Data Analysis for University Physics

Transcription

1 Data Analysis for University Physics by John Filaseta orthern Kentucky University Last updated on ovember 9, 004

2 Four Steps to a Meaningful Experimental Result Most undergraduate physics experiments have a cook book approach. Students follow a Procedure that involves recording measurements (as Raw Data). Students then make calculations from the data to obtain an experimental result. However, any experimental result only has meaning if it is appropriately analyzed. The four steps given below explain how to get a meaningful experimental result from recorded measurements. Three of these steps are completed as Data Analysis. The final (and perhaps the most important step) is completed as the Assessment and Conclusion. This final step is where student experimenters examine the validity of their results and requires the student to go beyond simply following a cook book ; and can only be completed successfully if the earlier steps on Data Analysis are done properly. ote: On occasion, Data Analysis Steps and 3 below can be replaced by plotting and fitting data (see Appendix B). Step 1: Find the mean value of each measured quantity and the total uncertainty (TU) associated with each mean. Measurements are repeated to calculate a mean value (X m ). The total uncertainty (TU) of the mean is found by combining instrumental uncertainty (IU) and random uncertainty (RU). IU = one half the smallest division on the instrument s scale OR one half the last recorded significant decimal place. s RU = where s = standard deviation of the sample of data for this measurement and = number of trials. (If repeated trials were not done, then ask instructor for an estimated value for RU.) If you are not sure how to calculate s (sample standard deviation), then read Calculating the Standard Deviation of a Sample in Appendix A. Combine IU and RU to find the total uncertainty in your mean measurement: TU = (IU) + (RU). ow record the mean measurement as X m ± TU. Step : Calculate the final experimental result as a function (f ) of one or more mean measurements. Find the total uncertainty of the final result (TU f ) by propagating the uncertainty of the mean measurements into the final result. Let x and y be mean measurements and TU x and TU y be their respective total uncertainties. Let a, b and the powers m and n be known constants. Propagation of uncertainty is done as follows: If f is found by adding/subtracting only: f = ax ± by then TU f = a (TU x ) + b (TU y ) If f is found by multiplication/division only: f = ax m y n m(tu then TU f = f x ) % $ ' + n(tu y ) % $ ' x & y & If the final result is a function (f ) of three mean measurements, then just add a 3 rd term under the radicals above; for example, if f = ax ± by ± cz, then add c (TU z ) under the radical. Likewise, if p(tu f = ax m y n z p then add the fraction z ) % $ ' under the radical. z & If the final result is a combination of adding/subtracting and multiplying/dividing, then use steps to solve. For example, suppose f = (ax ± by)(ax m y n ). Let f 1 = ax ± by and f = ax m y n then find the TU values for f 1 and f. ow use the multiplication approach above to get TU for f where f = f 1 f with f 1 and f being to the 1 st power.

3 3 Step 3: Compare the final result to a true value OR to the same result obtained by other methods. Compare the amount of error (or difference) to what is allowed by total uncertainty. First find the percentage amount of uncertainty (or fuzziness ) in your result: TU % TU f = 100% f & % ( $ f ' Then pick the appropriate path below: Path A: Your final result (f ± TU f ) is compared to a true (or accepted) value (f true ). Find the percent error (or deviation from true) of your result: $ f f % error =100% true ' & ) % f true ( Determine if the experimental error has been successfully minimized: If % error < %TU f, then error is less than the allowed uncertainty (or fuzziness ) and error has been successfully minimized. If % error > %TU f, but % error < %TU f then a small amount of error exists and/or the uncertainty was underestimated. If % error > %TU f then a significant amount of error exists, and either mistakes were made and/or the uncertainty was underestimated. Path B: If there is no true (or accepted) value available, then your final result (f ± TU f ) is compared to the result obtained by another method or experiment (f other ± TU other ). Find the average result: f ave = (f + f other )/ and then find the combined total uncertainty of the two methods (or experiments) and the percent difference: $ f f %TU combo = (%TU f ) + (%TU fother ) % diff = 100% other ' & ) % f ave ( Determine whether or not the two methods (or experiments) are consistent: If % diff < %TU combo then disagreement between the two methods is less than the allowed uncertainty (or fuzziness ) so consistency is good. If % diff > %TU combo, but % diff < %TU combo then a small amount of disagreement exists and/or the uncertainty was underestimated. If % diff > %TU combo then a significant amount of disagreement exists, and either mistakes were made and/or the uncertainty was underestimated. Step 4: Complete an assessment and reach a conclusion. Indicate all plausible sources of error and plausible additional sources of uncertainty in the result. If possible, explain the size and direction of change in the result for each source of error. Explain the amount of increase in uncertainty from the additional sources of uncertainty found. Possibly modify the result if errors can be calculated. Suggest improvements. Reach a conclusion as to the success of this experiment. This step should be completed regardless of the outcome of Step 3. That is, even if the % error (or % diff) is less than the allowed uncertainty, it is necessary to complete this step in a thorough manner.

