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Transcript

Experimental Uncertainties: A Practical Guide • What you should already know well • What you need to know, and use, in this lab More details available in handout ‘Introduction to Experimental Error’ in your folders. • In what follows I will use convention: – Error = deviation of measurement from true value – Uncertainty = measure of likely error Why are Uncertainties Important? • Uncertainties absolutely central to the scientific method. • Uncertainty on a measurement at least as important as measurement itself! • Example 1: “The observed frequency of the emission line was 8956 GHz. The expectation from quantum mechanics was 8900 GHz” • Nobel Prize? Why are Uncertainties Important? • Example 2: “The observed frequency of the emission line was 8956 ± 10 GHz. The expectation from quantum mechanics was 8900 GHz” • Example 3: “The observed frequency of the emission line was 8956 ± 10 GHz. The expectation from quantum mechanics was 8900 GHz ± 50 GHz” Types of Uncertainty • Statistical Uncertainties: – Quantify random errors in measurements between repeated experiments – Mean of measurements from large number of experiments gives correct value for measured quantity – Measurements often approximately gaussiandistributed • Systematic Uncertainties: – Quantify systematic shift in measurements away from ‘true’ value – Mean of measurements is also shifted ‘bias’ Examples • Statistical Errors: True Value 0.45 – Measurements gaussian0.4 distributed 0.35 – No systematic error (bias) 0.3 – Quantify uncertainty in 0.25 measurement with standard 0.2 deviation (see later) 0.15 – In case of gaussian-distributed 0.1 measurements std. dev. = s in 0.05 formula 0 – Probability interpretation -3 -2 -1 -0 1 2 (gaussian case only): 68% of x x 2 1 exp 2 measurements will lie within ± 1 2 2s 2s s of mean. 3 Examples • Statistical + Systematic Errors: True Value 0.45 – Measurements still gaussian0.4 distributed 0.35 – Measurements biased 0.3 – Still quantify statistical 0.25 uncertainty in measurement with 0.2 standard deviation 0.15 0.1 – Probability interpretation 0.05 (gaussian case only): 68% of 0 measurements will lie within ± 1 s -3 -2 -1 -0 1 2 of mean. x x 2 1 exp 2 2 – Need to quantify systematic error 2s 2s (uncertainty) separately tricky! 3 Systematic Errors • How to quantify uncertainty? • What is the ‘true’ systematic error in any given measurement? – If we knew that we could correct for it (by addition / subtraction) • What is the probability distribution of the systematic error? True Value 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 – Often assume gaussian distributed and quantify with ssyst. – Best practice: propagate and quote separately 0 -3 -2 -1 -0 1 2 3 Calculating Statistical Uncertainty • Mean and standard deviation of set of independent measurements (unknown errors, assumed uniform): 1 x0 x N i x; i 1 2 s x x i N 1 i 2 • Standard deviation estimates the likely error of any one measurement • Uncertainty in the mean is what is quoted: sx s 1 2 xi x N N ( N 1) i 1/ 2 . Propagating Uncertainties • Functions of one variable (general formula): df F X dx • Specific cases: x 2 2 xx nx x n n 1 x or sin x cos x x 1 ln x x x or x2 x 2 x x2 xn x n n x x Propagating Uncertainties • Functions of >1 variable (general formula): f 2 • Specific cases: 2 f f x y . x y 2 f= Apply equation Simplify x y xy f 2 x 2 y 2 f 2 x 2 y 2 x y f 2 y 2 x x 2 y 2 2 x f 2 2 y x2 2 4 y y 2 f f f f 2 2 y x x y 2 2 2 x y x y 2 Combining Uncertainties • What about if have two or more measurements of the same quantity, with different uncertainties? • Obtain combined mean and uncertainty with: x 2 x s i i i 1 s i 2 i 1 s 2 i 1 s 2 i • Remember we are using the uncertainty in the mean here: s si N Fitting • Often we make measurements of several quantities, from which we wish to 1. determine whether the measured values follow a pattern 2. derive a measurement of one or more parameters describing that pattern (or model) • • • This can be done using curve-fitting E.g. EXCEL function linest. Performs linear least-squares fit Method of Least Squares ln eta [cPs] • This involves taking measurements yi and comparing with the equivalent fitted value yif • Linest then varies the model parameters and hence yif until the following quantity is minimised: y N i 1 i yi f 2 In this example the model is a straight line yif = mx+c. The model parameters are m and c 1 0.5 0 -0.5 -1 -1.5 0.0026 • Linest will return the fitted parameter values (=mean) and their uncertainties (in the mean) 0.0028 0.003 0.0032 0.0034 0.0036 0.0038 1/T [1/K] In the second year lab never use the equations returned by ‘Add Trendline’ or linest to estimate your parameters!!! Weighted Fitting • Those still awake will have noticed the least square method does not depend on the uncertainties (error bars) on each point. • Q: Where do the uncertainties in the parameters come from? – A: From the scatter in the measured means about the fitted curve • Equivalent to: 1 2 s x x i N 1 i 2 • Assumes errors on points all the same • What about if they’re not? Weighted Fitting • To take non-uniform uncertainties (error bars) on points into account must use e.g. chi-squared fit. • Similar to least-squares but minimises: y i yi si i 1 N 2 f 2 • Enables you to propagate uncertainties all the way to the fitted parameters and hence your final measurement (e.g. derived from gradient). • This is what is used by chisquare.xls (download from Second Year web-page) this is what we expect you to use in this lab! General Guidelines Always: • Calculate uncertainties on measurements and plot them as error bars on your graphs • Use chisquare.xls when curve fitting to calculate uncertainties on parameters (e.g. gradient). • Propagate uncertainties correctly through derived quantities • Quote uncertainties on all measured numerical values • Quote means and uncertainties to a level of precision consistent with the uncertainty, e.g: 3.77±0.08 kg, not 3.77547574568±0.08564846795768 kg. • Quote units on all numerical values General Guidelines Always: • Think about the meaning of your results – A mean which differs from an expected value by more than 1-2 multiples of the uncertainty is, if the latter is correct, either suffering from a hidden systematic error (bias), or is due to new physics (maybe you’ve just won the Nobel Prize!) Never: • Ignore your possible sources of error: do not just say that any discrepancy is due to error (these should be accounted for in your uncertainty) • Quote means to too few significant figures, e.g.: 3.77±0.08 kg not 4±0.08 kg