Physics 1140 Fall 2013 Introduction to Experimental Physics

Transcription

1 Physics 1140 Fall 2013 Introduction to Experimental Physics Joanna Atkin Lecture 4: Statistics of uncertainty

2 Today If you re missing a pre-lab grade and you handed it in (or any other problem), talk to me! Random uncertainties minimizing through repeated measurements Statistics Bell curve/standard distribution/normal distribution/gaussian distribution Average/mean Standard deviation Standard error on the mean

3 Experiment 1.6 ± 0.3 arcseconds 1.98 ± 0.12 arcseconds Theory 1.75 arcseconds Example: light bending Average = x 1 + x 2 2 = = 1.79 Royal Observatory Greenwich Discrepancy x th x 1 = 0.15 Discrepancy x th x 2 = 0.23 Discrepancy x th x avg = 0.05

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5 Gaussian Distribution The fluctuation of balls around the center (for a moderately large number of trials) is approximated by the bell curve, a.k.a. the Gaussian (or normal ) probability density: G(x) = 1 2πσ exp (x µ) 2 2σ 2 normalization The most widely used distribution in statistics, applicable to a wide variety of phenomena in the social and natural sciences: Small fluctuations in stock prices Distribution of speeds of molecules in a room Distribution of grades in a large course 2 center position width

6 Estimation of random errors The Gaussian function models typical random fluctuations of a measurement. µ µ: tells us what the expected measurement of x should be. σ σ: the typical size of fluctuations in our measurement x about the mean.

7 Estimation of confidence intervals About 2/3 of the time (68.3%), a measurement of x will fall within 1σ of µ. This is the 1σ confidence interval.

8 Estimation of confidence intervals About 19/20 of the time (95.5%), a measurement of x will fall within 2σ of µ.

9 Estimation of confidence intervals About 99/100 of the time (99.7%), a measurement of x will fall within 3σ of µ.

10 Calculations of statistical errors For a measured quantity x with Gaussian (random) uncertainties, we want to know the true µ and σ of the possible population of x values. But we have only a finite set of measurements. We have to use the data sample recorded in the lab to estimate µ and σ. Define quantities: average and standard deviation of a sample.

11 Average (mean): estimator for µ Suppose we have a set of measurements for the light bending experiment, of 1.69, 1.78, 1.70, 1.93, and 1.86 arcseconds deflection. The average of a set is defined by So for this example: x N i = = 1 N x i θ = = 1.79

12 Clicker question 1 1. Find the average from the following list of measurements: (Don t worry about sig figs here) Measurement of length (m) A. 4.8 B. 2.4 C. 24 D. 48

13 How to improve your experimental results In the limit where there is no systematic error, the average of an infinite number of measurements is your target value. Even if you only take a few shots (5 is a good number), the average is usually closer to the target than most of the typical individual shots.

14 Standard deviation: estimator for σ To have a more precise estimate of the error in an experiment, we could calculate the average deviation of the data from the average: x i x but this turns out to be zero. ( ) Instead use ( x i x ) 2 So the sample standard deviation, σ, is defined as: σ = (x i x) 2 Note: N and N-1 often used interchangeably in the denominator; N-1 preferred. N i=1 N 1 bias-corrected

15 Standard deviation In Gaussian statistics, the standard deviation σ x measures the range of x in which 68% of measurements should occur. Pr ob(x ± σ) = x +σ x σ 1 2 σ 2π e (x x )/ 2σ dx NOTE: σ x (or σ x ) refers to the uncertainty in each individual measurement x. Example: using the 5 measurements of light bending: σ θ = ( ) 2 + ( ) ( ) 2... = = 0.10

16 Clicker question 2 Of the following sets of 5 points each, which one has the largest standard deviation? A. 1000, 999, 1000, 1002, 998 B. 0.1, -0.1, 0.2, 0.3, 0.0 C. -25, -20, -25, -30, -20 D. 5, 4, 6, 3, 5 E. -100, -99, -101, -102, -100

17 Standard error on the mean The spread in each individual measurement, ±σ, is often less interesting than the uncertainty of the mean of x. We expect the precision of our measurement of the mean value of x to improve with the size of our sample. So, we define the typical spread in measurements of the mean as the standard error on the mean We can work this out with error propagation: f (x 1, x 2 x N ) = x = f x i = 1 N, and δ x i = σ N x i=1 i N

18 Standard error on the mean δ f = f x 1 2 (δ x 1 ) 2 + f x 2 2 (δ x 2 ) = 1 N σ 2 = N 1 i=1 N N So we can define the standard error on the mean σ x = For our example with light bending again: σ θ = 2 sample std. dev. sample size = σ x N sample std. dev. = σ 2 = σ N

19 Clicker question 3 4. A data set of 25 measurements has a Standard Deviation of the Mean (Standard Error on the Mean) of 2.5. How many measurements must be made to reduce the Standard Deviation of the Mean to, at most, 0.2? A measurements B measurements C measurements D measurements E. All of the above will give the desired Standard Deviation of the Mean 12.5/0.2 = 62.5 = N, N = =

20 The Normal/Gaussian Distribution What is the probability of making a measurement somewhere between +0.5σ? Pr ob(x ± σ) = x +σ x σ 1 2 σ 2π e (x x )/ 2σ dx Can do integral, or just use look-up table of probability

21 Taylor Appendix A

22 The Normal/Gaussian Distribution t = 0.5 The Probability is 38.29% for 0.5σ.

23 Clicker question 4 What is the probability that we would make a measurement outside of 0.50σ? A % B % C % D %

24 Uncertainties and experiment Using the Gaussian function, we can estimate whether or not an experimental result is reasonable. Result is within 1σ of expected value: experiment and expectation are in good agreement. large probability of getting answer more than 1σ away from expected value (32%). Result is within 1σ-2σ of expected value: experiment and expectation are consistent. Probability of measurement more than 2σ away: 4.6% Result is within 2σ-3σ of expected value: experiment and expectation may disagree. Grounds for further investigation. Result is 3σ or more away from expected value: difference is statistically significant.

25 Trying to prove a negative result looking for a large discrepancy above the background. 5σ confidence interval: appearance of hump at 125 GeV very unlikely to have occurred by chance (1 in 3.5 million chance.) Higgs Boson CERN 2012

26 This week Starting second lab, M2-M6 this week. Don t forget to complete and hand in the prelab! Homework 2 due Wednesday in G2B66. Homework 3 will be posted on Wednesday.