Experimental Uncertainties: A Practical Guide

1. 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

2. 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?

3. 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”

4. 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 gaussian-distributed • Systematic Uncertainties: • Quantify systematic shift in measurements away from ‘true’ value • Mean of measurements is also shifted  ‘bias’

5. Examples True Value • Statistical Errors: • Measurements gaussian-distributed • No systematic error (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 ± 1s of mean.

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

7. Systematic Errors True Value • 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? • Often assume gaussian distributed and quantify with ssyst. • Best practice: propagate and quote separately

8. 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:

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

10. Propagating Uncertainties • Functions of >1 variable (general formula): • Specific cases:

11. 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:

12. Fitting • Often we make measurements of several quantities, from which we wish to • determine whether the measured values follow a pattern • 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

13. Method of Least Squares In this example the model is a straight line yif = mx+c. The model parameters are m and c • 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 the second year lab never use the equations returned by ‘Add Trendline’ or linest to estimate your parameters!!!

14. 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?

15. 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!

16. 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

17. 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