in k-th bin
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Limiting Distributions
as the number of measurements
approaches infinity, their distribution
approaches some definite continuous curve
the limiting distribution, f(x)
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Limiting Distributions
fk∆k = fraction of measurements in k-th bin
= fraction of measurements
that fall between x=a and x=b
= probability that any
measurement will give an
answer between x=a and x=b
f(x) dx = fraction of measurements
that fall between x and x+dx
= probability that any
measurement will give an
answer between x and x+dx
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Normalization Condition
normalization condition
limiting
distribution
N ∞
fraction of our N measurements
that gave the result xk
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The Mean and Standard Deviation
the mean is the sum of all different values xk, each weighted
by the fraction of times it is obtained, Fk = f(xk) dxk
standard deviation σx
_
2
σx is the average of the squared deviation (x - x)2
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The Normal Distribution
the limiting distribution for a measurement
subject to many small random errors is bell shaped
and centered on the true value of x
the mathematical function that describes the bell-shape
curve is called the normal distribution, or Gauss function
prototype function
σ – width parameter
X – true value of x
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The Gauss, or Normal Distribution
normalize
standard deviation σx = width parameter of the Gauss function σ
the mean value of x = true value X
after infinitely many trials
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The standard Deviation as 68%Confidence Limit
erf(t) – error function
the probability that a measurement
will fall within one standard deviation
of the true answer is 68 %
_ δx
x = xbest +
δx=σ
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Justification of the Mean as Best Estimate
probability of obtaining a
value in the interval near x1
probability of obtaining
the value x1
abbreviations
probability of obtaining
the set of N values
find maximum
principle of maximum likelihood
the best estimate for X and σ are
those values for which
ProbX,σ(x1, …, xN) is maximum
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Justification of Addition in Quadrature
consider X=Y=0
probability of getting any given value of x
probability of getting any given value of y
rewriting in terms
of x + y
probability of getting any
given x and any given y
probability of getting any
given x + y
the values of x + y are normally
distributed with width
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Standard Deviation of the Mean
error propagation
uncertainty
in the mean
_
_ σx
(value of x) = x +
_
average of 10 measurements, x, are normally
distributed about X with width σ_x
individual measurements of x are normally
distributed about X with width σx
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Acceptability of a Measured Answer
_σ
(value of x) = xbest +
xexp - expected value of x, e.g. based on some theory
xbest differs from xexp
by t standard deviations
< 5 % - significant discrepancy, t > 1.96
< 1 % - highly significant discrepancy, t > 2.58
boundary for unreasonable improbability
erf(t) – error function
the result is beyond the boundary
for unreasonable improbability
the result is unacceptable
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Rejection of Data
suspicious
erf(t) – error function
if < 0.5 the measurement is “improbable” and can be rejected according to Chauvenet’s criterion
Chauvenet’s criterion
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Weighted Averages
combining separate measurements: what is the best estimate for x ?
assume that measurements are governed
by Gauss distribution with true value X
probability that A finds xA
probability that A finds xA and B finds xB
find maximum of probability
principle of maximum likelihood
the best estimate for X is that value for
which ProbX (xA, xB) is maximum
chi squared – “sum of squares”
find minimum of χ 2
“method of least squares”
weighted average
weights
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Weighted Averages
x1, x2, …, xN - measurements of a single quantity x with uncertanties σ1, σ2, …, σN
weighted average
weights
uncertainty in xwav
can be calculated
using error propagation
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Example of Weighted Average
three measurements of a resistance
what is the best estimate for R ?
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Least
-Squares Fitting
Least-Squares
consider two variables x and y that are connected by a linear relation
y = A + Bx
graphical method of
finding the best straight
line to fit a series of
experimental points
x1, x2, …, xN
y1, y2, …, yN
find A and B
analytical method of finding the best straight line to fit a series of experimental
points is called linear regression or the least-squares fit for a line
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Calculation of the Constants A and B
probability of obtaining the observed value of y1
probability of obtaining the set y1, … , yN
find maximum of probability
chi squared – “sum of squares”
find minimum of χ 2
“least squares fitting”
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Uncertainties in yy,, A, and B
sum of squares - estimate for
uncertainty in the measurement of y
uncertainty in the measurement of y
uncertainties in the constants A and B
given by error propagation in terms
of uncertainties in y1, … , yN
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Example of Calculation of the Constants A and B
if volume of an ideal gas is kept constant,
its temperature is a linear function of its pressure
absolute zero
of temperature
A=?
absolute zero of temperature, A = - 273.150 C
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Covariance and Correlation
review of error propagation
estimate of uncertainty in q
when errors in x and y are
independent and random
justifies
if x and y are governed by independent
normal distributions, with standard
deviations σx and σy , q(x,y) are normally
distributed with standard deviation σq
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Covariance
find δq if δx and δy are not independent
σq for arbitrary σx and σy
σx and σy can be correlated
covariance σxy
when σx and σy are independent σxy=0
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Covariance
Schwarz inequality
≈
upper limit on the uncertainty
significance of original estimate for δq
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Coefficient of Linear Correlation
do N pairs of (xi , yi) satisfy a linear relation ?
linear correlation coefficient
or correlation coefficient
_
if r is close to +1
when x and y are linearly related
if r is close to 0
when there is no relationship between x and y
x and y are uncorrelated
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Quantitative Significance of r
calculate correlation coefficient
r = 0.8
N=10
probability that N measurements of two uncorrelated
variables x and y would produce r ≥ r0
= 0.5 %
it is very unlikely that
x and y are uncorrelated
correlation is “significant” if ProbN(|r| ≥ r0) is less than 5 %
correlation is “highly significant” if ProbN(|r| ≥ r0) is less than 1 %
Physics 2BL
it is very likely that
x and y are correlated
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Chi Squared Test for a Distribution
40 measured values of x (in cm)
16 %
34 % 34 %
are these measurements
governed by a Gauss distribution ?
deviation
~1?
expected size of its fluctuation
16 %
chi squared
observed and expected distributions
agree about as well as expected
significant disagreement between
observed and expected distributions
Ok – observed number
Ek – expected number
– fluctuations of Ek
<n
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no reason to doubt that the
measurements were governed
by a Gauss distribution
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Forms of Chi Squared
yi have known uncertainties σi
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Degrees of Freedom and Reduced Chi Squared
a better procedure is to compare χ2 not with the number of bins n
but instead with the number of degree of freedom d
n is the number of bins
c is the number of parameters that had to be calculated
from the data to compute the expected numbers Ek
c is called the number of constrains
d is the number of degrees of freedom
reduced chi squared
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Probabilities of Chi Squared
quantitative measure of agreement between observed data and their expected distribution
probability of obtaining
a value of ~χ2 greater or
~ 2 , assuming
equal to χ
0
the measurements are
governed by the expected
distribution
x
disagreement is “significant” if ProbN( ~
χ2 ≥ ~χ02 ) is less than 5 %
~ ~
disagreement is “highly significant” if ProbN( χ2 ≥ χ02 ) is less than 1 %
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reject the expected
distribution
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