1_Chapter 4 Probability and Statistics Intro.pdf

These basic probability definitions are the

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These basic probability definitions are the foundation for advanced statistics in data analysis Random fluctuations in data sets are assumed to follow a predictive distribution of occurrence (pdf) over a certain range of values around the central or mean value Outcome is a true value estimate with an uncertainty interval around it; interval associated with a probability
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Basic Statistics Values
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Mean and Standard Deviation Before applying sophisticated analyses, we must always determine the center and the spread of the data These are known as the mean (or average ) and the standard deviation Important note: No statistical analysis should proceed if the mean and standard deviations are not first estimated – These functions are available in all programming languages and business spreadsheets – The mean can only be formulated if the function is “stationary” (the mean does not change in the data set)
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Mean and Standard Deviation Infinite Data Set Continuous Form 1 Again defined differently for continuous functions (look at first) and discrete data sets For a continuous, known function x ( t ) that varies in time or space:... True mean value: Standard deviation: (True variance is ı 2 )
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Mean and Standard Deviation Infinite Data Set Continuous Form 2 Function x ( t ) is an arrangement of an infinite number of x values in time or space. The same values can be restated as x times the probability of occurrence of x : x times a pdf, p ( x ). The integral is transformed from an integral over time or space to an integral over the probability distribution. True mean value: True variance: Standard deviation: ı 2
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Common PDFs... Rectangular: random data falls within minimum value a and maximum value b with equal probability of occurrence If b - a = 1, probability that random data point lies within a and a + ½ is 50%, the area under the p( x ) curve between a and a + ½.
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Common PDFs... Rectangular: random data falls within minimum value a and maximum value b with equal probability of occurrence Data below is random, but follows rectangular distribution around y = 1.5.
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Common PDFs... Gaussian (or Normal): used for physical properties continuous in time that have variations due to random error Random values clustered around the true mean value; symmetric
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Common PDFs... Gaussian (or Normal): a lot of experimental data recorded has random fluctuations that follow this pdf Æ used for statistics in this course Data below has true mean of 1.5, ı = 0.25, notice clustering and no max/min boundary
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Higher Order Moments Although adequate for many types of data, the mean and the standard deviation statistical measures may not be enough The mean is a first order statistic The variance is a second order statistic, but others can be calculated up to 4th order with moment generating function. 3rd order: skew or skewness Æ information about whether the distribution is symmetric 4th order: kurtosis Æ information about peakedness There are examples where these low-order statistics can be misleading, i.e., give a false sense of the “truth”
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Mean and Standard Deviation Infinite Discrete Data Set Defined differently for discrete data sets For an infinite data set of N points : True mean value: True standard deviation: ݔ ே՜ஶ ͳ ± ෍ ݔ ௜ୀଵ ߪ ൌ ே՜ஶ ͳ ± ෍ሺݔ í ݔ ) ௜ୀଵ
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