Lecture 4 Propagation of errors

Transcription

1 Introduction Lecture 4 Propagation of errors Example: we measure the current (I and resistance (R of a resistor. Ohm's law: V = IR If we know the uncertainties (e.g. standard deviations in I and R, what is the uncertaint in V? Given a functional relationship between several measured variables (x,, z, Q = f (x,,z What is the uncertaint in Q if the uncertainties in x,, and z are known? To answer this question we use a technique called Propagation of Errors. Usuall when we talk about uncertainties in a measured variable such as x, we assume: the value of x represents the mean of a Gaussian distribution the uncertaint in x is the standard deviation (σ of the Gaussian distribution not all measurements can be represented b Gaussian distributions (more on that later Propagation of Error Formula To calculate the variance in Q as a function of the variances in x and we use the following: " #Q' #Q' #Q' Q = " x +" % #x ( + " x #Q ' % # ( % #x (% # ( If the variables x and are uncorrelated (σ x = 0, the last term in the above equation is zero. Assume we have several measurement of the quantities x (e.g. x, x...x and (e.g.,.... The average of x and : = " x i and µ = " i K.K. Gan L4: Propagation of Errors

2 define: Q i " f (x i, i Q " f (,µ evaluated at the average values expand Q i about the average values: Q i = f (,µ + (x i " #Q ' + ( % #x ( i " µ #Q ' + higher order terms % # ( µ assume the measured values are close to the average values neglect the higher order terms: Q i "Q = (x i " #Q ' + ( % #x ( i " µ #Q ' % # ( * Q = + (Q i "Q = (x i " µ #Q' + x + % #x ( ( i " µ #Q' + % # ( #Q' #Q' = * x +* % #x ( + #Q ' #Q' + % # ( % µ #x ( % # ( µ (x i " ( i " µ If the measurements are uncorrelated the summation in the above equation is zero " #Q' #Q' Q = " x +" % #x ( uncorrelated errors % # ( µ µ µ + (x i " ( i " µ #Q ' + % #x ( & % #Q # ' ( µ K.K. Gan L4: Propagation of Errors

3 If x and are correlated, define σ x as: " x = (x i # ( i # µ " &%Q &%Q & Q = " x ( + +" ' %x * ( + + %Q &%Q ( + ( + " ' % * ' µ %x * x ' % * correlated errors µ Example: Power in an electric circuit. P = I R Let I =.0 ± 0. amp and R = 0 ± Ω P = 0 watts calculate the variance in the power using propagation of errors " #P ' #P ' P = " I +" % #I ( R = " %#R ( I (IR +" R (I = (0. ( **0 + ( ( = 5 watts P = 0 ± watts I= R=0 If the true value of the power was 0 W and we measured it man times with an uncertaint (σ of ± W and Gaussian statistics appl 68% of the measurements would lie in the range [8,] W Sometimes its convenient to put the above calculation in terms of relative errors: " P P = " I #P ' P % #I ( + " R #P ' P = 4" I %#R ( I + " R R = 4 0. ' + ' = 0. (4 + % ( % 0( the uncertaint in the current dominates the uncertaint in the power current must be measured more precisel to greatl reduce the uncertaint in the power K.K. Gan L4: Propagation of Errors 3

4 Example: The error in the average. The average of several measurements each with the same uncertaint (σ is given b: µ = n (x + x +...x n " #µ ' µ = " x %#x ( +" x #µ ' %#x ( #µ ' +..." x n %#x n ( = " & ' % n( +" & ' % n( +..." & ' % n ( = n" & ' % n( " µ = " error in the mean n We can determine the mean better b combining measurements. The precision onl increases as the square root of the number of measurements. Do not confuse σ µ with σ! σ is related to the width of the pdf (e.g. Gaussian that the measurements come from. σ does not get smaller as we combine measurements. 00 Problem in the Propagation of Errors 0 In calculating the variance using propagation of errors we usuall assume the error in measured variable (e.g. x is Gaussian If x is described b a Gaussian distribution f(x ma not be described b a Gaussian distribution! K.K. Gan L4: Propagation of Errors 4 d/d σ = 4.0 µ = 0.0 σ µ = data points

5 What does the standard deviation that we calculate from propagation of errors mean? Example: The new distribution is Gaussian. Let = Ax, with A = a constant and x a Gaussian variable. µ = A and σ = Aσ x Let the probabilit distribution for x be Gaussian: % A µ ( ' * & A (xµ x " % ( (µ p(x,," x dx = " x # e " x ' * & A dx = e " A # A d = " # e " d = p(,µ," d The new probabilit distribution for, p(, µ, σ, is also described b a Gaussian. d/d = x with x = 0 ± σ = σ x = Start with a Gaussian with µ = 0, σ = Get another Gaussian with µ = 0, σ = 4 K.K. Gan L4: Propagation of Errors 5

6 Example: When the new distribution is non-gaussian: = /x. The transformed probabilit distribution function for does not have the form of a Gaussian. d/d = /x with x = 0 ± σ = σ x /x Start with a Gaussian with µ = 0, σ =. DO OT get another Gaussian! Get a pdf with µ = 0., σ = This new pdf has longer tails than a Gaussian pdf: Prob( > µ + 5σ = 5x0-3, for Gaussian 3x Unphsical situations can arise if we use the propagation of errors results blindl! Example: Suppose we measure the volume of a clinder: V = πr L. Let R = cm exact, and L =.0 ± 0.5 cm. Using propagation of errors: σ V = πr σ L = π/ cm 3 V = π ± π/ cm 3 If the error on V (σ V is to be interpreted in the Gaussian sense finite probabilit ( 3% that the volume (V is < 0 since V is onl σ awa from than 0! Clearl this is unphsical! Care must be taken in interpreting the meaning of σ V. K.K. Gan L4: Propagation of Errors 6