Measurement and Uncertainties


 Erick Robinson
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1 Phs LL Introducton Measurement and Uncertantes An measurement s uncertan to some degree. No measurng nstrument s calbrated to nfnte precson, nor are an two measurements ever performed under eactl the same condtons. The whole dea of measurement s to determne the true value of some quantt, but ths true value s alwas hdden to us all we have are our uncertan measurements. Because of ths, measurement s actuall a two part process: frst, the measurement tself, and second, a determnaton of the uncertant n that measurement. To put t n everda terms, f someone were to ask ou how long t takes to drve to the beach, ou mght sa, It took me about four and half hours, gve or take twent mnutes. The statement epresses both a measurement and an estmate of ts uncertant. Note n the eample that the true value of the travel tme remans unknown. The 4.5 hours s one partcular measurement, and ts value would depend greatl on how the measurement was made the route taken, weather condtons, other traffc, etc. To deal wth these ssues, scentsts have developed certan technques and conventons n data handlng and analss so that the can assess and communcate the accurac and the precson of ther measurements. (Accurac refers to how correct the measurement s how close t comes to the true value. Precson refers to how fnel scaled the measurement s n other words, to the sze of the uncertant.) These ssues are descrbed n the sectons below. Theor of Errors Eperence has shown that there are two major sources of uncertant n measurements: random error and sstematc error. (The word error s used n the sense of varaton, not n the sense of mstake.) Sstematc error results from defectve nstruments, mstaken procedures or false assumptons. In ths case, ever measurement s skewed b some unknown factor and the results are naccurate. No averagng or other data handlng technque wll mprove the results the onl cures for sstematc errors are to antcpate them and to avod them. Random error s the result of unpredctable effects from the surroundngs or uncontrollable varatons n the epermental apparatus or setup. Random errors are nevtable n an real eperment. But, snce the are nherentl statstcal n nature, random errors can be dealt wth b utlzng concepts from probablt theor, as descrbed below.
2 Uncertantes n a Set of Measurements To ncrease the accurac of our results, we measure the same thng man tmes and average the results. Ths ncreases accurac, snce random varatons wll tend to cancel out n the average. To estmate the uncertant n the average value, we can use the average devaton or the standard devaton of the measurements. These terms are defned below. or n measurements of quantt, the average or mean value s: n Ths s the most probable value for the true value of the quantt beng measured, assumng the varatons n the measurements are random and not sstematc. The devaton δ of each measurement from the mean s: δ The average devaton δ n a set of measurements s the average of the absolute values of the devatons: δ n δ The theor of errors tells us that there s a 57.5% probablt that the true value les wthn ± δ. The average devaton s a frst estmate of the range of error (uncertant) n the mean value. The standard devaton of the set of measurements s a better estmate of the range of error. Ths s found b averagng the squares of the devatons, and then takng the square root: σ n δ The theor of errors tells us that, as the number of measurements n approaches nfnt, 68.3% of them wll fall wthn ± σ. The standard devaton s a measure of the sze of the scatter of the results about the average value. The sample standard devaton s a modfcaton of the standard devaton that s approprate for a lmted number of measurements:
3 σ sample ( n ) δ Here we dvde b (n ) nstead of b the number of measurements n. The reason for the modfcaton s that the standard devaton (dvdng b n) refers to the entre set of possble measurements an nfnte number. In laborator work we take a lmted number of measurements. In other words, we onl take a sample of the entre set of possble measurements. The formula s derved n advanced tets on probablt theor. Here, we wll just get a sense of wh t s reasonable. rst, note that as the number of measurements ncreases ( n ), the sample standard devaton becomes closer and closer to the formal standard devaton. Now look at what happens when n s small. or one measurement, the average s just that one measurement and the average devaton s zero. But (n) s also 0 zero, so that σ sample, whch s undefned. Ths s as t should be, snce for one 0 measurement onl, we have no nformaton about the statstcal scatter of the measurements about the mean value. Were we to use the frst formula (dvdng b n) we would fnd thatσ 0, whch mples that there s no scatter and that we are completel