1. Measurement Uncertainties Physics 161 University Physics Lab I Fall 2007

2. Measurements What do we do in this lab? Perfect Measurement Measurement Techniques Measuring Devices Measurement Range

3. Uncertainties Types of Errors Random Uncertainties: result from the randomness of measuring instruments. They can be dealt with by making repeated measurements and averaging. One can calculate the standard deviation of the data to estimate the uncertainty. Systematic Uncertainties: result from a flaw or limitation in the instrument or measurement technique. Systematic uncertainties will always have the same sign. For example, if a meter stick is too short, it will always produce results that are too long.

4. Uncertainty Difference between uncertainty and error. Blunder is not an experimental error! Calibration

5. Mean and STD Multiple Measurements Average Standard Deviation

6. Reporting a Value To report a single value for your measurement use Mean and Standard Deviation Measured value= mean of multiple measurements ± Standard Deviation

7. Expressing Results in terms of the number of σ In this course we will use σ to represent the uncertainty in a measurement no matter how that uncertainty is determined You are expected to express agreement or disagreement between experiment and the accepted value in terms of a multiple of σ. For example if a laboratory measurement the acceleration due to gravity resulted in g = 9.2 ± 0.2 m / s 2 you would say that the results differed by 3σ from the accepted value and this is a major disagreement To calculate N σ

8. Example

9. Gaussian Distribution

10. Standard Deviation

11. Accuracy vs. Precision Accurate: means correct. An accurate measurement correctly reflects the size of the thing being measured. Precise: repeatable, reliable, getting the same measurement each time. A measurement can be precise but not accurate.

12. Accuracy vs. Precision Precision depends on Equipment A more precise measurement has a smaller σ. 9.4±0.7 m/s/s 9.5±0.1 m/s/s Accuracy depends on how close the measurement is to the predicted value. 9.4±0.7 m/s/s 9.5±0.1 m/s/s Which one is a better measurement? 9.4±0.7 m/s/s 9.5±0.1 m/s/s

13. Comparison of Measurements

14. Absolute and Percent Uncertainties (Errors) If x = 99 m ± 5 m then the 5 m is referred to as an absolute uncertainty and the symbol σ x (sigma) is used to refer to it. You may also need to calculate a percent uncertainty ( %σ x ): Please do not write a percent uncertainty as a decimal ( 0.05) because the reader will not be able to distinguish it from an absolute uncertainty.

15. Error Propagation

16. Propagation of Uncertainties with addition or Subtraction If z = x + y or z = x – y then the absolute uncertainty in z is given by Example:

17. Propagation of Uncertainties with Multiplication or Division If z = x y or z = x / y then the percent uncertainty in z is given by

18. Propagation of Uncertainties in mixed calculations

19. Error Propagation

22. Example Let x=4±.4 and y=9±.9 then

23. Special Functions z=sin(x) for x=0.90±0.03 Sin(0.93)=0.80 Sin(0.90)=0.78 Sin(.87)=0.76 Z=0.78±0.02

24. Least Square Fitting In many instances we would like to fit a number of data points in to a line. Ideally speaking the data points should be on a line; however, due to random and systematic errors they are shifted either up or down from their ideal points. Least squares is a statistical model that determines the slope and intercept for a line by minimizing the sum of the residuals.

25. Least Squares

27. Slope Intercept

28. Least Squares

29. Percent Difference Calculating the percent difference is a useful way to compare experimental results with the accepted value, but it is not a substitute for a real uncertainty estimate. Example: Calculate the percent difference if a measurement of g resulted in 9.4 m / s 2.

30. Significant Figures Addition Rule: –Use the least number of sig. figs after the decimal point. –12.3456 + 8.99= 21.3356  21.34 Multiplication Rule: –Use the same number of sig. figs as the number with least sig. figs. –12.447*2.31 =28.75257  28.8