Data Analysis for University Physics
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Data Analysis for University Physics by John Filaseta Northern Kentucky University Last updated on March 31, 2004 2 Data Analysis: Basic Concepts and Definitions The accuracy of an experiment reveals how close the experimental result comes to the "true" (or accepted) value. Reducing error will give more accurate results. The precision of a measurement or experimental result is determined by the amount of its uncertainty. Reducing uncertainty will give more precise (or certain) experimental results. Notice, an experimental result can be accurate but imprecise; and conversely, an experiment can be precise but inaccurate. Human Error implies the experimenter made mistakes. This error should always be avoided. Systematic Error is due to defects in equipment or procedure and results in measured values that are consistently away from the "true" value in one direction only (i.e., the measured values are always too high or always too low, but not both). For example, a faulty weighing scale that always reads one-half ounce too high is a source of systematic error. Systematic error can often be corrected whenever discovered (since the correction is the same for all trails), but nonsystematic error that occurs when error is in one direction for some trials but in another direction for other trials can be difficult to correct. The values of repeated measurement trials will often vary. This variation creates uncertainty in the measurements. There are two unavoidable sources of uncertainty:(a) Random Uncertainty (also called random error, but really is an uncertainty) results from fluctuating conditions (such as temperature, pressure, etc.), and small disturbances (mechanical vibrations, spurious electrical pick- ups, etc.). No experiment can be perfectly controlled to give exactly zero random uncertainty. But increasing the number of trials will reduce random uncertainty (or statistical uncertainty). (b) Instrumental Uncertainty results from the limitations of measuring devices and can be estimated as the fraction of the smallest division on an instrument. For example, a micrometer, which has smaller divisions than a meter stick, is more precise at measuring lengths than a meter stick. No experiment can be designed to have zero instrumental uncertainty. A good experiment will be both accurate (minimal error) and precise (minimal uncertainty). It is possible to have an experimental result that matches the expected value, but if that result has a very large uncertainty then the result becomes essentially meaningless. It is also possible to have a nearly certain result (i.e., very low uncertainty) with a large error (far from the accepted value). The experimenter should perform an experiment making every effort to reduce both uncertainty and error. When assessing the experimental result, a careful examination of all plausible sources or error and additional sources of uncertainty should be a critical part of any report of the results. For every plausible source of error, the experimenter should explain how this error would influence the final experimental result. For example, consider an experiment that has the goal of finding π by from ratios of measured circumferences over diameters. The measured diameters appear to be systematically low. This plausible error would cause the results (π values) to be systematically high. A good experimenter verifies the diameters are systematically low (such as due to a faulty measuring device); and re- measures all diameters; and re-calculates all π values. Questions: 1. Does large uncertainty imply there must be large error in the experimental result? Explain. 2. Does good precision imply the measured result must be accurate? Explain. 3. Can large error be justified in an experiment where large uncertainty is unavoidable? Explain. 4. Name three ways to reduce experimental uncertainty? Will these also reduce error in the result? 3 Assignment on Data Analysis: Basic Concepts and Definitions Questions: 1. What is the difference between precision and accuracy? What is the difference between uncertainty and error? 2. How does larger instrumental uncertainty affect precision? affect error? 3. You perform an experiment with many trials to verify a true value. How might you determine whether or not you encountered a systematic error? 4. What are two ways to reduce random uncertainty (RU)? 5. Name another type of uncertainty other than random uncertainty? Can this uncertainty be made zero? Explain. 6. Which type of error is usually preferred in an experiment: systematic error or nonsystematic error? Explain. 7. Other than the example of the weighing scale (given earlier), give an example of a systematic error that could occur in some experiments? Be specific as to the size and direction of the systematic error in your example. 8. Suppose the true value for X is known to be precisely 10. Three teams perform experiments to determine X. Team A reports X = 10 ± 5, and Team B reports X = 9.8 ± 0.3, and Team C reports 8.00 ± 0.05. Which team has the best results? Justify your answer. 9. Sketch a circle. Let the center of the circle represent the bull’s eye of a dartboard. The “true” value of an experiment that involves throwing darts is the bull’s eye. Answer below by marking your circle with a small X for each spot where a dart lands. a. Illustrate on your first circle (dartboard), an example of how the darts will land if the experiment has good accuracy and good precision. b. Illustrate on a second circle (dartboard), an example of how the darts will land if the experiment has good accuracy but poor precision. c. Illustrate on a third circle (dartboard), an example of how the darts will land if the experiment is poor accuracy but good precision. 4 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 the steps are completed as Data Analysis. The final (and perhaps the most important step) is completed as part of 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. Note: On occasion, data analysis Steps 2 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 (Xm). 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 N and N = 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: 2 2 TU = (IU) + (RU) . Now record the mean measurement as Xm ± TU. Step #2: Calculate the final experimental result as a function (f ) of one or more mean measurements. Find the total uncertainty of the final result (TUf) by propagating the uncertainty of€ the mean measurements into the final result. Let x and y be mean measurements and TUx and TUy 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 2 2 2 2 then TUf = a (TUx ) + b (TUy ) • If f is found by multiplication/division only: f = axmyn 2 2 m(TUx ) n(TUy ) then TUf = f + € x y If the final result is a function (f ) of three mean measurements, then just add a 3rd term under the 2 2 radicals above; for example, if f = ax ± by ± cz, then add c (TUz) under the radical. Likewise, if 2 m n p p(TU ) f = ax y z 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 m n m n to solve. For example, suppose f = (ax ± by)(ax y ). Let f1 = ax ± by and f2 = ax y then find the TU values for f1 and f2. Now use the multiplication approach above to get TU for f where f = f1f2 € st with f1 and f2 being to the 1 power. 5 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 f % TUf = 100%⋅ f Then pick the appropriate path below: Path A: Your final result (f ± TUf ) is compared to a true (or accepted) value (ftrue). € • 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 < %TUf, then error is less than the allowed uncertainty (or “fuzziness”) and error has been successfully minimized. €If % error>%TUf , but % error< 2%TUf then a small amount of error exists and/or the uncertainty was underestimated.