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# Experimental Uncertainties: A Practical Guide - PowerPoint PPT Presentation

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:

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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

• 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?

• 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”

• 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’

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.

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!

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

• 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:

• Functions of one variable (general formula):

• Specific cases:

• Functions of >1 variable (general formula):

• Specific cases:

• 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:

• 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

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!!!

• 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?

• 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!

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

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