certan of the result, whch s absurd. In these laboratores, we wll use the sample standard devaton. Wth ths understandng, we wll dspense wth the subscrpt and refer to the sample standard devaton smpl as the standard devaton, σ. To summarze the above, we should alwas report our epermental values for some measured quantt n the form: ± σ. Ths shows our best determnaton of the true value ( ) and our estmate of the uncertant (σ ). Uncertantes n Sngle Measurements or sngle measurements, we cannot appl statstcal technques. Instead we wll use the followng conventon: The uncertant n a sngle measurement s smpl the precson of that measurement. or nstance, f we fnd that the length of a rod usng a ruler s 8.4 centmeters, the uncertant n that measurement s at most ± mllmeter. We should report the measurement for the length L as:
4 L ( 8.4 ± 0.)cm In smbolc notaton, we wrte ths as: L Lmeasured ± L. Here, L measured s the value we determned wth the ruler and L s the uncertant. The smbol s used n the sense of varaton or dfference. Sgnfcant gures and Roundng When performng calculatons on measured quanttes, arthmetc results should be rounded to reflect the precson (the senstvt) of the measurng nstruments. The gudelnes to follow are summarzed n the table below. Percent Error, Percent Dfference, Percent Devaton A basc part of an eperment s to determne the sgnfcance of our measured results. We want to know f the results are reasonable or nonsense, or f the compare well to results found b others. We do ths b takng the rato of two quanttes to be compared, and epressng the result as a percent. Ths s summarzed n the second table below. An Eample Here are some fall tmes for a stone dropped from a heght of meters, usng a dgtal stopwatch. Tral Tme Devaton Squared Devaton (seconds) (seconds) (seconds squared) Average: Standard Devaton: 0.0 seconds All values have been rounded to an approprate number of sgnfcant fgures. We can report the tme of fall t as: t t ± σ 0.59 ±.0 seconds v
5 Table : Sgnfcant gures Sgnfcant gures The number of sgnfcant fgures n a number tells us how certan the result s, and where roundng has occurred. Addton & Subtracton Rule Multplcaton & Dvson Rule (Use also for square roots, trg functons & etc.) All dgts n a measurement are sgnfcant ecept zeros that are used as placeholders. Not all dgts n a calculated result are sgnfcant. See the rules below. When addng or subtractng, the result should have no more decmal places than the term wth the least number of decmal places. In the eample, the three length measurements sum to 4.3 cm, accurate to the nearest /0 cm. When multplng, dvdng, takng roots or usng trgonometrc functons, the result should have no more sgnfcant fgures than the factor wth the least number of sgnfcant fgures. Eamples: 0 meters has 3 sgnfcant fgures. 000 centmeters also has 3 sgnfcant fgures. Eample:.763 cm.5 cm cm cm (not cm) Eample:.763 cm.5 cm.0068 cm.7 cm 3 (not cm 3 ) Eamples: Sn(33).54 Sn(33.0).545 (not as gven b a calculator) v
6 Table : Percent Error & etc. Percent error Use when comparng an epermental result to a known or accepted value % Error Result Accepted Accepted Percent dfference Use when comparng two equall weghted values. % Dfference Value Value Average (take the absolute value) Percent devaton Use when comparng one value n a set of measurements to the average of the set. % Devaton Value Average Average v
7 Propagaton of Error Suppose we want to calculate some phscal quantt C based on epermental measurements of phscal quanttes A and B. In other words, C s a functon of A and B. or nstance, we mght measure a dstance and a tme for a movng object and calculate ts veloct: v /t. The veloct v s a functon of and t. Both A and B have uncertantes assocated wth them, and we need to calculate the resultng uncertant n C, so that we can report our results n ths form: C Cmeasured ± C where C measured s the value calculated from our measurements and C s the range of possble error. Ths subject s known as propagaton of error. We calculate C from A and B and ther uncertantes A and B usng the propertes of dervatves (snce C s a functon of A and B, and A and B are ncrements n these functons). The general method s outlned n the net secton. or man standard functons, the method can be tabulated and we need onl remember a few smple rules. Look over the Range of Error Calculatons table below, then follow through ths eample. Eample Suppose ou have the followng formula for the speed of a sound wave along a strng: v d In words, the veloct of the sound wave s the square root of the rato of the tenson on the strng to the mass per unt length d. To fnd v, we wll smplf and go pece b pece. The epresson can be rewrtten as: v d ollowng rule 3 n the table: v v ( / d) ( / d) But followng rule : ( / d) ( / d) d d v
8 So: ) ( d d v v And: ) ( d d v v To calculate Δv, we would plug our measured or calculated values for, d, v, Δ, and Δd nto the above epresson. v
