PHYS 352. On Measurement, Systematic and Statistical Errors. Errors in Measurement

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1 PHYS 352 On Measurement, Systematic and Statistical Errors Errors in Measurement when you make a measurement you should quote an estimate of the uncertainty or error all measurements have systematic errors some measurements have statistical errors e.g. measurement of the total solar neutrino flux by the Sudbury Neutrino Observatory [4.94 ± 0.21 (stat.) ± 0.36 (syst.)] 10 6 cm 2 s 1 1

2 Statistical Errors physicists consider them as the easy one to deal with statistical errors arise whenever a probability is involved radioactive decay when a sample of an ensemble is measured if the measurement is at the quantum level example: measuring very low levels of illumination the light source being measured is so weak that only single photons are arriving at the light detector (e.g. a photomultiplier tube) not all photons from the light source are detected i.e. there is some probability that any given photon makes it to the light detector and is subsequently detected the size of the electrical signal from the light detector (integrated over a fixed time interval) is a measure of the light source intensity and this measure has statistical error in some time intervals a few more photons are detected and some time intervals less the mean value and the statistical spread of the number of detected photons are precisely understood by physicists statistics determine the uncertainty in the measurement easy! Systematic Error from Sayer: Systematic error. This is the degree to which a measured value differs from the 'true' value because of errors inherent in the measurement. This may be due to an incorrect scale, a wrong calibration or an erroneous assumption. In instrumentation, changes in the original calibration of an instrument over time, or if the instrument is used under abnormal conditions are major sources of systematic error. Systematic errors are generally not statistical in nature and sometimes may be corrected by measurement of a standard. physicists often deal with systematic errors as though they are statistical because there isn t a better alternative meaning that we combine systematic errors in quadrature as though they were statistically independent; but much of the time they are not 2

3 How to Think About Systematics? how could my measurement be wrong? leads to: how could my measurement system be wrong? leads to: how could my transducer give a wrong value leads to: examining transducer characteristics four classes of transducer characteristics lead to four important classes of systematics accuracy/calibration repeatability linearity noise or backgrounds Calibration Systematics absolute calibration is wrong water was at 99 C when the thermometer tick for 100 C was marked calibration drifts with time transducer (and its associated electronics) are turned on and it outputs accurate readings; then, it starts to warm up (electrical heating) and its output changes with its internal temperature example: how to estimate calibration accuracy systematic? apply a reference standard as input and measure the transducer output value boil water (controlled) and temperature probe measures this as C make desired temperature measurements repeat measurement of reference standard boil water again, and temperature probe outputs 99.7 C the calibration correction factor is either 1/1.015 or 1/0.997; make the correction, split the difference, and use the difference as error you have bounded the calibration stability and accuracy between the times the reference was checked, one postulates the accuracy lies within the observed range 3

4 Linearity Systematic linearity curve (output y(x) versus input x) is not what you think; this is a possible systematic e.g. your assumption is perfect linearity whereas the transducer has small nonlinearity at the end of its range example how to estimate linearity systematic make multiple measurements of y versus x fit a straight line to y(x): get slope m±δm and intercept b±δb the uncertainties in the parameters for the linearity curve help estimate the systematic error due to non-linearity thus, for any measurement y (corresponding ideally to a single input x), there is a range of values for x that is consistent with the fitted linearity curve parameters millivolts thermocouple output Example: Repeated Measurement say you are making a measurement of some quantity, call it X you make a single measurement of that quantity and estimate the sources of error X ± ΔX you then take repeated measurements of the same quantity and find the scatter distribution of the measured X values is smaller than ΔX from a single measurement? perhaps you have overestimated the uncertainty? is the same as ΔX from a single measurement? perhaps you did a good job with your error estimate? is greater than ΔX from the single measurement? you could increase your original error estimate to account for the poor repeatability to do this, you could calculate the standard deviation of the mean (for example) and use that as the uncertainty; optionally a weighted average of multiple measurements and their errors determines the ensemble s error summary: unaccounted for systematics may lead to expanded uncertainties or use if different measurements have different uncertainties 4

5 Plot: Repeated Measurements Caution: Uncertainty from Scatter this is what we do as experimental physicists and it s correct (acceptable) to do this but we need to be aware of the perils we are applying statistical analysis to systematics something is causing the measured points to scatter something is causing the poor repeatability of the transducer that something could be an undiscovered effect or, that something could be easily corrected if you find a systematic effect, and you find out how to correct it, it is no longer a systematic error 5

6 What s the Point? this course is Measurement, Instrumentation and Experiment Design understanding the underlying concepts behind statistical and systematic errors is crucial and it comes from understanding the instrumentation (and measurement system) what you want to know: show me how to estimate systematic errors; what procedure do I follow? my answer: there is no fixed procedure for assigning systematic errors that s applicable in the realm of all possible measurements and I can t describe a procedure for you to follow that s correct every time; but I can tell you that the most important thing is to understand and think about the basics measurement is the reality; it is not always correct but there is always a reason why an instrument (transducer) is producing the measurement (accurate or offset) that it is the more you know about transducers, the better you will be at estimating the systematic errors they produce 6