Error Analysis (Analysis of Uncertainty)

1. Error Analysis(Analysis of Uncertainty) • Almost no scientific quantities are known exactly • there is almost always some degree of uncertainty in the value • Value ± Uncertainty • Values that are measured experimentally • Values that are calculated • from an equation • using other values which have their own uncertainty • A value might be determined both ways: calculate it and measure it

2. Uncertainty in Measured Values • Two components of Uncertainty • Measured value ± systematic errors ± random errors • Precision of a measurement (reproducibility) • variations due to random fluctuations • measure many times • power supply, angle of view of a meter, etc. • Accuracy of a measurement (total uncertainty) • includes uncertainty in precision • also includes systematic errors • incorrect experimental procedure, uncalibrated instrument, use a ruler with only 9 mm per cm, leakage of gas

3. Treatment of Random Errors • Assume that systematic errors have been eliminated • Simple Estimate • Analog Gauge or Scale • How finely divided is the readout, and how much more finely do you estimate that you can interpolate between those divisions? • Digital Readout • What is the smallest stable digit?

4. Statistical Treatment of Random Errors • Suppose you repeated the exact same measurement at the exact same conditions an infinite number of times • Not every measurement will be the same due to random errors • Instead there will be a distribution of measured values • Could use the results to construct a frequency distribution or probability function

5. Frequency Distribution orProbability Function • With a finite number of measurements you get a frequency distribution • Probability of a measurement falling within a given box is number in that box divided by total number • With an infinite number you get a probability function • Plot of P(x) versus x • P(x) is the probability of a measurement being between x and x + dx

6. Characteristics of the Probability Function • Certain kinds of experiments may naturally lead to a certain kind of probability function • For example, counting radioactive decay processes leads to a Poisson Distribution • Often, however, it is assumed that the errors are by a Normal Distribution Function •  is the mean (average) of the infinite number of measurements •  is the standard deviation of the infinite number of measurements

7. Use of the Probability Function • P(x-µ) is normalized: • That is, the total area under the P curve equals 1.0 • If you knew  and  (and so you knew P) you could find the limits between which 95% of all measurements lie. Insert plot with shaded area at left • Noting that P is symmetric about µ you could say with 95% confidence that the measured value lies between  -  and  +  • That is, the value is  ±  at the 95% confidence level

8. An Infinite Number of MeasurementsIsn’t Practical • You can only make a finite number of measurements • Therefore you do not know  or  • You can calculate the average and variance for your set of measurements

9. Average and Variance of the Data SetDo Not Equal  and  • Use Student’s t-Table to relate the two: • Pick a confidence level, 95% • Define degrees of freedom as N-1 • Read value of t • Be careful, t-Tables can be presented in two ways • One is such that 95% will be less than t • In this case if you want 95% between -t and t you need 97.5% less that t (the curves are symmetric) • Another is such that 95% will be between -t and t • Uncertainty limits are then found from the variance • value = average of the data set ± 

10. One Form of Student’s t-Table Add shaded bell curve here • The value of t from this form of the table corresponds to 95% of all measurements being less than +  and therefore 5% being greater than  +  • Note that if you want 95% of all measured values to fall between  -  and  +  • then 97.5% of all measured values must be less than  +  (or 2.5% will be greater than  + ) • and then due to the symmetry of P 97.5% will also be greater than  -  (or another 2.5% will be less than  - ) • so 95% will be between  -  and  +  Add abbreviated t-table here

11. Another Form of Student’s t-Table Add shaded bell curve here • The value of t from this form of the table corresponds to 95% of all measurements being between  -  and  +  • Therefore 5% are either • greater than  +  • or less than  -  Add abbreviated t-table here

12. Example • Add Problem Statement here • preferably use data from one of the experiments they are doing

13. Solution • Add solution here

14. Summary: Uncertainty in Measured Quantities • Measured values are not exact • Uncertainty must be estimated • simple method is based upon the size of the gauge’s gradations and your estimate of how much more you can reliably interpolate • statistical method uses several repeated measurements • calculate the average and the variance • choose a confidence level (95% recommended) • use t-table to find uncertainty limits • Next lecture • Uncertainty in calculated values • when you use a measured value in a calculation, how does the uncertainty propagate through the calculation • Uncertainty in values from graphs and tables