Introduction to experimental errors

Introduction to experimental errors

J W Cockburn

Department of Physics and AstronomyUniversity of SheffieldPhysics & Astronomy Laboratories

Outline of talk

• Why are errors important?• Types of error – random and systematic

(precision and accuracy)• Estimating errors • Quoting results and errors• Treatment of errors in formulae• Combining random and systematic errors• The statistical nature of errors

Why are errors important?

Two measurements of body temperature before and after a drug is administered

38.2C and 38.4C

Is temperature rise significant? – It depends on the associated errors

(38.20.01)C and (38.4 0.01)C - significant

(38.20.5)C and (38.4 0.5)C – not significant

Random errors• An error that varies between successive

measurements• Equally likely to be positive or negative• Always present in an experiment• Presence obvious from distribution of values

obtained• Can be minimised by performing multiple

measurements of the same quantity or by measuring one quantity as function of second quantity and performing a straight line fit of the data

• Sometimes referred to as reading errors

Systematic errors

• Constant throughout a set of readings.

• May result from equipment which is incorrectly calibrated or how measurements are performed.

• Cause average (mean) of measured values to depart from correct value.

• Difficult to spot presence of systematic errors in an experiment.

Random vs systematic errors

True value

Random errors only

Random + systematic

•A result is said to be accurate if it is relatively free from systematic error•A result is said to be precise if the random error is small

Quoting results and errors

• Generally state error to one significant figure (although if one or two then two significant figures may be used).

• Quote result to same significance as error

• When using scientific notation, quote value and error with the same exponent

• Value 44, error 5 445

• Value 128, error 32 13030

• Value 4.8x10-3, error 7x10-4 (4.80.7)x10-3

• Value 1092, error 56 109060

• Value 1092, error 14 109214

• Value 12.345, error 0.35 12.30.4

Don’t over quote results to a level inconsistent with the error 36.6789353720.5

Quoting results and errors

Estimating reading errors 1

Oscilloscope – related to width of trace3.8 divisions @ 1V/division = 3.8VTrace width is ~0.1 division = 0.1V(3.80.1)V

Digital meter – error taken as 5 in next significant figure(3.3600.005)V

Estimating reading errors 2

Analogue meter – error related to width of pointerValue is 3.25VPointer has width 0.1V(3.30.1)V

Estimating reading errors 3

Estimating reading errors 4

16 17

•Linear scale (e.g. a ruler)•Need to estimate precision with which measurement can be made•May be a subjective choice16.770.02

Estimating reading errors 5

16 17

•The reading error may be dependent on what is being measured.•In this case the use of greater precision equipment may not help reduce the error.

16 17

Treatment of errors in formulae

• In general we will calculate a result using a formula which has as an input one or more measured values.

• For example: volume of a cylinder

• How do the errors in the measured values feed through into the final result?

2A r h

Treatment of errors in formulae

• In the following A, B, C and Z are the absolute values

A, B, C and Z are the absolute errors in A, B, C and Z

• Hence A/A is the fractional error in A and (A/A)100 is the percentage error in A etc

A and A will have the same units• Assume errors in numerical or physical

constants (e.g. , e, c etc) are much smaller than those in measured values – hence can be ignored.

2 2 2 2

2 2 2 2

2 2 2 2

/

ln

exp

n m l

Z A B C Z A B C

Z A B CZ ABC or AB C etc

Z A B C

Z A B CZ A B C n m l

Z A B C

AZ A Z

AZ

Z A AZ

Treatment of errors in formulae

Example of error manipulation 1

2A r Where r=(50.5)m

A=78.5398m2

2 22

2 2

0.50.1

5

2 (2 0.1) 0.04

0.2 0.2 0.2 (78.5398 ) 16

r

r

A r

A r

Ahence A A A m m

A

Hence final result is A=(7916)m2

• P=2L+2W where L=(40.2)m and W=(50.2)m

• P=18m

•

• P=(18.00.3)m

2 2 2(0.2) (0.2) 0.08

0.28

P

P m

Example of error manipulation 2

2l

g l=(2.50.1)m, g=(9.80.2)ms-2

=3.1735s

22 21 1

2 2

l g

l g

2 241 0.1 1 0.2

5.04 102 2.5 2 9.8

/=0.022 hence =0.022x3.1735=0.070=(3.170.07)s

Example of error manipulation 3

Random + systematic errors

Combine random error and systematic error (if known) by adding the squares of the separate errors.

Example: A length is measured with a reading (random error) given by (892) cm using a rule of calibration accuracy 2%.

2 2

2

22

( )

( )

2(0.02) 0.000905

89

0.03

Total fractional error fractional reading error

systematic error

Total fractional error

Absolute error = 0.03x89=2.7cm

Value =(893)cm

The statistical nature of errors

• Because of the way in which errors are combined to generate the total error this does not give the maximum possible range of values.

• Instead the total error associated with a value provides information concerning the probability that the value falls within certain limits.

The statistical nature of errors

If a quantity has an associated error then• There is a 67% chance that the true value lies

within the range - to + • There is a 95% chance that the true value lies

within the range -2 to +2

Pro

babi

lity

of r

esul

t

22

√

Comparing values

• Need to look at overlap of distributions

• Case of two quantities A and B which differ by sum of errors A+B

• Probability of agreement ~2x1/36 = 6%Probability of agreement ~2x1/36 = 6%

Conclusions

• Systematic and random (reading) errors – accuracy and precision

• Quoting errors• Estimating reading errors• Manipulating and combining errors• The statistical nature of errors

Further reading: Document on website or any text book on practical physics e.g. ‘Experimental Methods’ L Kirkup or ‘Practical Physics’ G L Squires