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Chapter 6 The Normal Distribution Fundamental Statistics for the Behavioral Sciences, 5th edition David C. Howell ©2003 Brooks/Cole Publishing Company/ITP Chapter 6 The Normal Distribution Major Points • Distributions and area • The normal distribution • The standard normal distribution • Setting probable limits on an observation • Measures related to z 2 Chapter 6 The Normal Distribution Distributions and Area • The idea of a pie chart as representing area See next slide • Bar charts say the same thing See subsequent slide 3 Chapter 6 The Normal Distribution Where are the Bad Guys? 4 5 Chapter 6 The Normal Distribution Bar Chart of Bad Guys Chart 75 P e r c e n t a g e 50 25 0 Prison Jail Parole Probation 6 Chapter 6 The Normal Distribution Getting Closer 40 30 20 10 Std. Dev = 10.56 Mean = 49.1 N = 289.00 0 .5 81 .3 77 .1 73 .9 68 .7 64 .5 60 .3 56 .1 52 .9 47 .7 43 .5 39 .3 35 .1 31 .9 26 .7 22 .5 18 Behavior Problem Score 7 Chapter 6 The Normal Distribution The Normal Distribution • The general shape of the distribution See slide 9 • The formula for the normal distribution. 1 f (X ) e 2 ( X ) 2 2 2 Cont. Chapter 6 The Normal Distribution Normal Distribution--cont. • X is the value on the abscissa • Y is the resulting height of the curve at X • and e are constants 8 9 Chapter 6 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 Chapter 6 The Normal Distribution 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 10 11 Chapter 6 The Normal Distribution 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. 12 Chapter 6 The Normal Distribution 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 13 Chapter 6 The Normal Distribution z Table Mean to z Larger Portion Smaller Portion 0.000 .0000 .5000 .5000 0.100 .0398 .5398 .4602 0.200 .0793 .5793 .4207 1.000 .3413 .8413 .1587 1.500 .4332 .9332 .0668 1.645 .4500 .9500 .0500 1.960 .4750 .9750 .0250 z 14 Chapter 6 The Normal 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 Chapter 6 The Normal Distribution 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 15 16 Chapter 6 The Normal Distribution 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 17 Chapter 6 The Normal Distribution 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 18 Chapter 6 The Normal Distribution 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. Chapter 6 The Normal Distribution 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 19 20 Chapter 6 The Normal Distribution 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. Chapter 6 The Normal Distribution 21 Review Questions • Why do you suppose we call it the “normal” distribution? • What do we gain by knowing that something is normally distributed? • How is a “standard” normal distribution different? Cont. Chapter 6 The Normal Distribution Review Questions--cont. • How do we convert X to z? • How do we use the tables of z? • Of what use are probable limits? • If we know your test score, how do we calculate your percentile? • What is a T score and why do we care? 22