9 Range of Error Calculatons In the followng, C refers to some phscal quantt that s calculated from measured quanttes A and B. A and B are the uncertantes n A and B. These mght be: The standard devatons for these quanttes; Estmates, based on the calbraton of our measurng nstruments. C s the uncertant or range of error n the calculated result. If C equals: Then:. A B or A B C A B. AB (A tmes B) or A/B (A dvded b B) C/C A/A B/B 3. A m B n (A to the m power tmes B to the n power) C/C m (A/A) n (B/B) where m denotes the absolute value of m 4. Sn A C Cos A * A 5. Cos A C Sn A * A 6. ln A (natural log of A) C A/A 7. Other transcendental functons Use dfferentals (see Range of Error usng Dfferentals)
10 Range of Error usng Dfferentals Dfferentals and Increments A dfferental s an nfntesmal change n some functon or ndependent varable. A dervatve s just a rato of dfferentals: d/d. We could wrte the dfferental of the functon lke ths: d d d Read ths as: The dfferental of equals the dervatve of wth respect to, tmes the dfferental of. An ncrement s a small but fnte change n some functon or ndependent varable. If s a functon of and changes b an ncrement, then would change b an ncrement. How much changes can be appromated b modfng the above epresson for dfferentals to: d d d We can use ths to estmate the range of error n quantt. Here, would be some measured value and would be our uncertant n that value t mght be the standard devaton for. or nstance, supposed we measured the sde s of a square plate to be 5.6 cm / 0. cm. The area A of the plate would be s 43 cm, wth an appromate error of: A ss 6 cm (rounded to the unts place) (Note that f ou dvde ths epresson b A s, ou wll end up wth A/A s/s, whch s what ou get b followng rule 3 n the table.) Ths method can also be appled to trgonometrc functons. or nstance, f f snθ, then f cos θ θ. Dfferentals of unctons of Several Varables We can use the same method for functons of several varables. or nstance, suppose f s a functon of,, z,.w,... (an number of varables). The dfferental of f s: df d d dz dw... z w
11 The notaton sgnfes the partal dervatve of f wth respect to. A partal dervatve s a dervatve of a functon of several ndependent varables wth respect to one of them, treatng the other varables as constants. The operaton of takng a partal dervatve s eactl the same as for ordnar dervatves. The notaton smpl sgnfes that we are temporarl treatng the other varables as constants. Read the above epresson as, The dfferental of f equals the partal dervatve of f wth respect to tmes the dfferental of, plus the partal dervatve of f wth respect to, tmes the dfferental of, plus... As n the sngle varable case, we can estmate our range of error b usng ncrements n the above epresson: f z w... z w Here, f s the change n the functon f that results when the ndependent varables change b small amounts. If the ncrements,, etc. represent the errors n the ndependent varables, then f s an estmate of the uncertant n the calculated result. Rule: Uncertantes Alwas Add One caveat n rangeoferror calculaton: dervatves are sometmes negatve. or d( ) 3 nstance,. Uncertantes can be postve or negatve: ±. Alwas d choose the sgn for so that each term n the above epresson for f adds to the others, rather than subtracts. We want to estmate the mamum possble error, so we assume the worst case: that each error compounds the total uncertant rather than havng some errors offset others. In effect ths means ou should take the absolute value of each of the terms n the epresson. Two eamples The epresson for f  we ll call t the uncertant epresson s how the formulas for uncertantes were derved. or nstance, ou can show that for a functon C A/B, then C C A B A B (You wll need to use the caveat that uncertantes add).
12 or functons that can be broken down nto smple combnatons of power functons, t s easest just to remember the rules n the table. or complcated epresson, or for transcendental functons (trg functons, eponental functons, etc.) t s easest to take the dervatves. The followng eamples should llustrate the procedure. () Denst of a clnder. Denst s defned as mass dvded b volume. The volume of a clnder s gven b V πr L, where r s the radus and L the length. Suppose: r.65 /.05 cm L 7.33 /.05 cm m 446 g / 4 g Then ρ m V m.76gm / cm πr L 3 ρ ρ ρ ρ m r L m r L m m m r 3 πr L πr L πr L L m r ρ ( m r L ) L.5 g/cm 3 Note we changed the sgn of the second and thrd terms on the rght to follow the caveat. The result s the same as ou would get followng the rules n the table. () Drecton of a force, Gven the components of a force:.56 /.0 N 3.78 /.5 N Then the magntude s: The drecton of the force s: N θ Tan  ( / ) 55.9 degrees The uncertant n θ s: θ θ θ
13 θ θ sn cos.0366 radans.09 degrees In the above we used:, θ tan etc. ) ( Tan d d, and the chan rule.