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Lecture

4normalerrorcurve.pdf

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Department
Chemistry
Course
ENCH 213
Professor
Diane Beauchemin
Semester
Fall

Description
Statistical analysis Error propagation • must propagate error to find the overall error • errors can be estimated through statistical analysis following replicate measurements • by being careful: possible to get high accuracy and precision ◦ eliminate systematic errors ◦ minimize random errors Q: The word precision when referring to the data obtained from an analysis, means: a) how close the mean obtained from a set of data is to the true value of the composition of the sample analyzed. b) the relative error calculated from the mean of a set of data. c) how close the individual data points obtained from a set of data are to each other. Indeterminate or random errors • cause data to fluctuate randomly arounthe mean of the set (part of every physical or chemical measurement) • to visualize what happens, assume 2 random equal errors e and 1 , whi2h • can combine to give overall errors: combinations overall error # combinations +e 1e 2` +2e 1 +e -e ; -e +e 0 2 most frequent 1 2 1 2 -e1-e 2 -2e 1 If large number of individual errors: → gaussian curve Ways to look at the data • example: % PCB in soil (large sample)= 7.7, 4.5, 6.9, 4.9, 5.3, 6.9, 5.9, 1.1 % • arithmetic mean = = 5.42% ◦ n = 8 = number of measurements • median = middle value (in order) ◦ if even number of replicates: median = mean of middle 2 measurements ◦ 1.1<4.5<4.9< 5.3<5.9<6.9<6.9<7.7 % ◦ median= (5.3+5.9)/2=5.6 • spread or range = x max-xmin7.7 - 1.1 = 6.6 % • deviation d oi a single value x froi the mean: d = x - xi i • standard deviation about the mean: divide by n-1 b/c finite data set Expression of the result • %PCB in large sample= 5.4 ±2.1 % • %PCB in small sample= 6.3 ±9.4 % • to improve reliability, pool data from individual sets with similar s: ◦ n=nuiber of data in set i ◦ n =nsmber of data sets being pooled For infinite set of data • µ = population mean • σ = population standard deviation • on general, distribution of replicate measurements → gaussian curve: • distribution of the population for finite set of data: ◦ µ → x ◦ σ → s Normal error curve • if expressed vs • nomal error curve becomes: ◦ Gaussian curve where unit = 1 standard deviation ◦ describes all populations ◦ z only depends on the confidence level: how much of surface under curve are looking at • in statistical tables (Table 4-1), it is normalized to unity: ◦ for σ = 1 → ◦ area from -∞ to +∞ = 1 Properties of the normal error curve • mean (central point) has maximum frequency • symmetrical distribution of positive and negative deviations about maximum • exponential decrease in frequency as magnitude of deviations increases ◦ small random errors observed more often than large ones • regardless of its width ◦ 68.3% of area beneath normal error curve is within ± 1σ ◦ 95.5% of all data → within ± 2σ ◦ 99.7% → within ± 3σ • standard deviation = useful predictive tool ◦ 95.5% chance that the error on your single measurement is less than ± 2σ Q: If one has a Gaussian distribution of data points, a) approximately two thirds of the data points lie above the mean, and one third of the data points lie below the mean accounting for all of the data points. b) approximately 2/3 of the data points lie within plus or minus two standard deviations of the mean. c) approximately 2/3 of the data points lie within plus or minus one standard deviation of the mean. Q: A Gaussian distribution of data is symmetric if a) 4.5% of measurements lie outside the range defined by two standard deviations above the mean, and two standard deviations below the mean. b) 4.5% of measurements lie outside the range defined by two standard deviations above the mean, and two standard deviations below the mean with 2.25% of the values abov
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