Measurements and ErrorsIntroductory Lecture

Prof Richard Thompsonr.thompson@imperial.ac.uk

4th October 2007

Aim of Lecture

• To provide you with the basic set of mathematical tools you will need in the First Year Laboratory Course for the analysis of experimental errors

• This will be followed up by practical exercises on error analysis and presentation of data in the computer suite next week

• The Statistics of Measurement course next year will provide the detailed mathematical background

Resources

• This presentation on the web

– UG course materials – Year 1 – Measurements and Errors

– Or from the undergraduate labs site www.imperial.ac.uk/physicsuglabs

• Summary Sheet (on Laboratories website)

• Exercise session next week in the computer suite on errors and on good presentation of data

– Bring your calculator and lab book!

Books

• Practical Physics, G L Squires, 4th ed, Cambridge University Press, 2001

• Experimental Measurements: Precision, Error and Truth, N C Barford, 2nd ed, Wiley, 1985

• An introduction to Error Analysis, J R Taylor, University Science Books, 1982

A classic example of experimental error!

Experimental Measurement

• Aim of Experimental Physics is to test theoretical ideas and predictions about how the universe works – requires measurement

• The experiment must be good enough to provide a significant test of the theory

• For this we need to know how reliable a measurement is

• This is specified by quoting an error or uncertainty for every measurement

The Need for Error Statements

• Say that theory predicts x = 7.1• If I measure x = 5 does that agree with theory?

YES• If x = 5 3 there is

good agreement

NO• If x = 5.0 0.1 there is

no agreement

Except of course I have not stated what the error on the theory number is!

The moral is:

a measurement without an error is meaningless!

The Three Types of Error

1. Random errors– due to e.g.

• random fluctuations in apparatus or surroundings• noise in electrical circuits• inaccuracies in setting positions of components• inaccuracies in reading scales

– what we will mainly discuss today– Improve result by taking more measurements

• These errors average out because they are random

The Three Types of Error

2. Systematic errors – same for all results– e.g. zero offsets; calibration errors– No advantage in repeating measurements

0.3 V 1.8 V

Always measure or eliminate systematic errors where possible

3. And then there are Mistakes or Blunders– Be careful and observant– Check everything that you write down– Watch out for “rogue points” on graphs

The Three Types of Error

How to Quote a Measurement Result

• A measurement without an estimated error is meaningless– units are also important

• Quote final result in form a ± b units– e.g. c = (2.94 ± 0.04) x 108 m/s– This means the true value is likely to lie between (a

– b) and (a + b)

• Errors should be quoted normally to one (but no more than two) significant figures

Random Errors(signal from a telescope moving on and off a galaxy)

Repeated Measurements of a Quantity

A set of many measurements gives a distribution of results, centred on the mean (or average) value.

The spread of values obtained is measured by the sample standard deviation (s)

Here s is about 0.15V

2 s

mean

Best Estimate of True Value

• We want to use the set of measurements to find the true value of the quantity we are measuring

• Consider an experiment where n independent measurements, xi , are made of a quantity x

• Let x0 stand for the “true” value of x

• The best estimate of x0 is given by the arithmetic mean

1

1 n

ii

x xn

Best Estimate of Standard Deviation

• It’s also useful to know the standard deviation of the distribution we would get if we made an infinite number of measurements

• The best estimate of this from our sample (n values) is the sample standard deviation (s):

or

2

2 2

1 1

11

1n n

i ii i

s x xn n

2 2

1

11 ( )

n

ii

ns x x

Best Estimate of Random Error

• How accurately does the mean measure the true value?• The best estimate of the random error of the mean of n

independent measurements is given by

m is known as the standard error of the measurement

• More measurements improve the result if other sources of error are smaller

• Probability of true value lying within m of mean is ~2/3

m s n

Standard Error by Calculator

• Put your calculator into Statistics mode• Enter data – usually with M+ key

• Press n-1 or xn-1 – gives sample standard deviation

• Divide by n – gives standard error• Make sure you know how to calculate this with

your own calculator!

The Standard Calculator

Casio fx-83ES

Cost: around £10

This will be supplied for your use in all examinations here

Propagation of Errors

• Sometimes we need to know the error in something that is a function of what we actually measure

– e.g. z = 2x or z = x2

• How do we find z (the error in z) given x (the error in x)? We find:

– If z = k x then z = k x

– If z = xn then z = n xn-1 x

• or write this as (z/z) = n (x/x)

– In general if z = z(x) then z = |dz/dx| x

• Note: x here may be a standard error calculated from a set of values or an independently estimated error

Combination of Errors

• Sometimes we need to know the error in something that is calculated by combining two measured quantities

• e.g. If z = x+y or z = xy, then

• The errors add in quadrature• Example

– measure length L of room in two steps A and B

L = (A + B) ± L where

– If A = 13.3 ± 0.2m and B = 20.0 ± 0.3 m– then total length = 33.3 ± 0.4 m

2 2z x y

2 2L A B

Combination of Errors

• If z = xy or z = x/y, then

• The fractional errors add in quadrature• Example

– measure length L and breadth B of a field and want to calculate area A

A = LB ± A where

– If L = 304 ± 3m and B = 513 ± 4m then A = 156000 ± 2000m2 or (1.56 ± 0.02) x 105 m2 (but not 155952 ± 1961m2)

22

z x yz x y

2 2/A L BA L B

Measurements and Errors Session in the Computer Suite

• A practical session to gain practice in dealing with errors and in the presentation of results with Excel

• Don’t forget to bring your calculator• Don’t forget to bring your lab book

Finally….

• Remember to think about the error of each measurement that you make

• Be sensible and realistic in your treatment of errors

And do Enjoy the First Year Lab !!