Experimental Uncertainties A Practical Guide What you should

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’ (2 nd Year Web page). • In what follows I will use the word uncertainty instead of error, although in the literature the both are used: – Everybody uses the term “error-bar” in graphs.

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 ± 50 GHz”

Types of Uncertainty • Statistical Uncertainties (aka random error): – Quantify random uncertainties 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 (aka syst error, bias): – Quantify systematic shift in measurements away from ‘true’ value – Mean of measurements is also shifted ‘bias’

Examples • Statistical Uncertainties: – Measurements gaussiandistributed – No system. uncertainty (bias) – Quantify uncertainty in measurement with standard deviation (see later) – In case of gaussian-distributed measurements std. dev. = s in formula – Probability interpretation (gaussian case only): 68% of measurements will lie within ± 1 s of mean. True Value

Examples • Statistical + Systematic Uncertainties: – Measurements still gaussiandistributed – Measurements biased – Still quantify statistical uncertainty in measurement with standard deviation – Probability interpretation (gaussian case only): 68% of measurements will lie within ± 1 s of mean. – Need to quantify systematic uncertainty separately tricky! True Value

Systematic Uncertainties • How to quantify uncertainty? • What is the ‘true’ systematic uncertainty in any given measurement? – If we knew that we could correct for it (by addition / subtraction) • What is the probability distribution of the systematic uncertainty? – Often assume gaussian distributed and quantify with ssyst. – Best practice: propagate and quote separately True Value

Calculating Statistical Uncertainty • Mean and standard deviation of set of independent measurements (unknown errors, assumed uniform): • Standard deviation estimates the likely error of any one measurement • Uncertainty in the mean is what is quoted:

Propagating Uncertainties • Functions of one variable (general formula): • Specific cases:

Propagating Uncertainties • Functions of >1 variable (general formula): • Specific cases: f= Apply equation Simplify

Combining Uncertainties • What about if have two or more measurements of the same quantity, with different uncertainties? • Obtain combined mean and uncertainty with: • Remember we are using the uncertainty in the mean here:

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 • 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: • Linest will return the fitted parameter values (=mean) and their uncertainties (in the mean) In this example the model is a straight line yif = mx+c. The model parameters are m and c 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: • 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: • 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