Introduction to experimental errors

1. Department of Physics and Astronomy University of Sheffield Physics & Astronomy Laboratories Introduction to experimental errors J W Cockburn

2. 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

3. Why are errors important? Two measurements of body temperature before and after a drug is administered 38.2C and 38.4C Is temperature rise significant? – It depends on the associated errors (38.20.01)C and (38.4 0.01)C - significant (38.20.5)C and (38.4 0.5)C – not significant

4. 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

5. 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.

6. Random vs systematic errors Random errors only True value 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

7. 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

8. Quoting results and errors • Value 44, error 5  445 • Value 128, error 32  13030 • Value 4.8x10-3, error 7x10-4  (4.80.7)x10-3 • Value 1092, error 56  109060 • Value 1092, error 14  109214 • Value 12.345, error 0.35  12.30.4 Don’t over quote results to a level inconsistent with the error 36.6789353720.5 

9. Estimating reading errors 1 Oscilloscope – related to width of trace 3.8 divisions @ 1V/division = 3.8V Trace width is ~0.1 division = 0.1V (3.80.1)V

10. Estimating reading errors 2 Digital meter – error taken as 5 in next significant figure (3.3600.005)V

11. Estimating reading errors 3 Analogue meter – error related to width of pointer Value is 3.25V Pointer has width 0.1V (3.30.1)V

12. 17 16 Estimating reading errors 4 • Linear scale (e.g. a ruler) • Need to estimate precision with which measurement can be made • May be a subjective choice • 16.770.02

13. 16 17 16 17 Estimating reading errors 5 • 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.

14. 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?

15. 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)100is 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.

16. Treatment of errors in formulae

17. Example of error manipulation 1 Where r=(50.5)m A=78.5398m2 Hence final result is A=(7916)m2

18. Example of error manipulation 2 • P=2L+2W where L=(40.2)m and W=(50.2)m • P=18m • P=(18.00.3)m

19. Example of error manipulation 3 l=(2.50.1)m, g=(9.80.2)ms-2 =3.1735s /=0.022 hence =0.022x3.1735=0.070 =(3.170.07)s

20. 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 (892) cm using a rule of calibration accuracy 2%. Absolute error = 0.03x89=2.7cm Value =(893)cm

21. 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.

22. 2 2 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 √  Probability of result  

23. 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%

24. 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