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by: Dr. Simeon Wiza


Dr. Simeon Wiza
GPA 3.96

David Pengra

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David Pengra
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This 10 page Class Notes was uploaded by Dr. Simeon Wiza on Wednesday September 9, 2015. The Class Notes belongs to PHYS 431 at University of Washington taught by David Pengra in Fall. Since its upload, it has received 20 views. For similar materials see /class/192432/phys-431-university-of-washington in Physics 2 at University of Washington.




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Date Created: 09/09/15
Notes on Data Analysis and Experimental Uncertainty Prepared by David B Penyra University of Washington and L Thomas Dillman Ohio Wesleyan University This set of notes has been adapted from ones given to Ohio Wesleyan University physics students in the introductory laboratories Although they are pitched at a relatively elementary level they contain many hints that may be useful even to advanced students The topics discussed below may be supplemented with pertinent sections in the references listed at the end of the article the text by Bevington and Robinson see reference gives a good account of basic statistical theory and practical computer methods most often used in analyzing data from physics experiments 1 Types of Uncertainty There are two basic kinds of uncertainties systematic and random uncertainties Systematic un certainties are those due to faults in the measuring instrument or in the techniques used in the experiment Here are some examples of systematic uncertainty o If you measure the length of a table with a steel tape which has a kink in it you will obtain a value which will appear to be too large by an amount equal to the loss in length resulting from the kink On the other hand a calibration error in the steel tape itselfian incorrect spacing of the markingsiwill produce a bias in one direction o If you measure the period if a pendulum with a clock that runs too fast the apparent period will be systematically too long The stiffness of many springs depends on their temperature If you measure the stiffness of a spring many times by compressing and decompressing it the internal friction inside the spring may cause it to warm You may see this by a systematic trend in your data set for example each data point in a data set will be smaller than the previous one Random uncertainties are associated with unpredictable variations in the experimental conditions under which the experiment is being performed or are due to a de ciency in de ning the quantity being measured Here are some examples of random uncertainty 0 Electrical noiseifrom nearby circuits or equipment thermal effects or imperfect connectionsi may cause random uctuations in the magnitude of a quantity measured by a voltmeter o The length of a table may depend on which two points along the edge of the table the measurement is made The length is imprecisely de ned in such a case 0 Repeated measurements of the period of a pendulum which are made with a stopwatch vary because it is hard for a person to start and stop the watch at exactly the same point in the pendulum s swing Note however that if the experimenter always starts the watch late but stops it early this will lead to a systematic error Of these two types of uncertainties random uncertainties are much easier to deal with and to quantify There is no general procedure for the magnitude of uncertainties as there is for random uncertainties Only an experimenter whose skill has come through long experience can consistently detect systematic uncertainties and prevent or correct them If an experiment has low systematic uncertainty it is said to be accurate If an experiment has low random uncertainty it is said to be precise Obviously an experiment can be precise but inaccurate or accurate but imprecise When thinking about uncertainty it is important to remember these associations so they are worth repeating 0 Random uncertainty decreases the precision of an experiment 0 Systematic uncertainty decreases the accuracy of an experiment These distinctions are illustrated in Fig 1 You should avoid falling into the trap of thinking that because the uncertainty of a measurement is always the same then it is systematic Systematic uncertainty does not mean that the uncertainty is repeatable What it means is that the uncertainty involves physics that has not been accounted for in the analysisitwo very different ideas 0 O Precise Precise Not precise Not precise Not accurate Accurate Not accurate Accurate Figure 1 A bulls eye plot showing the distinction between precision and accuracy in a measure ment The black dots represent data points taken in a measurement of a quantity whose true value is at the center of the circles Before proceeding further it may be useful to point out that blunders are not a source of uncertainty They can always be eliminated completely by careful work In your laboratory reports never list misreading the instrument or getting the wrong units as a source of uncertainty 2 The Mean Standard Deviation and Standard Deviation of the Mean Random