1. 1 2 nd Pre-Lab Quiz 3 rd Pre-Lab Quiz 4 th Pre-Lab Quiz

2. 2 Today ’ s Lecture: Issues from last week lab sections Discussion of data Interpretation & Error Analysis Statistical analysis –The Mean, Standard Deviation and Standard Deviation of the Mean Histograms and Distributions, The Limiting Distribution Introduction to the Normal (Gauss) distribution

3. 3 Quick recap from last lecture…

4. 4 Propagating Errors for Experiment 1 Formula for density. Take partial derivatives and add errors in quadrature shorthand notation for quadratic sum: Or, in terms of relative uncertainties:

5. 5 Propagating Errors for R e basic formula Propagate errors (use shorthand for addition in quadrature) Note that the error blows up at h 1 =h 2 and at h 2 =0.

6. 6 h l  ==> Find h. always use radians when calculating the errors on trig functions Given that: Example

7. 7 1.Statistical precision 2.Limitations of measurement tools 3.Limits due to experiment design - choice of tools - definition of measured quantity Limiting Factors in Experiments

8. Repeated Measurements Improve accuracy of measurement errors have to be random Examples: - thickness of a page in a book - period of a pendulum Estimate uncertainties Not always clear what is the single measurement uncertainty –Examples?? Variation between measurements can be used! errors have to be random

9. 9 the best estimate for x → the average or mean sigma notation N measurements of the quantity x common abbreviations The Mean deviation of x i from xThe mean minimizes: Will prove later

10. 10 uncertainty in any one measurement of x →  x =  x 68% of measurements will fall in the range x true ±  x x true +  x x true –  x x true x standard deviation of x RMS (route mean square) deviation average uncertainty of the measurements x 1, …, x N The Standard Deviation (RMS) approx Saw that the mean minimizes deviations of x i from the mean. How do we distinguish between different patterns?

11. 11 Standard Deviation - Single measurement uncertainty If you measure a quantity x, many times, approximately 68% of the results will fall within one  x of the true value (assuming all errors are random). In other words,if you make a single measurement, there is a 68% probability that the result will fall within  x of the correct value. This is a measure of uncertainty, and we will adopt as our definition of it. Note, that there is nothing magical about setting the uncertainty at 68% (or roughly 2/3). It is just a convention.

12. 12 Repeated Measurements - Mean & S.D.

13. 13 uncertainty in x is the standard deviation of the mean _ based on the N measured values x 1, …, x N we can state our final answer for the value of x: (value of x) = x best ±  x The Standard Deviation of the Mean (or: uncertainty of the mean) Used often to reduce random uncertainties. - Examples?

14. 14 In other words… If you make a single measurement, there is a 68% probability that the result will fall within one  x of the correct value. But what is the error associated with our best estimate from N measurements? It turns out (and we will prove this later), that the error is reduced by square root of N (the number of measurements). Thus, the uncertainty in our best estimate, which is called the standard deviation of the mean (SDOM), is given by:

15. 15 Mean, RMS and Standard Deviation of the Mean You made a set of measurements of the same quantity X. You summarized your measurements in form of the “ average ” and the “ spread ” of your measured values. –Mean –RMS or standard deviation You determined the uncertainty of the Mean value –Standard Deviation of the Mean (SDOM) NOTE the distinction between Standard Deviation and SDOM

16. 16 Example We make measurements of the period of a pendulum 3 times and find the results: T = 2.0, 2.1, and 2.2 s. (a) What is the mean period? (b) What is the RMS error (the standard deviation) in the period? (c) What is the error in the mean period (the standard deviation of the mean)? (d) What is the best estimate for the period and the uncertainty in the best estimate.

17. 17 Don ’ t get confused Repeated measurements –Repeat same measurement N times –Compute mean to get best estimate –Use SDOM to estimate uncertainty on the mean ==> gain 1/sqrt(N) in accuracy WRT single measurement error (SD) Measurement of many-at-a-time –For example: n pages, n periods of a pendulum… –Repeat N times and compute mean for best estimate –Use SDOM to estimate uncertainty on the mean ==> Gain 1/n in accuracy WRT measurement error (SDOM)

18. 18 In the lab Measure: T 10 1, T 10 2, T 10 3,…, T 10 N  Compute mean and S.D.: &  T10 Hence best answer is: &  T10 /Sqrt(N) Now we can get the best values for a single period: = / 10  T =  T10 /(10*Sqrt(N))

19. 19 Introducing Histograms & “ Limiting Distributions ” You made a set of measurements of the same quantity X. You summarized your measurements in form of the “ average ” and the “ spread ” of your measured values. –Mean & RMS You determined the uncertainty of the Mean value –Standard Deviation of the Mean As you take more and more measurements you want to visualize the distribution of measurements you make A convenient way to do so is to plot the measured values in a binned histogram As the number of measurements becomes very large you see a clear shape emerging for the distribution of measured values --> the “ limiting distribution ”

