statistics-w-solutions

# statistics-w-solutions - ME250 Design and Manufacturing I...

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ME250 Design and Manufacturing I Statistics and Tolerances

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Outline Introduction Random variables Normal distribution Absolute and statistical tolerances Summary 2
Introduction What is the correct value? 2.01” 1.98” 2.03” 3

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Introduction Reality: OD will not be exactly the same For each measurement For each products Sources of variations anufacturing: cutting condition tool wear material Manufacturing: cutting condition, tool wear, material irregularity. . Measurement: measured location, measured devices. . Environment: temperature, humidity… So what can we do? pecify a dimension with an allowable range commonly as: Specify a dimension with an allowable range, commonly as: upper wer or nominal tolerance 4 lower
Introduction Given measurements of n sample products… 2.01 1.98 2.03 2.00 1.99 2.00 How likely the next OD fall within the target range, say 2.00 ± 0.01? ??? 5

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Random variables Random variable : variable whose value varies for every sample istogram: equency plot of sampled alues Histogram: frequency plot of sampled values sample id 1 2 3 4 5 6 od [in] 2.01 1.98 2.03 2.00 1.99 2.00 3 4 enc y 1.99 2.01 x  0 1 2 Frequ e 1.96 1.98 2.00 2.02 2.04 More OD [in] “bins” 6 Excel Help: http://office.microsoft.com/en-us/excel/HA102382521033.aspx
Normal distribution What happens to histogram if ? n  1 2 3 4 Frequency 0 1.96 1.98 2.00 2.02 2.04 More OD [in] 7

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Normal distribution Gauss says… 4 y 0 1 2 3 1.96 1.98 2.00 2.02 2.04 More D [in] Frequenc OD [in] n  “bell” shape 8
Normal distribution Gauss says… (simply put) If the values of a random variable x is due to the effects of any other random variables then the histogram of ith many other random variables, then the histogram of x with n samples approaches to the normal distribution as n  2 () x x 2 2 1 2 x x fx e  Denoted as : mean (“horizontal shift”) (, ) x xN x x : standard deviation (“flatness”) Excel Tip: NORMDIST x 9

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Normal distribution Example: Which line has the highest mean? Which line has the highest stdev? 10
Normal distribution Example: Which line has the highest mean? Red , Green , and Blue Which line has the highest stdev? Blue 11

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Normal distribution Estimators of and from n sample data 22 2 11 ) nn x x n x  x x 1 n x Excel tip: AVERAGE, STDEV () xi i ii xx       1 i i n Estimators of and from bin x i and frequency n x x i 1 m x nx 2 1
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## This note was uploaded on 09/12/2011 for the course MECHENG 250 taught by Professor Stevis during the Winter '09 term at University of Michigan.

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statistics-w-solutions - ME250 Design and Manufacturing I...

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