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Data analysis

Data analysis. An Introduction to Error Analysis. The study of uncertainties in physical measurements. Data analysis. CONTENTS. Preliminary Description of Error Analysis How to Report and Use Uncertainties Propagation of Uncertainties Statistical Analysis of Random Uncertainties

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Data analysis

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  1. Data analysis An Introduction to Error Analysis The study of uncertainties in physical measurements

  2. Data analysis CONTENTS • Preliminary Description of Error Analysis • How to Report and Use Uncertainties • Propagation of Uncertainties • Statistical Analysis of Random Uncertainties • The Normal Distribution

  3. Data analysis 4. Statistical Analysis of Random Uncertainties 4.1 Random and Systematic Errors

  4. Data analysis 4. Statistical Analysis of Random Uncertainties 4.1 Random and Systematic Errors

  5. The best estimate would be the average or mean. Data analysis 4. Statistical Analysis of Random Uncertainties 4.2 The Mean and Standard Deviation For a set of measured values : xi what is the best estimated x : xbest ? To be confirmed later.

  6. Data analysis 4. Statistical Analysis of Random Uncertainties 4.2 The Mean and Standard Deviation Standard deviation An estimate of the average uncertainty of the measurements xi.

  7. Data analysis 4. Statistical Analysis of Random Uncertainties 4.2 The Mean and Standard Deviation Standard deviation An estimate of the average uncertainty of the measurements xi.

  8. Another definition Data analysis 4. Statistical Analysis of Random Uncertainties 4.2 The Mean and Standard Deviation Standard deviation An estimate of the average uncertainty of the measurements xi. Sample standard deviation There is a theoretical reason for the difference, but it is insignificant when N is sufficiently large. Population standard deviation

  9. “uncertainty” Data analysis 4. Statistical Analysis of Random Uncertainties 4.3 The Standard Deviation as the Uncertainty in a Single Measurement sx : 68% of the results fall within a distance sx on either side of xtrue. dx = sx “with 68% confidence”

  10. Measure a spring constant k for the spring A : 10 times 86, 85, 84, 89, 85, 89, 87, 85, 82, 85 kavr = 85.7 N/m sk = 2.16  2 N/m Measure a spring constant k for the spring B : 1 time 71 B k = 71 ± 2 N/m with 68% confidence Data analysis 4. Statistical Analysis of Random Uncertainties 4.3 The Standard Deviation as the Uncertainty in a Single Measurement A

  11. SDOM Best value of x=xavr± dx = xavr± sx/N Data analysis 4. Statistical Analysis of Random Uncertainties 4.4 The Standard Deviation of the Mean -sx xavr +sx How certain is this mean value ?

  12. Data analysis 4. Statistical Analysis of Random Uncertainties 4.4 The Standard Deviation of the Mean SDOM Best value of x=xavr± dx = xavr± sx/N Proven in Chap.5 Std : average uncertainty in the individual measurement SDOM : uncertainty of the average

  13. Measure a spring constant k for the spring A : 10 times 86, 85, 84, 89, 85, 89, 87, 85, 82, 85 kavr = 85.7 N/m sk = 2.16  2 N/m 4. Statistical Analysis of Random Uncertainties Data analysis 4.4 The Standard Deviation of the Mean SDOM : skavr = sk/10 = 0.7 N/m k = 85.7 ± 0.7 newtons/meter

  14. Data analysis 4. Statistical Analysis of Random Uncertainties 4.4 The Standard Deviation of the Mean SDOM

  15. Data analysis 4. Statistical Analysis of Random Uncertainties 4.6 Systematic Errors dx : dxrandom, dxsystematic neglected so far. No simple theory ! • Do best for calibration of instruments. • Do best for obtaining a specifications of equipments on precision.

  16. Data analysis 4. Statistical Analysis of Random Uncertainties 4.6 Systematic Errors • Causes and sources : • Instruments/equipments • Incorrect value for some parameter to be used in calculation • Flaw in designing the experiment

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