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

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 38.2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

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 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

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

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 

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

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

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

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 choice 16.770.02

Estimating reading errors 5 16 17 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.

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?

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.

Treatment of errors in formulae

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

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

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

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

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

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%

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