Let us first define the standard deviation s. Suppose weperform N measurements 1, 2, · · · , N with the average¯. Then the deviation of each measurement is given byδ = − ¯ with = 1, 2, · · · , N. The standard deviation s iss =√√√ ∑N=1(δ)2N − 1When we report the average value of N measurements,the uncertainty we should associate with this averagevalue is the standard error.σ = spN =√√√√∑N=1(δ)2N(N − 1)The standard error is smaller than the standard deviationby a factor of 1/pN, since the statistical uncertainty canbe reduced by large number of measurements. Also it isuseful to write σ2 = δ2 ≡ 1N(N − 1)N∑=1δ2 .Suppose we want to determine a quantity = ƒ (, ),which depends on and . We want to know the error in = ƒ (, ) if we measure and with errors σ and σ .Using the Taylor expansion, we can obtain the law of theerror propagation as follows(δ)2 = ∂ƒ∂2(δ)2 + ∂ƒ∂2(δ)2 + 2 ∂ƒ∂∂ƒ∂(δδ)If the measurements of and are uncorrelated, then,on the average, we should expect to find equal distribu-tions of positive and negative values for this term, and weshould expect (δδ) = 0. At the end of the day, using thedefinition of the standard error σ, we can obtainσ =√√√ ∂ƒ∂2σ2 + ∂ƒ∂2σ2Exercise problems: Now find the standard error σ in = ƒ (, ) as a function of the errors in σ and σ for thefollowing functions:(a) = + (0.5 pts)You can find the absolute uncertainty of the sum(or difference) is the root square sum of theindividual absolute uncertainties when adding (orsubtracting).(b) = × (0.5 pts)(c) = / (1 pt)You can find that the relative uncertainty of the prod-uct (quotient) is the root square sum of the individualrelative uncertainties.(d) = 2 (1 pt)(e) = exp(c) with c constant. (0.5 pts)(f) = 1/
Question
Let us first define the standard deviation s. Suppose weperform N measurements 1, 2, · · · , N with the average¯. Then the deviation of each measurement is given byδ = − ¯ with = 1, 2, · · · , N. The standard deviation s iss =√√√ ∑N=1(δ)2N − 1When we report the average value of N measurements,the uncertainty we should associate with this averagevalue is the standard error.σ = spN =√√√√∑N=1(δ)2N(N − 1)The standard error is smaller than the standard deviationby a factor of 1/pN, since the statistical uncertainty canbe reduced by large number of measurements. Also it isuseful to write σ2 = δ2 ≡ 1N(N − 1)N∑=1δ2 .Suppose we want to determine a quantity = ƒ (, ),which depends on and . We want to know the error in = ƒ (, ) if we measure and with errors σ and σ .Using the Taylor expansion, we can obtain the law of theerror propagation as follows(δ)2 = ∂ƒ∂2(δ)2 + ∂ƒ∂2(δ)2 + 2 ∂ƒ∂∂ƒ∂(δδ)If the measurements of and are uncorrelated, then,on the average, we should expect to find equal distribu-tions of positive and negative values for this term, and weshould expect (δδ) = 0. At the end of the day, using thedefinition of the standard error σ, we can obtainσ =√√√ ∂ƒ∂2σ2 + ∂ƒ∂2σ2Exercise problems: Now find the standard error σ in = ƒ (, ) as a function of the errors in σ and σ for thefollowing functions:(a) = + (0.5 pts)You can find the absolute uncertainty of the sum(or difference) is the root square sum of theindividual absolute uncertainties when adding (orsubtracting).(b) = × (0.5 pts)(c) = / (1 pt)You can find that the relative uncertainty of the prod-uct (quotient) is the root square sum of the individualrelative uncertainties.(d) = 2 (1 pt)(e) = exp(c) with c constant. (0.5 pts)(f) = 1/
Solution
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