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Experimental Uncertainties: A Practical GuidePowerPoint Presentation

Experimental Uncertainties: A Practical Guide

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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

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 gaussian-distributed

- Systematic Uncertainties:
- Quantify systematic shift in measurements away from ‘true’ value
- Mean of measurements is also shifted ‘bias’

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.

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

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

- 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:

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
- 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

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

- 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:

- 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!)
- 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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