4 4 A Detailed Example of Four Steps to a Meaningful Experimental Result Consider a simple but somewhat crude experiment to measure π. Objective: Measure π as the ratio of the circumference to diameter using a cylindrical coffee cup. Theory: The ratio of circumference to diameter should give the value of π as Procedure: Take a cylindrical coffee cup and measure the circumference by winding a string around the base of the cup and marking one winding as precisely as possible. The string is then stretched out and the length of one winding (i.e., the circumference) is measured using a ruler that has millimeter markings for its smallest divisions. The diameter of the base is measured by squeezing the teeth of a vernier caliper across the diameter of the cylinder s base. A vernier caliper has 0.1 mm markings for its smallest division. Repeat to complete 5 trials. Raw Data: Measured with a ruler ( 1 mm = 0.1 cm = smallest division): Trials on coffee cup: circumference (cm): 6.0, 5.83, 6.4, 5.95, 6.38 ote: the last significant digit recorded is estimated since the end of the string falls between two millimeter markings on the ruler. Measured with a vernier caliper ( 0.1 mm = 0.01 cm = smallest division): Trials on coffee cup: diameter (cm): 8.5, 8.135, 8.33, 8.099, ote: the last significant digit recorded is estimated since the teeth separation falls between two 0.1 mm markings on the vernier caliper. Data Analysis: Step 1: Find the mean value of each measured quantity and the total uncertainty (TU) associated with each mean. Mean of Circumference: C = 6.16 cm (rounding off to the same number of significant digits). IU for C = 1/ the smallest division on the ruler = 0.5 mm = 0.05 cm. s RU for C = = 0.1 cm where = 5 and the sample standard deviation s = 0.6 cm which can be found by an application (such as Graphical Analysis) or calculated as follows: s = i=1 (X i X m ) 1 = ( ) + ( ) +L+ ( ) 5 1 = 0.6 cm TU for mean C value: TU = (IU) + (RU) = (0.05) + (0.1) = 0.13 cm So the mean circumference is C ± TU C = 6.16 ± 0.13 cm Mean of Diameter: D = cm (keeping the same number of significant digits). IU for D = 1/ smallest division on the vernier caliper = 0.05 mm = cm. s RU for D = = 0.06 cm where = 5 and sample standard deviation s = cm (found by Graphical Analysis application). TU for mean D value: TU = (IU) + (RU) = (0.005) + (0.06) = 0.06 cm

