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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:

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Experimental uncertainties a practical guide
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
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 important1
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
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’


Examples
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.


Examples1
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!


Systematic errors
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


Calculating statistical uncertainty
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
Propagating Uncertainties

  • Functions of one variable (general formula):

  • Specific cases:


Propagating uncertainties1
Propagating Uncertainties

  • Functions of >1 variable (general formula):

  • Specific cases:


Combining uncertainties
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
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


Method of least squares
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!!!


Weighted fitting
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 fitting1
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
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 guidelines1
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


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