Experimental Uncertainties: - PowerPoint Presentation

Slide1

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.Slide2

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?Slide3

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”Slide4

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

gaussian-distributed

Systematic Uncertainties (

aka syst error, bias

):

Quantify systematic shift in measurements away from ‘true’ value

Mean of measurements is also shifted

 ‘bias’Slide5

Examples

Statistical Uncertainties:

Measurements

gaussian-distributed

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 ValueSlide6

Examples

True Value

Statistical + Systematic Uncertainties:

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 ± 1

s

of mean.

Need to quantify systematic uncertainty separately

 tricky!Slide7

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

s

syst

.

Best practice: propagate and quote separately

True ValueSlide8

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:Slide9

Propagating Uncertainties

Functions of one variable (general formula):

Specific cases:Slide10

f

=

Apply equation

Simplify

Propagating Uncertainties

Functions of >1 variable (general formula):

Specific cases:Slide11

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:Slide12

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 fitSlide13

Method of Least Squares

This involves taking measurements

y

i

and comparing with the equivalent fitted value

y

i

f

Linest then varies the model parameters and hence

y

if 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

y

i

f = 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!!!Slide14

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?Slide15

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!Slide16

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 valuesSlide17

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