lec04_stat

lec04_stat - Error Analysis - Statistics Accuracy and...

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Slide 1 Error Analysis - Statistics Accuracy and Precision Individual Measurement Uncertainty Distribution of Data Means, Variance and Standard Deviation Confidence Interval Uncertainty of Quantity calculated from several Measurements Error Propagation Least Squares Fitting of Data
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Slide 2 Accuracy and Precision Accuracy Closeness of the data (sample) to the “true value.” Precision Closeness of the grouping of the data (sample) around some central value.
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Slide 3 Accuracy and Precision Inaccurate & Imprecise Precise but Inaccurate Relative Frequency X Value True Value X Value True Value No. of darts hitting particular location Total No. of darts Location on the target
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Slide 4 Accuracy and Precision Accurate but Imprecise Precise and Accurate Relative Frequency X Value True Value X Value True Value
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Slide 5 Accuracy and Precision Q: How do we quantify the concept of accuracy and precision? -- How do we characterize the error that occurred in our measurement?
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Individual Measurement Statistics Take N measurements: X 1 , . . . , X N Calculate mean and standard deviation: What to use as the “best value” and uncertainty so we can say we are Q% confident that the true value lies in the interval x best x. Need to know how data is distributed. N i i X N x 1 1   N i x i x X N S 1 2 2 1 Slide 6 2 var stdev x x S S Population mean
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Slide 7 Population and Sample Parent Population The set of all possible measurements. Sample A subset of the population - measurements actually made. Population Bag of Marbles Handful of marbles from the bag Samples
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Slide 8 Histogram (Sample Based) Histogram A plot of the number of times a given value occurred. Relative Frequency A plot of the relative number of times a given value occurred. Histogram 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 X Value (Bin) Number of Measurements Relative Frequency Plot 0 0.05 0.1 0.15 0.2 0.25 0.3 30 35 40 45 50 55 60 65 70 75 80 X Value (Bin) Relative Frequency ∆x=5 (27.5~32.5] (32.5~37.5] (37.5~42.5] (42.5~47.5] (47.5~52.5] : weight of marbles
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Slide 9 Probability Distribution Function ( P(x) ) Probability Distribution Function is the integral of the pdf, i.e. Q: Plot the probability distribution function vs x. Q: What is the maximum value of P(x)? Probability Distribution (Population Based) Probability Density Function (pdf) ( p(x) ) Describes the probability distribution of all possible measures of x . Limiting case of the relative frequency. x X     dx x p x P x Probability Density Function 0 0.05 0.1 0.15 0.2 0.25 0.3 30 35 40 45 50 55 60 65 70 75 80 x Value (Bin) Probability per unit change in x   ] [ x X P x P Probability that ∆x 0
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Slide 10 Ex: is a probability density function.
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lec04_stat - Error Analysis - Statistics Accuracy and...

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