Ch4 - Quantitative Chemical Analysis Chapter 4: Statistics...

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Quantitative Chemical Analysis Quantitative Chemical Analysis Chapter 4: Statistics Statistics
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1 All measurements have an associated uncertainty (error). Statistics gives us a way to quantitate error. The ensuing discussion assumes all errors are purely random, not systematic. A histogram of many measurements (say, n times) of a single quantity takes on a bell shape as n becomes very large. Lifetimes of 4,768 Light Bulbs (hours) The bell-shaped distribution is called a Gaussian distribution . x The curve superimposed on the histogram is the Gaussian distribution based on the average (mean) value, , and the standard deviation, s.
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2 Arithmetic mean (aka mean or average) Median : In an ordered series of numbers, the middle value. Mode : the value that is observed the greatest number of times. Standard deviation : Where n is the number of data points and n -1 is the number of degrees of freedom (DOF)
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3 Histogram for Serum Cholesterol in 953 American 17-Year Olds x The above curve superimposed on the histogram is the Gaussian distribution based on and s. ( = 176, s = 30) x x
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Histogram of Cation Current Passing through Individual Channels of a Frog Muscle Cell (922 Measurements) 4
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5 2s 2 ( x x ) 2 e y = c The Gaussian distributions superimposed on the above histograms have the form where c is a constant to fit the curve to the histogram. Now, as n →∞ , µ is the “True” value of what is being measured and σ is the standard deviation. and, mean = median = mode
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6 The above curve is related to the normal distribution curve y = 1 σ 2 π e ( x µ ) 2 2 2 The pre-exponental factor is the normalization factor that insures that the integral from x = −∞ to + = 1. is the “True” value of what is being measured and is the standard deviation.
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7 Normal Distribution Curves as a Function of σ y = 1 2 π e ( x µ ) 2 2 2 = 0 at left Area = 1 in all cases The width of the curve (at its inflection points), 2 , depends on the amount of scatter in the data. The integral from to is always 0.3413 from + to – is 0.6826 These are the probability that then next measurement will fall in that range.
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8 Normal distribution curve applies to all problems where error is random. Integral from + σ to – is always 0.6826 so, usually plotted with abscissa in units of rather than x .
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Ch4 - Quantitative Chemical Analysis Chapter 4: Statistics...

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