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Unformatted text preview: 10/6/11 1 Gaussian Distribution If the errors for a set of measurements are random, then the measurements can be represented by a Gaussian distribution Two quantities are used to describe a Gaussian curve The Mean the arithmetic average of all measurements The Standard Deviation a measure of the spread in the measurements Gaussian Distribution Mean: n = total number of measurements Standard deviation the factor n-1 in the denominator is called the degrees of freedom = = n i i x n 1 x 1 = = N I I ) X X ( ) N ( s 1 2 1 1 Gaussian Distribution Measurement Value Measurement Value 0.340 9 0.3430 0.3350 0 0.340 3 0.3470 0.3560 4 0.3590 0.3500 5 0.3530 3 0.3630 6 0.3460 4 0.3530 7 0.3470 5 0.3480 8 0.3460 x = .3486 s = .0073 1 Gaussian Distribution For a small, finite number of measurements, the mean and standard deviation are only approximations of the true mean ( ) and standard deviation of the sample ( ) For a large number of measurements: = lm N = s lm N Gaussian Distribution The mathematical expression for a Gaussian distribution is: = 2 2 2 2 1 ) ( ep y The factor 1/ (2 ) 1/2 is a normalization constant and assures that the area under the curve for the Gaussian function equal unity Gaussian Distribution...
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- Fall '09