Propagation of errors - Instituto Tecnológico de Aeronáutica · Physics 2BL 7 Histograms and...

Physics 2BLPhysics 2BL 1

Propagation of errorsPropagation of errors

Compute error in q(x,y) from errors in x and y

22

q x yq qx y

σ σ σ⎛ ⎞∂ ∂⎛ ⎞= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

propagation of errors using the RMS

σ = (sigma) = RMS error = Standard Deviation

δq δx δy

root mean square

q=q + δq_

y=y + δyx=x + δx

__ independent, random errors

δq σq


Statistical analysisStatistical analysisStatistical analysis

uncertainty

period of pendulum2 lTg

π=

error propagation statistical analysis


The meanThe meanThe mean

the best estimate for x the average or mean

sigma notation

N measurements of the quantity x

common abbreviations


The standard deviationThe standard deviationThe standard deviation

deviation of x from xi

_

standard deviation of x

RMS (route mean square) deviation

average uncertainty of the measurements x1, …, xN

uncertainty in any one measurement of x δx = σx

68% of measurements will fall in the range xtrue + σx_


The standard deviation of the meanThe standard deviation of the meanThe standard deviation of the mean

uncertainty in x isthe standard deviation of the mean

_

based on the N measured values x1, …, xN we can state our final answer for the value of x:

(value of x) = x + δx__

xbest = x_


Systematic errorsSystematic errorsSystematic errors

random component systematic component


Histograms and DistributionsHistograms and DistributionsHistograms and Distributions

different valuesnumber 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

normalization condition

x is sum of xk weighted by Fk

_Fk specify the distribution of xk


Histograms and DistributionsHistograms and DistributionsHistograms 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


Limiting DistributionsLimiting DistributionsLimiting Distributions

as the number of measurements approaches infinity, their distribution approaches some definite continuous curve

the limiting distribution, f(x)


Limiting DistributionsLimiting DistributionsLimiting Distributions

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

= 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

fk∆k = fraction of measurements in k-th bin


Normalization ConditionNormalization Condition

normalization condition

limiting distribution N ∞fraction of our N measurements

that gave the result xk


The Mean and Standard DeviationThe Mean and Standard DeviationThe 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 σxσx

2 is the average of the squared deviation (x - x)2_


The Normal DistributionThe Normal DistributionThe Normal Distribution

the limiting distribution for a measurement subject to many small random errors is bell shapedand 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 parameterX – true value of x


The Gauss, or Normal DistributionThe Gauss, or Normal DistributionThe Gauss, or Normal Distribution

normalize

standard deviation σx = width parameter of the Gauss function σthe mean value of x = true value X

after infinitely many trials


The standard Deviation as 68%Confidence LimitThe standard Deviation as 68%Confidence LimitThe standard Deviation as 68%Confidence Limit

erf(t) – error function

the probability that a measurement will fall within one standard deviationof the true answer is 68 %x = xbest + δx δx=σ_


Justification of the Mean as Best EstimateJustification of the Mean as Best EstimateJustification of the Mean as Best Estimate

probability of obtaining avalue in the interval near x1probability of obtaining the value x1

abbreviations

probability of obtaining the set of N values

principle of maximum likelihoodthe best estimate for X and σ are those values for which ProbX,σ(x1, …, xN) is maximum

find maximum


Justification of Addition in QuadratureJustification of Addition in Justification of Addition in QuadratureQuadratureconsider X=Y=0

probability of getting any given value of x

probability of getting any given value of y

probability of getting any given x and any given y

probability of getting any given x + y

rewriting in terms of x + y

the values of x + y are normally distributed with width


Standard Deviation of the MeanStandard Deviation of the MeanStandard Deviation of the Mean

error propagation

(value of x) = x + σx__

uncertainty in the mean

individual measurements of x are normallydistributed about X with width σx

average of 10 measurements, x, are normallydistributed about X with width σx

__


Acceptability of a Measured AnswerAcceptability of a Measured AnswerAcceptability of a Measured Answer

(value of x) = xbest + σ_

xexp - expected value of x, e.g. based on some theory

xbest differs from xexpby t standard deviations

< 5 % - significant discrepancy, t > 1.96

< 1 % - highly significant discrepancy, t > 2.58

boundary for unreasonable improbability


the result is beyond the boundary for unreasonable improbability

the result is unacceptable


Rejection of DataRejection of DataRejection of Data


Chauvenet’s criterion

if < 0.5 the measurement is “improbable” and can be rejected according to Chauvenet’s criterion

suspicious


Weighted AveragesWeighted AveragesWeighted Averagescombining separate measurements: what is the best estimate for x ?