5 So the mean diameter is D ± TU D = ± 0.06 cm 5 Step : Calculate the final experimental result as a function (f ) of one or more mean measurements. Find the total uncertainty of the final result (TU f ) by propagating the uncertainty of the mean measurements into the final result. The final result is a ratio: π expt = C/D using mean measured values. So, the result for π expt can be expressed as a function f = C 1 D -1 = (6.16 cm) 1 (8.168 cm) -1 = 3.0. ow use the formula for propagation that involves just multiplication and division to propagate the uncertainty of the circumference (6.16 ± 0.13 cm) and the diameter (8.168 ± 0.06 cm) into the result (f = π expt ) as follows: TU f = f $ m(tu x ) x % ' & + n(tu y) % $ ' y & = C D $ = (TU C ) C $ 1(0.13) 6.16 % ' + (1(TU D) % $ ' & D & % ' + (1(0.06) % $ ' = & & giving the final experimental result as f ± TU f or equivalently π expt ± TU π. Final Experimental Result: π expt ± TU π = 3.0 ± 0.0 where rounding off keeps the appropriate number of significant digits. Step 3: Compare the final result to a true value OR to the same result obtained by other methods. Compare the amount of error (or difference) to what is allowed by total uncertainty. The percentage amount of uncertainty in the final result is: TU %TU f = 100% f % $ f & $ TU ' ( = 100%,exp t & ' % ) = 100% ±0.0 & % ( = ± 0.63% exp t ( $ 3.0 ' This uncertainty represents the amount of fuzziness in the experimental result for π. ow, the percentage error in the result is: $ % error = 100% & % f f true f true ' % ) = 100% exp t $ ( $ ' * = 100% & ( & ) % ' ) = 1.85% ( This error represents the possibility that mistakes were made. ext we compare % TU to % error. The values above indicate %error > %TU. This implies a significant amount of mistakes were made and/or the uncertainty was underestimated. ote: This does not necessarily imply that the experiment (or experimenter) failed. The experiment (or rather the experimenter) can still be successful if the experimenter can explain: plausible sources of error (and perhaps even correct the results after verifying these errors exist); and/or plausible additional sources of uncertainty that will increase the % TU of the result. This is why the next section, Assessment and Conclusion, is critical in determining the level of success of this experiment.

6 6 Assessment and Conclusion: Step 4: Complete an assessment and reach a conclusion. Indicate all plausible sources of error and plausible additional sources of uncertainty in the result. If possible, explain the size and direction of change in the result for each source of error. Explain the amount of increase in uncertainty from the additional sources of uncertainty found. Possibly modify the result if errors can be calculated. Suggest improvements. Reach a conclusion as to the success of this experiment. Plausible sources of error in this experiment (and corrections to error if possible): Markings on the meter stick or vernier caliper may have either an offset from zero or improper separation. We checked the zero setting of our vernier caliper and found no detectable offset. We compared our vernier caliper to others, and our meter stick to others and could find no detectable differences in marking separation. Although we can not for certain rule out this source of error, it appears to be negligible. The coffee cup may not have a circular base. We attempted to check this source of error by measuring the diameter at various locations around the cup base. The maximum teeth separation did not seem to shift when rotating the vernier caliper around the base. We concluded that this error was likely negligible; but it is possible that error at the level of 1% or so is not easily detected. There may have been some slack in the string when in it winds around the base. Also the plane of the winding may rise slightly above the plane of the circular base. Both of these errors give systematically high values for the circumference, which in turn would give systematically high values for our experimental value for π. Looking over the trials, we notice one could take essentially any one of our circumference trials and divide it by any one of our diameter trials and the result will be a value greater than true π. (The one exception is to take the lowest circumference trial divided by the highest diameter trial.) This source of error would explain the systematically high values for π. We have some suggestions (below) that may help minimize this error. The string may have not been taut enough when measuring the circumference by stretching out the length of the string. This would create a systematically low value for the circumference and π. Our data suggests this error was successfully minimized. In the procedure we were careful to make the string taut when stretched out lengthwise, but the errors on winding discussed above seemed more unavoidable. The vernier caliper teeth separation may be less than the circle s diameter if the teeth of the vernier caliper fall on a secant of the circle. If this happened on one or more trials then the diameter mean would be lower than true. And this will result in an experimental value for π that is higher than true. This explains another systematic error in that measurements for diameter could only be at the diameter or less than the diameter. One suggestion to minimize this error is to only measure that diameter by rotating the vernier caliper around the circular base and then using the maximum of the teeth separation as the recorded diameter trial. This assumes a perfect circular base. We went back to our coffee cup and tried this. This approach gave a diameter mean that was slightly higher. Using our new mean D value of 8.3 we get a value for π as 3.18 (which has 1.6% error). (continues on next page)

