The Central Limit Theorem Author: John M. Cimbala, Penn State University Latest revision: 20 August 2007 Introduction •It is rare that one can measure something for an entire population– instead, a sample(or several samples) of the population is measured, and the population statistics are estimated from the sample. •The Central Limit Theoremis an extremely useful tool when dealing with multiple samples. Multiple samples and the Central Limit Theorem •Consider a population of random variable x(we assume that variations in xare purely random – in other words, if we would plot a PDF of variable x, it would look Gaussian or normal). •The population mean μand the population standard deviation σare not known, but are instead estimated by taking several samples. •We take Nsamples, each of which contains nmeasurements of variable x, as indicated in the sketch to the right. •We define the sample meanfor sample Ias 11nIiixxn==∑, where index I= 1, 2, 3, … N(one for each sample). In other words, we calculate a sample mean in the
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This note was uploaded on 04/05/2008 for the course ME 345 taught by Professor Staff during the Spring '08 term at Penn State.