uncertainty is often associated with the concept of standard deviation This is best il lustrated by an example Suppose ten students each measure the diameter of a steel ball with a micrometer caliper For a variety of reasons we do not expect all the measurements to be identical The sources of error include 0 some students tighten the micrometer caliper more than others 0 the steel ball may not be perfectly round some students may not exercise care to be sure they are measuring a great diameter ithe ball is not centered between the jaws the temperature of the steel ball may change with time as the ball is handled and hence its diameter may change slightly through thermal contraction or expansion there may be varying amounts of corrosion on the steel ball Exercise 1 Which of the above sources of error contribute to systematic uncertainty Which contribute to random uncertainty Explain how you came up with your answers The obvious question to ask is What is the best value for the diameter of the steel ball77 If the sources of error are random that is they give values for the diameter which vary randomly above and below the true77 value but do not skew all of the values in one particular direction then an obvious procedure to get the best value for the diameter is to take the average or arithmetic mean The mean of a set of numbers is de ned as the sum of all the numbers divided by the number of them In mathematical language if we have N observations and mi represents any one of the observations ie i can have any integer value from 1 to N then the arithmetic mean which we designate by the symbol E is given by N 7 w1w2wN 1 1 Having obtained a mean or best value i it is important to have a way of stating quantitatively how much the individual measurements are scattered about the mean For a precise experiment we expect all measurements to be quite close to the mean value The extent of scatter about the mean value gives us a measure of the precision of the experiment and thus a way to quantify the random uncertainty A widely accepted quantitative measure of scatter is the sample standard deviation 5 For the special case where all data points have equal weight the sample standard deviation is de ned by the equation Although this equation may not be intuitive inspection of it reveals that 5 becomes larger if there is more scatter of the data about the mean This is because 7 if for any particular i will on the average increase with greater scatter of the data about the mean so that 290239 7 if increases Note that s has the same units as an or i since the square root of the sum of squares of differences between mi and E is taken The standard deviation 5 de ned by Eq 2 provides the random uncertainty estimate for any one of the measurements used to compute s lntuitively we expect the mean value of the measurements to have less random uncertainty than any one of the individual measurements It can be shown that the standard deviation of the mean value of a set of measurements om sigma em when all measurements have equal statistical weight is given by Note that om is necessarily smaller than 5 When we speak of the uncertainty 0 of a set of measure ments made under identical conditions we mean that number om and not s It is most important that the student distinguish properly between standard deviation associated with individual data points 5 and standard deviation of the mean of a set of data points om Exercise 2 Five students measure the mass of an object by making two separate measurements each These measurements in grams 980 987 989 995 991 998 992 1005 997 984 1 Calculate the mean the standard deviation and standard deviation of the mean using your calculator and the above formulas Show how you made the calculations M Do the same calculations as in part 1 but using the statistical package on your calculator Refer to your calculator s manual for instructions If you have lost your manual you may be able to nd the instructions at the manufacturer s website for example Texas Instruments has copies of their manuals at http wwwti com Usually in lab you will use your calculator to nd means and standard deviations rather than doing the calculations by hand so it is important to know how this is done Write the results down along with a brief description of how you performed them 3 Stating Results with Uncertainty There are two common ways to state the uncertainty of a result in terms of a 0 like the standard deviation of the mean om or in terms of a percent or fractional uncertainty for which we reserve the symbol 5 epsilon The relationship between e and o is as follows Let the quantity of interest be x then by de nition 5 2 7 4 When stating a result and its uncertainty in a report one typically uses the form x i oz with the units placed last For example if the mass of an object is found to be 92 g and the uncertainty in the mass is 03 g one would write in 92 i 03 g When using scienti c notation the factor of ten multiplier should come after the signi cant digits and uncertainty Write in 93 i 03gtlt10 3 kg not in 93 X 103 i 03 X 103 kg WRONG and certainly not in 93 X 103 kg i 30 X 104 kg WRONG Sometimes one will present uncertainty