7 7 Plausible sources of additional uncertainty in this experiment (and amount of increase in the total uncertainty of the result): o o The uncertainty in lengths measured should be higher than 1/ the smallest division since both ends of the length are uncertain. Both ends of the stretched string must be matched with the mm markings on a meter stick, so both locations are uncertain. A similar argument can be made for the vernier caliper. This would give a slightly higher IU value. Since the uncertainty would result from the addition of two ends we would have: IU meter stick = (0.05cm) + (0.05cm) = 0.07 cm. Likewise, IU vernier = cm. But this would increase the TU values for C and D only slightly since most of the uncertainty is random. Indeed our calculations show that this has no affect on the TU we obtained for our π value since we rounded-up anyway. The circle used for measuring the circumference from one winding of the string may not be the same circle used when measuring the diameter (with the teeth of the vernier caliper). This difference will occur since the circle formed by the winding of the string may not intersect with the locations where the teeth of the vernier caliper squeeze the cup. Random uncertainty should take into account fluctuations from these differences, but we point out that this uncertainty might be eliminated with a better procedure. Suggestions for improvements: Taking more trials would reduce the random uncertainty. Team members could individually record measurements for circumference and diameter while keeping these trials blind from the teammates until all trials are taken. Once completed the trials can be shared and recorded by all. This would reduce potential biases in measurements. Use a variety of cylindrical objects and choose object known to be more uniformly circular, such as jar lids, washers and coins. With a variety of circular objects one would reduce random uncertainty. Use computer generated circles on paper and then cut out the circles. Thread could trace out the circles on paper. And a vernier caliper can be used to measure the diameters drawn by computer as well. Error would be reduced using these more perfect circles. With a variety of circles being used one could plot C versus D, and the slope would represent the best experimental value for π. Conclusion: Using the circular base of a coffee cup, we obtained a value for π as the ratio of circumference to diameter. Even though we used a rather crude procedure, our experimental value for π deviated from the true value by only 1.85%. Still this percent error significantly exceeded our allowed percent total uncertainty (0.63%). Much of the error in π is systematically high due to two affects: the exaggerated circumference obtained by slack in the winding and the underestimated diameter found from the occasional measurement on a circle s secant near the diameter. We were able to improve on the diameter by using a procedure that would give the mean diameter as the maximum vernier caliper teeth separation around the base. This reduced our error to only 1.6% and gave us an experimental value for π as 3.18.

8 8 Examples on Propagation of Uncertainty (Step ) 1. A measurement of elapsed time is given as t = 3. ± 0.3 s. The experimental result is given by the function f = 5t 4. Find the result and its total uncertainty (TU). Find %TU. Solution: The function (result) is f = 5t 4 = 5(3.) 4 = 54.3s 4. To get TU f we notice the function involves only multiplication, so we will use the propagation formula that involves only multiplication: TU f = f $ m(tu x ) x % ' + n(tu y ) % $ ' & y & where f = ax m y n. Here let a = 5, x = t, m = 4, and n = 0, 4(TU giving TU f = f t ) % $ ' + 0(TU y ) % $ ' = 54.3s 4 4(0.3s) % $ ' = 196.6s 4. t & y & 3.s & And %TU = 100%(TU f /f) = 100%(196.6/54.3) 37.5% So f ± TU = 54.3 ± s 4, and %TU 37.5%. A distance L = 5x - 3y can be found from measured values of x and y. Given: x = 3.31 ± 0.04 cm and y = 1.7 ± 0.5 cm. Find L ± TU. Find %TU for L. Solution: The function (result) is f = L = 5x 3y = 5(3.31 cm) 3(1.7 cm) = 11.4 cm. To get TU f we notice the function involves only subtraction, so we will use the propagation formula that involves only subtraction: TU f = a (TU x ) + b (TU y ) where f = ax by. Here let a = 5 and b = 3, giving TU f = 5 (0.04cm) + 3 (0.5cm) 1.5 cm. And %TU f = 100%(TU f /f) = 100%(1.5/11.4) 13.1%. So f ± TU = L ± TU = 11.4 ± 1.5 cm, and %TU 13.1% 3. Given the ratio R = kx /y. Let x = 3.31 ± 0.04 cm, y = 1.7 ± 0.5 cm, and the constant k = 5/cm. Find R ± TU. Find %TU for R. Solution: The function (result) is f = R = kx /y = kx y -1 = (5/cm)(3.31 cm) (1.7 cm) -1 = 3.. To get TU f we notice the function involves only multiplication (and division), so we will use the propagation formula that involves only multiplication: TU f = f $ m(tu x ) x % ' + n(tu y ) % $ ' & y & where f = ax m y n. Let a = k = 5/cm, m =, and n = -1, (TU giving TU f = f x ) % $ ' + (1(TU y ) % $ ' = 3. $ x & y & And %TU = 100%(TU f /f) = 100%(9.5/3.) 9.5% So f ± TU = R ± TU =3. ± 9.5, and %TU 9.5% (0.04cm) 3.31cm % ' + (1(0.5cm) % $ ' = 9.5. & 1.7cm &

