Judah Levine JILA S (2-7785) Monday, 2-4 pm Also by appointment Usually ok without an appointment

Transcription

1 Judah Levine JILA S (2-7785) Monday, 2-4 pm Also by appointment Usually ok without an appointment Judah Levine, 2150L3 1

2 Clicker Stuff I will not count one absence from lecture in your clicker scores. Score will be based on average of other lectures These clicker numbers are not registered to any student: 44 4F 55 5E CD EA C0 ED D4 D1 46 5A D4 F3 48 CA DD A E0 8A 22 Judah Levine, 2150L3 2

3 The outlier problem Clicker Question Suppose x= 20, 65, 70, 70, 75 Mean = 60±23 Is it ok to ignore 20 as an outlier and report 70 ± 4? A:Yes B: No If these were the grades on an exam, what would you report as the class average? Judah Levine, 2150L3 3

4 Handing an outlier Drop it. It was a blunder that you did not catch at the time It differs from the other data in some unknown way Keep it. No justification for ignoring data after the fact. Professional solution Ignore values more than 4 Std. Dev. Probability < 0.01% Administrative decision, not a strict statistical decision Judah Levine, 2150L3 4

5 Clicker Question y= a 1 x 1 ± a 2 x 2 ± ± a n x n δδyy = aa 1 δδxx aa 2 δδxx x=100±2, y=250 ± 5 z= 50 ±1 q= 2x + 3y - 4z δq= A: 16 B: 5.5 C: 650 D: 25.5 Answer is A Judah Levine, 2150L3 5

6 Uncertainty of the mean NN xx = 1 NN xx ii ii=1 δδxx = NN δδxx 2 ii=1 ii NN = NN δδxx ii 2 NN = NNδδxx ii NN = δδxx ii NN Uncertainty of the improves with increasing N but improvement is slow Assumes same uncertainty for all x i Judah Levine, 2150L3 6

7 Systematic Errors Contributions to the uncertainty that are not improved by averaging Often called Type B errors in the literature Example: calibration of instruments Typically estimated by secondary measurements Systematic errors often vary slowly Judah Levine, 2150L3 7

8 Calibration Errors Compare measuring tool with a standard What are the uncertainties in this comparison measurement? How do I know that the standard is correct? This is a recursive problem What does it mean for the ultimate standard to be wrong? Judah Levine, 2150L3 8

9 Definition of the kilogram - 1 Louis XVI in 1791, 1795, 1799 Mass of 1 L of water at 4 C Large systematic uncertainties Kilogram of the Archives (KA) Platinum sponge Measurement uncertainty about 2 mg KA is effective legal definition at that time Judah Levine, 2150L3 9

10 Definition of the kilogram - 2 Treaty of the meter 1885 International Prototype Kilogram (IPK) 90% platinum, 10% iridium Mass set equal to KA in 1880 Uncertainty mg Legal definition in official copies made at the same time Kept at the International Bureau of Weights and Measures near Paris Judah Levine, 2150L3 10

11 The troubles - 1 In 1939 m(ka)= m(ipk) 0.43 mg What changed? In 1946 comparison of IPK with its official copies M(IPK) loses mass with respect to the 6 official copies Judah Levine, 2150L3 11

12 The troubles - 2 Masses of official copies slowly increase because of a film of oil and dirt? Can be minimized but not eliminated Official washing procedure After washing, copies of the kilogram still disagree 50 µg in about a century δm/m = /10 3 = Cause is not known Judah Levine, 2150L3 12

13 US Standards of Time and Frequency Maintained at NIST in Boulder NIST standard frequency drifted About δf/f = in 45 years Unknown causes Corrected by periodic calibrations Uncertainty in calibration process ~ 1% Residual frequency offset ~ Frequency error published on web every Wednesday Judah Levine, 2150L3 13

14 The Bottom Line Estimating and removing systematic errors is hard Much harder than random stuff Some sources are never known Estimation process has its own uncertainties Laboratory notebooks are very important It s an imperfect world Judah Levine, 2150L3 14

15 A random quantity - 1 x n cannot be determined even with perfect knowledge of x i, i=1,, n-1 Requirement is necessary but not sufficient Common limitation: Observable is the integral or derivative of a random variable Random forces generate random accelerations Observable is the velocity Judah Levine, 2150L3 15

16 Expectation value <x>= xxxx xx dddd p(x)dx is probability of finding value between x and x+dx Integration is over all possible values of x For N values which are equally probable, p(x)= 1/N Expectation value is same as the mean Judah Levine, 2150L3 16

17 A random quantity - 2 xx jj = xx 0 + εε jj x 0 is true value of the observable x j is the measured value and ε j is the random noise contribution εε jj = 0 εε jj εε kk = 0 jj kk εε jj εε jj = ss 2 x 0 is the true mean, s is the true std. dev. Judah Levine, 2150L3 17

18 Gaussian, Normal Distribution If the contribution to the uncertainty has a mean of zero and If the contribution to the uncertainty is not correlated from one measurement to the next one, and If the measured quantity has a large number of possible values Not a coin-toss with only two values, and If the measurement is repeated a very large number of times then Judah Levine, 2150L3 18

19 Gaussian Distribution - 1 Probability that a measurement will give a value between x and x + dx is pp xx dddd = KKee xx xx 0 2ss 2 Every measurement must produce some value, therefore pp xx dddd = 1 2 So that KK = 1 2ππss Judah Levine, 2150L3 19

20 Gaussian Distribution - 2 pp xx dddd = 1 xx xx 0 2ππss ee 2ss 2 2 Probability is a maximum at x=x 0 Probability decreases symmetrically about this value Width is determined by s Parameters x 0 and s are the true values of the distribution Generally not known Judah Levine, 2150L3 20

21 Gaussian Distribution - 3 pp xx dddd = 1 xx xx 0 2ππss ee 2ss 2 2 If all of the fine print is true, then for a large number of measurements Estimate of the mean approximates x 0 Estimate of the std. dev. approximates s pp xx dddd = 1 2ππσσ ee xx xx 2 2σσ 2 Judah Levine, 2150L3 21