in terms of e but in this case 5 is usually multiplied by 100 so that one would say The mass of the object is 92 grams with an uncertainty of 3 percent77 Unless otherwise instructed you should state all of your measurements following the rst form using 0 There is one important distinction between 0 and 5 when stating results oz always has the same units as a while 5 is always unitless Failure to be conscious of this difference typically costs students many points 4 Comparing Quantities with Uncertainty Frequently one wants to know whether two numbers obtained by two different methods but hypo thetically referring to the same physical quantity agree The term agreement means something very speci c in an experiment lf uncertainties for one or both numbers expressed by an associated o have been calculated one can say that the two numbers agree with each other if they overlap within their uncertainties For example if a theory predicts that the density of an object should be 100i01 gcm3 and a measurement gives a value of 98i03 gcm3 then we can say the two values agree within the experimental uncertainty But if the measurement gave instead 981 i 002 gcm3 then we would be forced to admit that the two values did not agree In the case of disagreement the experimenter faces a problem what effects have not been accounted for There could be a source of additional random error that has not been appreciated or more vexing there may be a source of systematic error that is fouling the accuracy of the measurement Generally sources of random error are easier to track down and rectify but in so doing one may uncover other sources of systematic error that were previously invisible You will often be asked to determine what the dominant source of error is in a particular experiment In general this is a subtle problem as there is no general method for determining systematic error However one important clue can be used when comparing measurements with each other or with theory if the measured quantity including the uncertainty calculated from random sources of error does not overlap with another expected value either from another experiment or theory then you can assume that the systematic error in the 04110 391 e t J t the 04110 391 e tal error This is especially true when comparing against theoretically calculated values as the theory almost always assumes some simpli cations in order to make the calculation reasonable for example neglecting the weight of a string or assuming that friction is zero To reiterate systematic error comes into an experiment when the experimenter neglects some important physics in the analysis In quick measurements we may not always calculate uncertainties for the quantities we measure In these cases the best we can state is that two values disagree by some amount This disagreement is usually presented as a percent of the value of the quantity For example if we did not have uncertainties calculated for the above two density values we could say that they disagree by X 100 2 5 98 7 100 100 The general rules for comparing results in lab reports are these o If uncertainties exist state the quantities with their uncertainty and see if they overlap If they do they agree If not they don t and you should try to explain why that is discuss the physics of the experiment and try to come up with some sources of systematic error o If uncertainties do not exist calculate a percent disagreement If the percent disagreement is less than a few percent the results are probably in agreement If the disagreement is more than ten percent they are probably not in agreement and you should try to explain why Exercise 3 The manufacturer of the mass that was measured by the students in Exercise 2 claims that the mass is 10 g within 04 Is this a valid claim Discuss whether your result agrees with the manufacturer s claim following the guidelines above 5 Signi cant Digits Most students learn the idea of signi cant digits in high school at about the same time that they learn scienti c notation But the results of uncertainty analysis complicates matters What if the uncertainty is very large What are the signi cant digits for o itself What is the uncertainty of a result that is measured repeatedly with a digital instrument like a voltmeter and the same number is recorded every time These questions cause much confusion Here are some guidelines 0 The uncertainty 0 in the nal result should have at most 2 digits and more commonly 1 digit Remember all uncertainty calculations are estimates there is no such thing as an exact uncertainty Use this rule if the rst digit of o is 1 use 2 digits for sigma eg oz 014 g or am 03 g but not oz 034 g The result itself should be stated to the same precision as am For example you should write 95 i 03 g or 952 i 014 g but not 952 i 03 g o If a is especially large you will lose signi cant digits For example suppose that multiple measurements are made with an instrument that is precise to 3 digits and mean value of 952 g is found but for other reasons the data points varied so that the standard deviation of the mean was 2 g The result would have to be reported as 9 i 2 g If the measurement is so bad that o is larger