9 9 Assignment on Experimental Uncertainty Topics: Calculating Uncertainties (Step 1), Propagating Uncertainties (Step ), and Comparing Results from Two Different Experimental Methods (Step 3). Questions: 1. True or False: A result calculated from measured values is essentially useless without assigning an appropriate uncertainty to the result. Justify your answer.. The formula for calculating total uncertainty (TU) involves what two types of uncertainty? Why is it appropriate for this formula to ignore systematic error (whenever it exists)? 3. Why is random uncertainty also called statistical uncertainty? 4. In data analysis, sometimes % diff is used rather than % error. Why? Problems: 1. The smallest division on a certain measuring tool is 0.1 unit. Joe and Jane (team JJ) make four measurements of X with this tool. They record their measurements as: X i = 6.3, 6.35, 6.18, 6.7 units where i = 1,,3 or 4 (the trial number). Answer the following: a. What is the IU for this tool? Are measurements with two digits passed the decimal point appropriate even though the tool s smallest division is 0.1 unit? Explain. b. Find the mean value X m and find RU. c. Find the TU of X m. Represent X m ± TU as X JJ ± TU JJ to give the best measured value of X from team JJ. d. Suppose a second team, called team CC (Curt and Carol), uses a much less precise measuring tool and obtains X CC ± TU CC as 6.4 ± 0.1 units. Compare the two teams results by finding the % diff and the combined uncertainty %TU combo. Are the results from the two teams consistent? Justify your answer. (Hint: Read Step 3 in the section, Four Steps to a Meaningful Experimental Result).. Given L = 5x - 3y + z where x = 3.31 ± 0.06 cm, y = 1.7 ± 0.1 cm, and z =.8 ± 0.09 cm. Find L, TU L and %TU for L. 3. Given the ratio R = kx /y 3, where x = 4.1 ± 0.06 cm, y = 3.06 ± 0.05 cm, and the constant k = 6. cm. Find R, TU R and %TU for R. 4. Two different methods for measuring X (with appropriate uncertainties) give the following results: X 1 = 10 ± and X = 7.0 ± 0.5. Find the combined total uncertainty from %TU combo and compare %TU combo to % diff to determine whether or not these two methods are consistent within experimental uncertainty.

10 10 Appendix A: Calculating the Standard Deviation of a Sample The standard deviation of sample (s) can be found as a function on most calculators or can be found from software applications (such as Graphical Analysis or Excel). One way to define the standard deviation of a sample is: s = i=1 (X i X m ) 1 where = the number of trials, X i = value of the i th trial, X m = mean value. The random uncertainty on the mean (RU = s ) shrinks as more trials are taken (since increase). So, as the number of trials increases, the more certain one can be about the mean value. By the way, for large the standard deviation of the sample (s) does not shrink when increasing the number of trials. Why?

