PHYSICS 2150 LABORATORY

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1 PHYSICS 2150 LABORATORY Instructors: Noel Clark James G. Smith Eric D. Zimmerman Lab Coordinator: Jerry Leigh Lecture 2 January 22, 2008

2 PHYS2150 Lecture 2 Announcements/comments The Gaussian distribution What it looks like Where and why it shows up Mean, sigma, and all that Statistical and systematic error

3 A COMMENT: YOUR FIRST LAB We hope you enjoyed the first experiment. Your first lab reports are due next week! Recall the advice in the syllabus and first lecture on your lab reports. More info next week when you are deeper into writing!

4 GAUSSIAN DISTRIBUTION Shows up just about everywhere Synonyms: Normal Distribution, Bell Curve Most basic form is unit Gaussian: centered at zero, unit integral, unit σ: F (x) = 1 2π exp ( x2 This is the probability density for a continuous, normally distributed random variable with mean zero and standard deviation of 1. 2 ) F(x) Unit Gaussian µ = 0! = 1 µ µ"! µ+! As with any probability density function, + dxf (x) = 1. x

5 GAUSSIAN DISTRIBUTION What can you do to the unit gaussian, and still keep it a gaussian? Change its mean from zero to µ Change its width from 1 to σ (while increasing height by 1/σ) Change its integral but then it s not a normalized probability distribution anymore. So this isn t allowed here. F (x) = 1 [ (x µ) 2 σ 2π exp 2σ 2 ] F(x) General µ normalized 0.5 Gaussian µ = ! = 0.7 µ"! µ+! As with any probability density function, this still integrates to 1. x

6 WHERE IT SHOWS UP If you don t have a clue what the probability distribution of a random quantity is (say, the circumferences of cows at the EDZ Ranch), it s highly likely to be approximately gaussian! This is due to the Central Limit Theorem: a sum of a large enough number of random numbers has a gaussian distribution, no matter what the initial distribution shapes might have been. Aside: Distribution of counts of a process with a uniform rate in a finite amount of time is Poisson-distributed (see a later lecture) but is approximately gaussian in the high-number limit. This is another example of the Central Limit Theorem. If there is no systematic bias (more on that later), then the mean of measurements of a quantity is the best estimate of its true value.

7 MEAN, SIGMA, AND ALL THAT Say we measured 150 cows, and have the histogram of results. Calculate the mean circumference <c>: Standard deviation is calculated by: σ c = c = 1 N N i=1 (c i c ) 2 N 1 i c i = 4.39 m = 0.70 m Read up in Taylor on definitions and uses of Entries Cows/0.2 / 0.02 cm m tries variance, standard deviation. WARNING: This is a histogram (a plot showing how many times the result fell in each bin), NOT a normalized probability distribution. Note that its integral is not Bovine Measured Circumference length (m) cm

8 MEAN, SIGMA, AND ALL THAT So, we can now say that the mean circumference is 4.39 m and the standard deviation is 0.70 m. What do we know? 68% of cows have circumference between ( ) and ( ) m. This is because the integral of the gaussian from µ σ to µ +σ is How well do we know the mean circumference? Need std. dev. on the mean: σ c = σ c N = 0.06 Entries Cows/0.2 / 0.02 cm m tries Bovine Measured Circumference length (m) cm So <c>=(4.39 ± 0.06) m, where ± means 1 sigma, or 68% probability that the true mean is within that interval (68% confidence level ). Note that we know the mean to much better than σ of individual measurements.

9 USING THE GAUSSIAN Can use the same mathematics to describe the results of repeated measurements of the same quantity, where there is random error/resolution in the instrument. The distribution will be centered on a mean (assume for now that this is the correct value) The distribution will have a standard deviation Can fit this to a gaussian (Lecture 4,5) or just calculate mean, sigma directly Uncertainty on the mean is now σ µ = σ N Entries / 0.02 cm ries σ Measured length cm µμ Note: More measurements smaller error on the mean! Also means better determination of error.

10 WHAT DOES SIGMA MEAN? First if the measurements are from a gaussian Fµμ, (x), then the probability of measuring a value in the range (a,b) is b P = a dxf µ,σ(x) µ+σ For a normalized gaussian, µ σ dxf µ,σ(x) = 0.68 The general integral can t be expressed analytically. Use error function (erf(x)) tables for values other than 1. So, saying J=5.4±0.9 means one can say the true value of J is between 4.5 and 6.3 with 68% confidence level.