than the value itself you may have no signi cant digits but only know the order of magnitude This case is most common when the quantity in question is expected to be close to zeroisuch measurements may only give an upper or lower bound on the quantity If a is calculated to be much smaller than the smallest digit of your measurement then assume that o is equal to 1 of the smallest digit For example if a measurement of a mass gives exactly 952 g ten times the result should be stated as m 952 i 001 g Thus you may need to round your uncertainty up to the least signi cant digit in your measurement 0 Do not confuse round off errors with uncertainty With calculators and computers there is no reason to prematurely truncate a result just because it is found to be uncertain lf properly used the formulas for propagating uncertainty will take care of the uncertainty in the nal result So keep your extra digits as you go at most one or two extra if calculating by hand but make sure to adjust the nal result when you present your measurements for comparison 639 Propagation of Uncertainty The method of computing the uncertainty in a result which depends on several variables each with its own uncertainty is called propagation of uncertainty or casually error propagation Suppose we have measured the length and width of a table and have computed the standard deviation of the mean value for both the length and the width Our aim is to determine the area of the table and an associated standard deviation of the area It can be shown that the best estimate of the area is simply the mean length times the mean width What uncertainty should we associate with this same area The answer is not obvious and in fact we can distinguish two distinct extreme cases Table 1 Common formulas for propagating uncertainty These equations can be combined in the cases of more complicated formulas or the student may work directly from equation Functional Form Formula Uncertainty formula Product or Quotient f any or f wy 5f 4 mg e SumorDifference fwyorfw7y af4aga Product of factors raised to powers f wmy 5f 4 aneg n25 Constant multipliers f Kw K constant 0f Kaz Logarithmic functions f log8w 0f ex f long w 0f log10eez 04343530 Exponential functions f ea 5f ax f 10z 5f 1og810az 2303az o The uncertainties in the length and width are completely independent 0 The uncertainties in the length and width are completely dependent If the uncertainties are completely independent the possibility of compensation occurs That is if the uncertainty in the length causes the area to be too large then the uncertainty in width may be such as to cause the area to be too small On the average the total uncertainty in the area will be algebraically less than the sum of the separate contributions to the uncertainty in the area On the other hand for completely dependent uncertainties we must take into account the fact that the uncertainties are always correlated This leads to complications involving a quantity called the covariance of two correlated quantities which we do not discuss in this elementary account The case of completely independent uncertainties is nearly approached in many experimental situations and we con ne our attention to this case Without going into the derivations see 1 pp 36741 for further details the theory of error analysis gives a general formula for the uncertainty when a result is found by a calculation from a collection of measurements The formula is based on the idea of a rst order Taylor series expansion of functions of many variables It is valid when the various uncertainties 7239 of the 2 different variables are small compared to the values of the quantities and on the requirement that the uncertainties are uncorrelated with each other Speci cally if the desired result is a well behaved function fwyz of the physical variables wyz which have uncertainties canal0 then the uncertainty in the value of the result 0f is given by the formula 8f 2 8f 2 8f 2 2 2 2 2 7 7 7 6 Tf adage 2a 2a l where the partial derivatives are all evaluated at the best known values of wy 2 We give the equations required to propagate uncertainty for a number of simple cases All of the formulas in Table 11 may be derived from equation 6 and the functional form that is listed See 1 pp 44747 for derivations We expect students to learn how to propagate uncertainty for simple cases which are covered by the above equations For example suppose you have an equation for some physical quantity say F which is related to another physical quantity say r by the formula 27rK T2 F 7 7 where K is a physical constant with no uncertainty This equation is of the form f xmy where x corresponds to 27rK and y corresponds to r in this case in 1 and n 72 so the uncertainty formula is 6F WZEgWK 77262 12 e 02 72 e 6 26 8 Note that because 27rK has no uncertainty MK 0 and it drops out of the equation for 5F To nd op you would then simply calculate op The quantities of and 5f are always positive as should be evident from the de ning formulas 3 and The formulas in Table 1 are not just math we can read some physics into their form For example in the case of constant multipliers if one scales each data point by a