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Lecture 4, PHYS1140 Statistical uncertainty. Given a set of repeated measurements of the same quantity, with each measurement having some random error, what is the best estimate of that quantity and what is the uncertainty? Measure same quantity (x) several times (N trials). Get results: xi = x1, x2, x3 … xN What is the best estimate of the true value of the quantity? x best Answer: x avg What is the uncertainty in xbest? x x i N i (IF all measurements equally good.) x best ?? Answer (all to be justified later!): x best x avg mean N x standard deviation = i x standard deviation of the mean = SDOM 2 i N 1 If you make a histogram of the results of the repeated measurements, you almost always find that the distribution of results has a special shape, called the “gaussian” or “normal” distribution. frequency x 2 The standard deviation is a measure of the width of this distribution. You can think of as the uncertainty of a single measurement of x. If you make one measurement of x (call it x1) then the uncertainty in x1 is That is, after 1 measurement, your best estimate of the true value of x is x = x1 ± s . Generally, the standard deviation is unknown unless you make lots of measurements and then compute it with the formula above. frequency frequency less precise measurement: larger x more precise measurement: smaller x For a given experiment, the standard deviation does not change as you take more and more measurements. is a measure of the uncertainty of the measurement apparatus, and that only depends on the apparatus, not on how many time you use the apparatus. is the uncertainty in the average (or mean) N value of all N trials. After N trials, the best estimate of the true value of x is x = x ± s mean Notice that mean gets smaller and smaller as the number of trials increases. You get a more precise average when you average lots of trials. The standard deviation of the mean mean The exact shape of the gaussian or normal distribution is given by f x,s (x) = 2 1 - (x- x ) /(2s 2 ) e s 2p The values of the mean, x , and the standard deviation, , depend on the details of the experiment: x depends on what’s being measured and depends on the measurement apparatus. But the shape of the curve is universal. It can be shown that this shape will always be found if the error in the measurements is due to the net effect of a large number of smaller, uncorrelated errors. (This is called the “Central Limit Theorem” in statistics – to be discussed later.) 3 A subtle point about the formula for the standard deviation: The underlying distribution function f(x) has some “true” mean and “true” standard deviation. The formula estimate of the “true” value of . Just as x x x i x i N 1 2 is our best i i N is our best estimate of the “true” value of x. +¥ This function f(x) is “normalized” to 1, meaning ò f (x)dx = 1 - ¥ f(x) area under part between x + = 0.68 total area under curve = 1 x- x x+ x The normal function f gives the probability distribution of a measurement of x: f(x) x1 x2 x x2 Probability(single measurement gives value in the range x1 < x < x2) = ò f (x) dx x1 Note that mean (standard deviation of the mean is always) less than standard deviation . Uncertainty in average of several measurements is always less than the uncertainty of a single measurement. 4 Proof that uncertainty of average = s N : Regard xavg as a function of (x1 , x2, x2, ... xN ): x x i i N 1 x1 x 2 x 3 ... x N N Apply master formula for error propagation: 2 2 2 æ¶ x ö æ¶ x ö æ ö ¶ x ÷ ÷ çç çç ÷ ÷ ÷ dx = ççç dx1 ÷ + d x + K + d x N÷ ÷ ÷ çè¶ x ÷ èç¶ x 2 2 ø ÷ ÷ è¶ x 1 ø ø N ¶x 1 = for any i. ¶ xi N So dx = 1 dx12 + dx 22 + K + dx2N N = 1 s2 + s2 + K + s2 N Remember: is the uncertainty in one measurement, and all measurements are assumed equally good, so dx1 = dx2 = ... = dx N = s . Notice there are N terms are under the square-root sign, so dx = 1 N s2 N = s N 5 PHYS1140 Males, N = 588 trials avg = 1.060, stdev=0.085, SDOM=0.003 Frequency 120 100 80 60 40 20 1.27 1.21 1.15 1.09 1.03 0.97 0.91 0.85 0.79 0.73 0 normalized time (1.00 = perfect) PHYS1140 Females, N = 177 trials avg = 1.058, stdev=0.092, SDOM=0.007 Frequency 35 30 25 20 15 10 5 1.27 1.21 1.15 1.09 1.03 0.97 0.91 0.85 0.79 0.73 0 normalized time(1.00 = perfect) Results of time trials: All time measurements were normalized to 1. If student estimated 12.6 s, and correct time was 11.7 s, then student normalized estimate is 12.6 s/11.7 s = 1.08 Men: average normalized time = tavg = 1.060 + 0.003 Women: tavg = 1.058 + 0.007 (larger uncertainty due only to smaller N) Men and women have same tavg within uncertainties. 6 Mathcad plot of (normalized) frequency of male times, and gaussian function (solid curve) with same mean and same standard deviation. Notice! Same systematic error in both men and women results: everyone estimated about 6% too high (Do you know why?) Both men and women have the same mean and same standard deviation. (No gender difference in estimating times) Smaller number (N) of trials for women, so larger mean (SDOM) for the women. Bigger number (N) of trials for men, so have better statistics, better fit to normal distribution. Bigger number of trials gives smaller random fluctuations away from normal distribution.