11 ANALYZING ERROR ON A QUANTITY You are in a car on a bumpy road on a rainy day, and are trying to measure the length of the moving windshield wiper with a shaky ruler. You measure it 37 times. The histogram of your results is at right. It doesn t look very gaussian. But, with only 37 measurements plotted in lots of bins, distributions often look ratty. 1σ error on mean You calculate the mean to be 56.2 cm, and the standard deviation to be 6.8 cm. The 1σ uncertainty on the mean is 1.1 cm. If we assume the underlying distribution is nevertheless gaussian and centered on the true value, we can turn this into a confidence level: the wiper length is between 55.1 and 57.3 cm with 68% confidence.

12 ANALYZING ERROR ON A QUANTITY Stop the car, go outside and measure the wiper properly: it is 61.0 cm long! true We said the wiper length was between 55.1 and 57.3 cm with 68% confidence. 1σ error on mean We are off by over 4 sigma. This is shockingly unlikely! Clearly there is a systematic shift. The distribution does not center on the true value. More data points won t get us any closer to the true value. We need to make better measurements, or find the source of the error and apply a correction to the data.

13 STATISTICAL (RANDOM) vs SYSTEMATIC UNCERTAINTIES STATISTICAL SYSTEMATIC NO PREFERRED DIRECTION BIAS ON THE MEASUREMENT: ONLY ONE DIRECTION (THOUGH OFTEN DON T KNOW WHICH) CHANGES WITH EACH DATA POINT: TAKNG MORE DATA REDUCES ERROR ON THE MEAN STAYS THE SAME FOR EACH MEASUREMENT: MORE DATA WON T HELP YOU! GAUSSIAN MODEL IS USUALLY GOOD (EXCEPT COUNTING EXPERIMENTS WITH FEW EVENTS) GAUSSIAN MODEL IS USUALLY TERRIBLE. BUT WE USE IT ANYWAY IF DON T HAVE A BETTER MODEL. Keep statistical, systematic errors separate. Report results as something like: g = [965 ± 30(stat) ± 12(syst)] cm/s 2 Add in quadrature (note that this assumes gaussian distribution) to compare with known values: g = [965 ± 32(total)] cm/s 2

14 PROPAGATION OF ERRORS Often, we aren t measuring directly the quantity our experiment is after: we measure some lab quantities and our final physics result is a function of them. Kaon experiment: we measure curvature of tracks, and from them calculate the momentum of the pions, and then calculate the mass of the kaon from that. We know the errors on the lab quantities. How do we find the error on the final physics result? This is a specific case of the general problem of finding the error on a quantity that is a function of random (uncertain) variables.

15 PROPAGATION OF ERRORS The general formula for error on a function q of random variables x,y,z,...: δq(x, y, z,...) =. ( ) 2 q x δx + ( ) 2 q y δy + ( ) 2 q z δz +... Special case 1: addition (with, say, multiples). Let q= x + y + 2z. ) 2 δq = ( q x δx ) 2 + ( q y δy = (δx) 2 + (δy) 2 + (2δz) 2 = (δx) 2 + (δy) 2 + 4(δz) 2 ) 2 + ( q z δz

16 PROPAGATION OF ERRORS. Special case 2: multiplication. Let q=xyz. δq = ( ) 2 q x δx + ( ) 2 q y δy + ( ) 2 q z δz δq q = (yzδx) 2 + (xzδy) 2 + (xyδz) 2 (δx ) 2 ( ) 2 ( ) 2 δy δz = + + x y z We can add fractional errors in quadrature to get the fractional error on the final result! (This works for division too. Derive it yourself!)

17 COMPARING WITH KNOWN VALUE: Measure: g = [965 ± 32] cm/s 2 = x± x Often negligible Known value: cm/s 2 = x0± x0 Discrepancy is (x-x0)±δ(x-x0), where [δ(x-x0)] 2 = (δx) 2 + (δx0) 2 (add in quadrature) Discrepancy in units of sigma (often called significance of discrepancy) is x x 0 = x x 0 σ x x0 σ 2 x + σx 2 0 Discrepancy here is (16 ± 32) cm/s 2, or 0.5σ. Use erf table to determine agreement confidence level: 62% agreement: good!

18 ANOTHER EXAMPLE: e/m

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