constant then the uncertainty in the mean value should also scale up proportionally More interestingly if you were to measure the area of a square by two independent measurements of the length x and the width y you would use the product or quotient formula to propagate the uncertainty as A xy In this case the errors in the two measurements may be opposite leading to a better nal result But if you used only the length measurement x and assumed that the width y were the same as the length then the possibility of two measurement errors working against each other would be lost and you would have to use the formula A x2 In this case the exact same measurement is used twice so the errors in width versus length can t cancel To propagate the uncertainty you would use the formula for the product of factors raised to powers which gives a larger nal uncertainty than in the former case Exercise 4 In an experiment with an air track an experimenter wishes to determine the average speed of an air track cart between two photogates The distance Ax between the photogates is given by Ax 1000 i 0003 m and the time of travel At between these two points is At 23 i 01 s Calculate the average speed s AxAt the fractional uncertainty es and the absolute uncertainty as given these data Show your work and state the results using correct signi cant digits and following the format given in the section Stating Results with Uncertainty Exercise 5 1 To measure the density of a rectangular object an experimenter measures the object s volume and mass The volume is given by the formula V LWH where L is the length W is the width and H is the height The density 0 is given by 0 mV where m is the object s mass If the measurement of the mass is uncertain by 2 and each of L W and H is uncertain by 4 what is the uncertainty in percent of the density 0 Show your work M The experimenter conducts the same density measurement with a second sample that is spher ical in shape The mass is again uncertain by 2 The diameter d of the sphere is measured to a precision of 4 The volume V of a sphere is given by the formula What is the percent uncertainty of the density 0 in this case Show your work Why is the uncertainty in this case di erent than in the case of a rectangular object What is the underlying reason not just how are the formulas di erent 7 Least Squares Curve Fits The method of least squares is often applied to determine the best curve through a set of data points that is suspected to exhibit a functional relationship If all the points have nearly the same weightuncertainty then we can try to arrange the curve so that as many points fall above the line as below However it is not always clear how to eyeball for the best curve thus analytical techniques become necessary The least squares technique may be described as follows Suppose you have a functional form fc a b which you would like to represent the data set as well as possible and where a b are adjustable parameters that can be varied in order to produce the best t curve The function may be a line mac b where the parameters are m and b a higher order polynomial with more parameters or some complicated function like a sine curve with the amplitude frequency and phase as parameters For each data point the leastsquares technique is to compute yi 7 ag a b and then to calculate a quantity known as X2 chi square which is given by 7 1 04 2 X2Zy2 27 7277 7 9 2 72 where the oi is the uncertainty of each data point The best t is found by adjusting the parameters a b and calculating X2 until the minimum value is achieved If there are N data points and n adjustable parameters one can calculate the reduced chi square 2 2 2 X 7 X i 7 10 XV V N 7 n gt The quantity 1 is known as the degrees of freedom in the problem If one is able to adjust the parameters so that xi m 1 then a good t can be asserted in this case the difference between the t curve and the data is on average about as big as the uncertainty in the data itself The general theory of curve tting is very subtle and beyond the scope of this article But we do have computer programs that can make such ts at our disposal These programs not only nd the best t parameters but also produce the uncertainties in the parameters When propagating uncertainty you should use the values of a given by the computer programs in the error propagation formulas References 1 Bevington Philip R and D Keith Robinson Data Reduction and Error Analysis for the Physical Sciences 3rd edition McGraw Hill New York 2003 E Barford N C Experimental Measurements Precision Error and Truth Addison Wesley Publishing Company Inc Reading Massachusetts 1967 E Beers Yardly Introduction to the Theory of Error Addison Wesley Publishing Company Inc Reading Massachusetts 1953 E Hawkins C E and Niewahner J H Data Analysis Graphing and Report Writing 1st ed Mohican Publishing Co Loudonville Ohio 1983 5 Meyer7 Stuart L7 Data Analysis for Scientists and Engineers7 John Wiley and Sons7 Inc7 New York7 1975 6 Young7 Hugh D7 Statistical Treatment of Experimental Data7 McGrawiHill Book Company7 Inc7 New York7 1962 uncertainty otesiex W Updated 6 January 2009


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