Presentation on theme: "Introduction to experimental errors"— Presentation transcript:
1 Introduction to experimental errors Department of Physics and AstronomyUniversity of SheffieldPhysics & Astronomy LaboratoriesIntroduction to experimental errorsJ W Cockburn
2 Outline of talk Why are errors important? Types of error – random and systematic (precision and accuracy)Estimating errorsQuoting results and errorsTreatment of errors in formulaeCombining random and systematic errorsThe statistical nature of errors
3 Why are errors important? Two measurements of body temperature before and after a drug is administered38.2C and 38.4CIs 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
4 Random errors An error that varies between successive measurements Equally likely to be positive or negativeAlways present in an experimentPresence obvious from distribution of values obtainedCan 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 dataSometimes 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 onlyTrue valueRandom + systematicA result is said to be accurate if it is relatively free from systematic errorA 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 errorWhen using scientific notation, quote value and error with the same exponent
8 Quoting results and errors Value 44, error 5 445Value 128, error 32 13030Value 4.8x10-3, error 7x10-4 (4.80.7)x10-3Value 1092, error 56 109060Value 1092, error 14 109214Value , error 0.35 12.30.4Don’t over quote results to a level inconsistent with the error 0.5
9 Estimating reading errors 1 Oscilloscope – related to width of trace3.8 1V/division = 3.8VTrace width is ~0.1 division = 0.1V(3.80.1)V
10 Estimating reading errors 2 Digital meter – error taken as 5 in next significant figure(3.3600.005)V
11 Estimating reading errors 3 Analogue meter – error related to width of pointerValue is 3.25VPointer has width 0.1V(3.30.1)V
12 Estimating reading errors 4 1617Linear scale (e.g. a ruler)Need to estimate precision with which measurement can be madeMay be a subjective choice16.770.02
13 Estimating reading errors 5 16171617The 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 cylinderHow 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 valuesA, B, C and Z are the absolute errors in A, B, C and ZHence A/A is the fractional error in A and (A/A)100 is the percentage error in A etcA and A will have the same unitsAssume errors in numerical or physical constants (e.g. , e, c etc) are much smaller than those in measured values – hence can be ignored.
17 Example of error manipulation 1 Where r=(50.5)mA= m2Hence final result is A=(7916)m2
18 Example of error manipulation 2 P=2L+2W where L=(40.2)m and W=(50.2)mP=18mP=(18.00.3)m
19 Example of error manipulation 3 l=(2.50.1)m, g=(9.80.2)ms-2=3.1735s/=0.022 hence =0.022x3.1735=0.070=(3.170.07)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 (892) cm using a rule of calibration accuracy 2%.Absolute error = 0.03x89=2.7cmValue =(893)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 The statistical nature of errors If a quantity has an associated error thenThere 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√Probabilityof result22
23 Comparing values Need to look at overlap of distributions Case of two quantities A and B which differ by sum of errors A+BProbability of agreement ~2x1/36 = 6%
24 ConclusionsSystematic and random (reading) errors – accuracy and precisionQuoting errorsEstimating reading errorsManipulating and combining errorsThe statistical nature of errorsFurther reading: Document on website or any text book on practical physics e.g. ‘Experimental Methods’ L Kirkup or ‘Practical Physics’ G L Squires