probability that A finds xA

find maximum of probabilityprinciple of maximum likelihoodthe best estimate for X is that value for which ProbX (xA, xB) is maximum

assume that measurements are governed by Gauss distribution with true value X

probability that A finds xA and B finds xB

chi squared – “sum of squares”find minimum of χ 2“method of least squares”

weighted averageweights


Weighted AveragesWeighted AveragesWeighted Averages

x1, x2, …, xN - measurements of a single quantity x with uncertanties σ1, σ2, …, σN

weighted averageweightsuncertainty in xwav

can be calculated using error propagation


Example of Weighted AverageExample of Weighted AverageExample of Weighted Average

three measurements of a resistancewhat is the best estimate for R ?


Least-Squares FittingLeastLeast--Squares FittingSquares Fitting

x1, x2, …, xNy1, y2, …, yN

consider two variables x and y that are connected by a linear relationy = A + Bx

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

graphical method of finding the best straight line to fit a series of experimental points


Calculation of the Constants A and BCalculation of the Constants A and BCalculation 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”


Uncertainties in y, A, and BUncertainties in Uncertainties in yy, A, and B, A, and B

uncertainty in the measurement of y

uncertainties in the constants A and B

sum of squares - estimate for uncertainty in the measurement of y

given by error propagation in terms of uncertainties in y1, … , yN


Example of Calculation of the Constants A and BExample of Calculation of the Constants A and BExample of Calculation of the Constants A and B

absolute zero of temperature, A = - 273.150 C

if volume of an ideal gas is kept constant, its temperature is a linear function of its pressure

absolute zero of temperatureA = ?


Covariance and CorrelationCovariance and CorrelationCovariance and Correlation

review of error propagation

when errors in x and y are independent and random

estimate of uncertainty in q

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

justifies


CovarianceCovarianceCovariancefind δq if δx and δy are not independent

σq for arbitrary σx and σyσx and σy can be correlated

covariance σxy

when σx and σy are independent σxy=0


CovarianceCovarianceCovariance

Schwarz inequality

upper limit on the uncertainty≈

significance of original estimate for δq


Coefficient of Linear CorrelationCoefficient of Linear CorrelationCoefficient of Linear Correlation

do N pairs of (xi , yi) satisfy a linear relation ?

linear correlation coefficientor 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 yx and y are uncorrelated

_


Quantitative Significance of rQuantitative Significance of rQuantitative Significance of r

r = 0.8N=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

it is very likely that x and y are correlated

calculate correlation coefficient

correlation is “significant” if ProbN(|r| ≥ r0) is less than 5 %

correlation is “highly significant” if ProbN(|r| ≥ r0) is less than 1 %


Chi Squared Test for a DistributionChi Squared Test for a DistributionChi Squared Test for a Distribution

40 measured values of x (in cm) are these measurements governed by a Gauss distribution ?

Ok – observed numberEk – expected number

– fluctuations of Ek

deviationexpected size of its fluctuation

16 % 16 %34 % 34 %

chi squared

~ 1 ?

observed and expected distributions agree about as well as expected

significant disagreement betweenobserved and expected distributions

no reason to doubt that the measurements were governed by a Gauss distribution< n


Forms of Chi SquaredForms of Chi SquaredForms of Chi Squared

yi have known uncertainties σi


Degrees of Freedom and Reduced Chi SquaredDegrees of Freedom and Reduced Chi SquaredDegrees of Freedom and Reduced Chi Squared

n is the number of binsc is the number of parameters that had to be calculated

from the data to compute the expected numbers Ekc is called the number of constrainsd is the number of degrees of freedom

a better procedure is to compare χ2 not with the number of bins nbut instead with the number of degree of freedom d

reduced chi squared


Probabilities of Chi SquaredProbabilities of Chi SquaredProbabilities of Chi Squaredquantitative measure of agreement between observed data and their 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 %~~

~~

probability of obtaining a value of χ2 greater or equal to χ0

2 , assumingthe measurements are governed by the expected distribution

~~

reject the expected distribution

Propagation of errors - Instituto Tecnológico de Aeronáutica · Physics 2BL 7 Histograms and...

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