chem-201-chapter-4 - Gaussian Distribution Gaussian...

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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 1 0.3410 9 0.3430 2 0.3350 10 0.3420 3 0.3470 11 0.3560 4 0.3590 12 0.3500 5 0.3530 13 0.3630 6 0.3460 14 0.3530 7 0.3470 15 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 ( μ ) and standard deviation of the sample ( σ ) For a large number of measurements: μ = x lim n σ = s lim n Gaussian Distribution The mathematical expression for a Gaussian distribution is: ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ σ μ π σ = 2 2 2 2 1 ) x ( exp y The factor 1/ σ (2 π ) 1/2 is a normalization constant and assures that the area under the curve for the Gaussian function equal unity Gaussian Distribution Measured Value 30 40 50
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