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The Normal Distribution The Distribution 1200 1000 800 600 400 200 Std. Dev = 1.00 Mean = -.01 N = 10000.00 0 50 3. 00 3. 50 2. 00 2. 50 1. 00 1. 0 .5 00 0. 0 -.5 0 .0 -1 0 .5 -1 0 .0 -2 0 .5 -2 0 .0 -3 0 .5 -3 0 .0 -4 X The Standard Normal Distribution • We simply transform all X values to have a mean = 0 and a standard deviation = 1 • Call these new values z • Define the area under the curve to be 1.0 z Scores • Calculation of z Xμ z σ where is the mean of the population and is its standard deviation This is a simple linear transformation of X. Tables of z • We use tables to find areas under the distribution • A sample table is on the next slide • The following slide illustrates areas under the distribution Normal Distribution Cutoff at +1.645 1200 1000 z= 1.64545 45 800 Area = .05 .05 600 400 200 0 50 3. 00 3. 50 2. 00 2. 50 1. 00 1. 0 .5 00 0. 0 -.5 0 .0 -1 0 .5 -1 0 .0 -2 0 .5 -2 0 .0 -3 0 .5 -3 0 .0 -4 z Using the Tables • Define “larger” versus “smaller” portion • Distribution is symmetrical, so we don’t need negative values of z • Areas between z = +1.5 and z = -1.0 See next slide Calculating areas • Area between mean and +1.5 = 0.4332 • Area between mean and -1.0 = 0.3413 • Sum equals 0.7745 • Therefore about 77% of the observations would be expected to fall between z = -1.0 and z = +1.5 Converting Back to X • Assume = 30 and = 5 • 77% of the distribution is expected to lie between 25 and 37.5 z X Therefore X z X 30 1.0 5 25 X 30 1.5 5 37.5 Probable Limits • X=+z • Our last example has = 30 and = 5 • We want to cut off 2.5% in each tail, so z = + 1.96 X z X 30 1.96 5 39.8 X 30 1.96 5 20.2 Cont. Probable Limits--cont. • We have just shown that 95% of the normal distribution lies between 20.2 and 39.8 • Therefore the probability is .95 that an observation drawn at random will lie between those two values Measures Related to z • Standard score Another name for a z score • Percentile score The point below which a specified percentage of the observations fall • T scores Scores with a mean of 50 and a standard deviation of 10 Cont.