CHAPTER 2: Fractional Uncertainties

1 1 CHAPTER 2: Fractional UncertaintiesAbsolute vs Fractional UncertaintiesWe have seen that the correct reporting of a physical measurement requires thatone write the best value with a quoted error uncertainty. In general then, wewriteMeasuredx=xbest xIn this case xis the absolute uncertainty of the measurement . However, it isoften more clear to write thefractional uncertaintyof the measurement instead ofthe absolute uncertainty. The idea is that a measurement with a relatively largefractional uncertainty is not as meaningful as a measurement with a relativelysmall Fractional of Fractional UncertaintyThe Fractional uncertainty is just the ratio of the absolute uncertainty, xto thebest valuexbest: Fractional Uncertainty xxbestIn general, the absolute uncertainty xwill be numerically less than the measuredbest valuexbest.

2 Otherwise the measurement is generally not worth only exception to this rule is in the case where one is trying to make aso-called null measurement . In that case only the absolute uncertainty all non-null measurements which have their absolute Uncertainties less thanthe measured quantity itself, then it is standard practice to quote the fractionaluncertainty as a percentage. For example, suppose one measures a lengthlas50 cm with an uncertainty of 1 cm. Then the absolute quote isl= 50 1 cmwhile the Fractional uncertainty isFractional Uncertainty = ll=150= the result can also be given asl= 50 cm 2%Lecture 3: Fractional Uncertainties ( CHAPTER 2) and Propagation of Errors ( CHAPTER 3)2 Propagation of ErrorsIntroduction to Propagation of ErrorsIn determining a physical quantity it is only very rarely that we make a directexperimental measurement on the quantity itself.

3 Much more often it is thecase that we make direct measurements on quantities which are mathematicallyrelated to the unknown physical quantity. Then by a series of either simple orcomplicated mathematical steps, we arrive at the unknown quantity of we can think ofdirectly measuredphysical quantities andindirectly measuredphysical quantities. The directly measured physical quantities will have errorsassociated with them as we discussed in the opening lectures, which errors re-flect the measurement apparatus or techniques used. The indirect, or calculatedphysical quantities will also have errors associated, and these errors will be thepropagatederrors from the direct measurement errors.

4 We have already seen inChapter 2 the first examples of error propagation involving addition or subtrac-tion and multiplication or division. The book s rules are:In adding or subtracting physical quantities, theabsolutemeasurement errors ofthe individual quantities are added to obtain the absolute error in the multiplying or dividing physical quantities, thefractionalmeasurements er-rors of the individual quantities are added to obtain the Fractional error in thecalculated professional physicists rules are much the same except that we add thesquares of the individual errors and take the square roots of that in Direct MeasurementsWe have seen that the error quotes for direct measurements are generally asso-ciated with how well can one read the measurement device.

5 The classic exampleis a meter stick calibrated in millimeter gradations where a half millimeter errorquote would be quite reasonable. However, one may have a digital device, suchas a digital stop watch, which is capable of giving readings in the might then be tempted to quote time errors to second. However,this would probably be a mistake in most introductory mechanics labs. Theactual physical process itself, such as the fall time in a gravity experiment or anAtwoods machine, is likely to vary by more than a milli-second even under verycontrolled conditions. Hence a quote of seconds would be more it is important to distinguish between the precision of the measuring deviceand the precision associated with the measurement 3: Fractional Uncertainties ( CHAPTER 2) and Propagation of Errors ( CHAPTER 3)3 Uncertainties in Direct MeasurementsCounting ExperimentsA very common type of physical measurement is simple a counting experiment.

6 The typical example is the decay of a long-lived (years) radioactive source forwhich the emission of particles is completely random over a very short timeinterval (say milli-seconds), but has a definite average rate over a longer timeinterval (say minutes).In such experiments, one counts the emitted particles for a fixed time such asa few minutes and records the number of counts. That number, divided by thetime interval, constitutes the count rate. One may count again for the sameamount of time and find a slightly different number of counts with a slightlydifferent count rate. Here we see an example of adirectly measuredphysicalquantity (the counts) and a derived physical quantity (the count rate).

7 The question which arises is what is the error associated with this kind of count-ing experiment. The answer is remarkably simple. The error associated with acounting measurement is simply the square root of the number of counts. Thisis the prime example ofstatistical error. Specifically, counting experiments arepart of what is know asPoisson Distributionwhich is fully discussed in Error in Counting Experiments and Count Rate ErrorsTo recapitulate the discussion above for counting experiments, if one has anexperiment whereNcounts are measured, then the uncertainty Nin thatmeasurement is given by the Poisson Statistics formula: N= NSo, to take an easy number, say we measure 100 counts in a 2 minute we can say that the error in that measurement is 100 = 10, and we canthen quoteN= 100 10 However, one will typically not quote the actual number of counts, but ratherthe rate of counts in a given time period.

8 Let s give the rate quantity the symbolR, so obviously in this simple exampleR=NT=1002= 50 counts/minuteNow what is the error associated with the rate quantityR?Lecture 3: Fractional Uncertainties ( CHAPTER 2) and Propagation of Errors ( CHAPTER 3)4 Error Propagation ExamplesCount Rate Errors and Error PropagationThe answer to the question of the error in a count rate is an example of errorpropagation. Here we have a directly measured physical quantity, counts, dividedby a fixed constant, time (2 minutes), in order to obtain a derived quantity. Insuch a case, the absolute error in the derived quantity is the absolute error inthe measured quantity divided by the fixed in the case at errorR=NT= R= NT= NTIn the present example, then R= 100/2 = 5 counts/minute.

9 So we wouldquote the result asR= 50 5 counts per General Rule for Error Propagation of Calculated QuantitiesThebook (page 5) gives the first general rule according to the following formula. Ifthere is a quantityqwhich is calculated as the product of a constantBand ameasured quantityxq=Bxand the measured quantity has an error x, then the error qin the calculatedquantity is given as q=B xIn our count rate example above we actually hadB= 1/Tand were dividingby a fixed quantity instead of multiplying. But mathematically, it makes way of stating this same rule is that the Fractional error in the derivedquantity is the same as the Fractional error in the directly measured Law and an Error PropagationA second general rule about error propagation applies to a power law depen-dence.

10 Take for exampleq=xnwherenmay or may not be an integer. Thenthe error qis given as qq=n xxIfnis an integer, you can think of this as adding upntimes the Fractional errorinxsinceqis the product 3: Fractional Uncertainties ( CHAPTER 2) and Propagation of Errors ( CHAPTER 3)5 Propagation of Errors with One VariableArbitrary Function of One VariableSuppose we have a calculated physical quantityqwhich depends upon a mea-sured physical quantityxaccording to the general functionq=q(x)In error analysis, we want to know that how much uncertainty qdo we attachtoqwhen the uncertainty inxis given as xThe answer to this question comes directly from calculus. You should haveseen in elementary calculus that it is always possible to expand a well-behavedfunction about some point in terms of increasing orders of derivativesq(x) =q(x0) + (x x0)dqdxx=x0+ (x x0)2d2qdx2x=x0+.