11 11 Appendix B: Linear Fits to the Data Experiments are often designed to test or challenge a theory. If the theory predicts a linear relationship between two variables being measured, then the experimenter should plot these variables to determine the likelihood that a linear relationship actually exists. Of course, nonlinear relationships can also exists and fitting can be done for any mathematical relationship, but here we will just examine the ideas behind a simple linear fit. Consider a theory that predicts a linear relationship between two measured variables: x and y. Repeated trials have been completed on these measurements. It appears that variable y depends on x and plotting will be done to determine if the relationship is likely a linear one. Since y is the dependent variable it will be plotted as a function of x (the independent variable). A linear relationship would give: y = mx + b where m is the slope and b is the y-intercept. Even with an exact linear relationship between x and y, the measured values may not fall exactly on a line due to unavoidable experimental uncertainty. The point of a linear fit is to find the line that best matches the data. The technique involved is called a least squares fit. Details of this fitting approach can be found in many references (for example, An Introduction to Error Analysis by James R. Taylor; Data Reduction and Error Analysis for the Physical Sciences by Philip R. Bevington and D. Keith Robinson). The main objective of the fit is to minimize the chi squared that is defined by: % ($y) i=1 where is the total number of trials and δy = y i y line is obtained for each trial as the difference between the measured y value (y i ) and the theoretical line being tested (y line = mx i +b). otice that δy can be negative, so squaring assures that each contribution to the sum is positive. The fitting is complete when this sum of differences squared is minimized. The theoretical line that minimizes is then plotted giving the slope and intercept with their respective uncertainties. These uncertainties from the fitted line can be used instead of calculating TU values (as done in Steps and 3 in the section: Four Steps to a Meaningful Experimental Result.) If the slope has a theoretical (or true ) value then the experimenter should calculate and compare the % error and % TU of the slope. Similarly, one can compare the fitted y-intercept to the theoretical (or true ) value. There are a number of excellent graphing applications that will essentially automate the fitting process for you. Your instructor will illustrate at least one. Some of these applications will give a linear correlation coefficient or coefficient of regression (C.O.R) rather than a. When is zero the fit is perfect (no further minimization is possible). When C.O.R is exactly one, then the fit is perfect and no further minimization is possible.

12 1 Appendix C: Format for a Formal Lab Report Your instructor will likely require a particular format for you to follow when writing lab reports. Below is a typical format: 1. Objective: State in one or two sentences the goal of the experiment.. Theory: Explain the physics theory being challenged or tested in this experiment. If the results of the experiment require a derivation from these theories (or principles), then you should complete such a derivation here. Do not discuss the procedure here. 3. Procedure: You do not need to write a lengthy procedure, but you can attach or refer to any write-ups given on the lab. Drawings of the lab set-up should be included. Any changes or additions your team implemented to the write-up procedure should be stated. Between the write-ups and your procedure section, anyone should be able to replicate what you did. 4. Raw Data: Record all trials without any modification to the direct measurements. Indicate the instruments used to make the measured trials (and the instrumental uncertainty). 5. Data Analysis: Complete Steps 1, and 3 of Data Analysis as given in this manual. ote, on occasion Steps and 3 may be replaced by plotting linear fits (see Appendix B). Show all work for determining IU, RU and TU. Explicitly show how you propagate the uncertainty into the final result. 6. Assessment and Conclusion: Complete Step 4 as given in this manual. Thoroughness of this section is often important in receiving a good grade. (See example in this manual.) 7. Answers to Post-Lab Questions: These questions are often designed to see if you really understood the physics behind the experiment.

13 13 Appendix D: Terminology, otation and References This manual is intended to be a brief yet practical introduction to Data Analysis. A good student of error analysis may notice some differences between this manual and other popular references (see below) on the subject. Some of these differences occur in the use of terminology. This manual gives the steps involved in propagating uncertainties, but most references on this subject refer to this topic as propagation of error even though the word error here does not imply mistakes. To avoid confusion between uncertainties (either random or instrumental in nature) and error (deviation from true that occur because of mistakes), the approach in this manual makes clear the distinction between uncertainty and error. Recall, that precision and accuracy have two different meanings. Propagation calculations are done to determine the level of precision (or certainty) in the final result (not the accuracy). Similarly some references will refer to random (or statistical) uncertainty as random (or statistical) error. The other differences between this manual and popular references on the subject are notational. Popular references make use of the symbol σ or δ when referring to uncertainty. This manual uses the symbols IU, RU and TU when referring to uncertainties since such notation more clearly reveals the source of the uncertainty. Popular references on Data Analysis: Bevington, P.R. and D.K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (3 rd Ed.), McGraw-Hill, ew York (003) Taylor, J.R., An Introduction to Error Analysis: The Study of Uncertainty in Physical Measurements ( nd Ed.), University Science Books, Sausalito, Calif. (1997) Lyons, L., A Practical Guide to Data Analysis for Physical Science Students, Cambridge University Press, Great Britain (1991).