20. Constructing a Histogram

21. 21 x = 26, 24, 25, 24, 23, 26, 24, 26, 28, 25 Taylor, J.R. An Introduction to Error Analysis, 2 nd ed. 1997. x = 26.4, 23.9, 25.1, 24.6, 22.7, 23.8, 25.1, 23.9, 25.3, 25.4 Histograms and Distributions - 1 f k  k = fraction of measurements in k-th bin Where: F k= n k /N

22. 22 Histograms and Distributions - 2 weighted sum – each value of x k is weighted by the number of times it occurred, n k fraction of our N measurements that gave the result x k normalization condition x is sum of x k weighted by F k _ F k specify the distribution of x k Lets write down how we would compute the mean from a histogram:

23. 23 Repetitive measurement - limiting distribution

24. 24 AB = f(x) ??? A B Limiting distribution As the number of measurements increases (e.g. as the exam is given to more students), the histogram is expected to converge to some well-defined curve: Limiting Distribution - f(x). What fraction of data (e.g. grades) falls within some range?

25. 25 Note: We choose normalization of limiting distribution: = 1 To be able to interpret: As the probability of obtaining a measured value between A and B. A B

26. Gauss ’ Distribution also known as Normal Distribution

27. 27 The Gaussian Distribution A bell ‐ shaped distribution curve that approximates many physical phenomena - even when the underlying physics is not known. Assumes that many small, independent effects are additively contributing to each observation. Defined by two parameters: Location and scale, i.e., mean and standard deviation (or variance,  2 ). Importance due (in part) to central-limit theorem: The sum of a large number of independent and identically ‐ distributed random variables will be approximately normally distributed (i.e.,following a Gaussian distribution, or bell ‐ shaped curve) if the random variables have a finite variance.

28. 28 the limiting distribution for a measurement subject to many small random errors is bell shaped and centered on the true value of x the mathematical function that describes the bell-shape curve is called the normal distribution, or Gauss function prototype function  – width parameter X – true value of x The Gauss, or Normal Distribution

29. 29 The Gauss, or Normal Distribution normalize standard deviation  x = width parameter of the Gauss function  the mean value of x = true value X

30. 30 X=45 X=93 Gauss distribution: changing X Gauss distribution has its maximum at x = X. Therefore, changing X, changes position of the maximum, but does not change the shape of the distribution.

31. 31 X=45  = 7  = 15 Gauss distribution: changing   appears in the equation twice and sets both height and width of the distribution. Large  corresponds to a wider distribution with a lower peak. The area under the distribution is always preserved, because of the normalization

32. 32  = X -  X +  = 0.68 Gauss distribution: the meaning of  The area under a segment from X -  to X+  accounts for 68% of the total area under the bell-shaped curve. That is, 68% of the measured points fall within  from the best estimate

33. 33   t  t  What about the probabilities to find a point within 0.5  from X, 1.7  from X, or in general t  from X ? To find those probabilities we need to calculate Unfortunately, we cannot do it analytically and have to look it up in a table

34. 34 t = 1

35. 35 t = 1.47

36. 36 Compatibility of a measured result(s): t-score Best estimate of x: Compare with expected answer x exp and compute t-score: This is the number of standard deviations that x best di ff ers from x exp. Therefore,the probability of obtaining an answer that differs from x exp by t or more standard deviations is: Prob(outside t  ) = 1-Prob(within t  )

37. 37 “ Acceptability ” of a measured result Conventions 1.96 2.58 boundary for unreasonable improbability erf(t) – error function If the discrepancy is beyond the chosen boundary for unreasonable improbability, ==> the theory and the measurement are incompatible (at the stated level) Large probability means likely outcome and hence reasonable discrepancy. “ reasonable ” is a matter of convention… We define:

38. 38 Example: Confidence Level Two students measure the radius of a planet. Student A gets R=9000 km and estimates an error of  = 600 km Student B gets R=6000 km with an error of  =1000 km What is the probability that the two measurements would disagree by more than this (given the error estimates)? ==> Define the quantity q = R A -R B = 3000 km. The expected q is zero. Use propagation of errors to determine the error on q. Compute t the number of standard deviations from the expected q. Now we look at Table A ==> 2.56  corresponds to 98.95% So, The probability to get a worse result is 1.05% (=100-98.95) We call this the Confidence Level, and this is a bad one.

39. 39 Next Lecture… Introduction to experiment #2 –Non-destructive measurement of variation –Strategy –Discussion of data Interpretation & Error Analysis – Confidence Level – Compatibility of measured results –T-score Read Taylor Chap 6

40. 40 A student measures a quantity x many times and calculates the mean as x = 10 and the standard deviation as  x = 1. What fraction of his readings would you expect to find between 11 and 12? _ Another Example (if time permits): The probability of a measurement to be between X + t 1  and X + t 2  Table Appendix B x X X + t 1  X + t 2 