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Unformatted text preview: CHEM 321 – Quantitative Analysis Ch. 4 – Statistics 41 Gaussian Distribution 42 Confidence Intervals 43 Comparison of Means with Student’s t 46 Grubbs Test for an Outlier 47 Method of Least Squares 41 Gaussian Distribution Requirements • Multiple (replicate) experimental measurements • Assumes only random errors in measurements ( no determinate errors ) Results • Histogram (# occurrences versus measured value) should give symmetrical, bellshaped curve • Data will “cluster” or center around some value ( mean ) • Data will show some “spread” ( uncertainty ) Problems • Inability to make sufficient number of replicate measurements ( n is finite ) • Nonrandom errors may be present (determinate errors, poor accuracy) Mean Standard Deviation Mean is best estimate of center of distribution (estimate of “true” value ) Standard deviation is the best estimate of how closely data are clustered about the mean (estimate of uncertainty or precision of the measurement) EXCEL formulas Mean: “ = average(A1:A5)” Std dev: “ = stdev(A1:A5)” = ∑ n x x n i i ( 29 1 2 = ∑ n x x s i i n – 1 = “ Degrees of freedom ” ( 29 1 2 = ∑ n x x s i i Theoretical Gaussian curve (true mean and uncertainty are unknown) Can estimate mean and uncertainty from experimental data – x = 845.2 s = 94.2 2 Gaussian curves with same mean but different standard deviations Consider case of 2 students analyzing the SAME unknown OR an analyst using 2 different methods to analyze the SAME unknown Comparing means and standard deviations is useful Its impossible to judge accuracy from this information ( μ not known, no info as to which data set is more correct) BUT one can compare precision of two data sets (s) In reporting a result, we typically give the mean and the standard deviation Remember to follow the “real rule” for SF here s x ± Example mean value and its uncertainty 845.2 ± 94.2 SF incorrect (8.5 ± 0.9) x 10 2 # SF clear (follow real rule for SF) Data that conform to a Gaussian distribution are said to be “normally distributed” A Gaussian distribution is an ideal, theoretical curve that is symmetrical about the mean Gaussian distribution equation ( 29  = 2 2 2 exp 2 1 σ μ π σ x y exp = base of natural log (2.7183) σ = population standard deviation μ = population mean (“true” value) Keep in mind that...
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 Spring '11
 P.Palmer
 Normal Distribution, Standard Deviation, µ

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