Chapter 6 The Normal Distribution Fundamental Statistics for the Behavioral Sciences, 5th edition David C. Howell © 2003 Brooks/Cole Publishing Company/ITP

2Chapter 6 The Normal Distribution Major Points Distributions and areaDistributions and area The normal distributionThe normal distribution The standard normal distributionThe standard normal distribution Setting probable limits on an observationSetting probable limits on an observation Measures related to zMeasures related to z

3Chapter 6 The Normal Distribution Distributions and Area The idea of a pie chart as representing areaThe idea of a pie chart as representing area XSee next slide Bar charts say the same thingBar charts say the same thing XSee subsequent slide

4Chapter 6 The Normal Distribution Where are the Bad Guys?

5Chapter 6 The Normal Distribution Bar Chart of Bad Guys

6Chapter 6 The Normal Distribution Getting Closer

7Chapter 6 The Normal Distribution The Normal Distribution The general shape of the distributionThe general shape of the distribution XSee slide 9 The formula for the normal distribution.The formula for the normal distribution. Cont.

8Chapter 6 The Normal Distribution Normal Distribution --cont. X is the value on the abscissaX is the value on the abscissa Y is the resulting height of the curve at XY is the resulting height of the curve at X and e are constants and e are constants

9Chapter 6 The Normal Distribution The Distribution

10Chapter 6 The Normal Distribution The Standard Normal Distribution We simply transform all X values to have a mean = 0 and a standard deviation = 1We simply transform all X values to have a mean = 0 and a standard deviation = 1 Call these new values zCall these new values z Define the area under the curve to be 1.0Define the area under the curve to be 1.0

11Chapter 6 The Normal Distribution z Scores Calculation of zCalculation of z Xwhere is the mean of the population and is its standard deviation XThis is a simple linear transformation of X.

12Chapter 6 The Normal Distribution Tables of z We use tables to find areas under the distributionWe use tables to find areas under the distribution A sample table is on the next slideA sample table is on the next slide The following slide illustrates areas under the distributionThe following slide illustrates areas under the distribution

13Chapter 6 The Normal Distribution z Table

14Chapter 6 The Normal Distribution z = Area =.05.05

15Chapter 6 The Normal Distribution Using the Tables Define larger versus smaller portionDefine larger versus smaller portion Distribution is symmetrical, so we dont need negative values of zDistribution is symmetrical, so we dont need negative values of z Areas between z = +1.5 and z = -1.0Areas between z = +1.5 and z = -1.0 XSee next slide

16Chapter 6 The Normal Distribution Calculating areas Area between mean and +1.5 = Area between mean and +1.5 = Area between mean and -1.0 = Area between mean and -1.0 = Sum equals Sum equals Therefore about 77% of the observations would be expected to fall between z = -1.0 and z = +1.5Therefore about 77% of the observations would be expected to fall between z = -1.0 and z = +1.5

17Chapter 6 The Normal Distribution Converting Back to X Assume = 30 and = 5Assume = 30 and = 5 77% of the distribution is expected to lie between 25 and % of the distribution is expected to lie between 25 and 37.5

18Chapter 6 The Normal Distribution Probable Limits X = + zX = + z Our last example has = 30 and = 5Our last example has = 30 and = 5 We want to cut off 2.5% in each tail, soWe want to cut off 2.5% in each tail, so Xz = Cont.

19Chapter 6 The Normal Distribution Probable Limits --cont. We have just shown that 95% of the normal distribution lies between 20.2 and 39.8We 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 valuesTherefore the probability is.95 that an observation drawn at random will lie between those two values

20Chapter 6 The Normal Distribution Measures Related to z Standard scoreStandard score XAnother name for a z score Percentile scorePercentile score XThe point below which a specified percentage of the observations fall T scoresT scores XScores with a mean of 50 and a standard deviation of 10 Cont.

21Chapter 6 The Normal Distribution Review Questions Why do you suppose we call it the normal distribution?Why do you suppose we call it the normal distribution? What do we gain by knowing that something is normally distributed?What do we gain by knowing that something is normally distributed? How is a standard normal distribution different?How is a standard normal distribution different? Cont.

22Chapter 6 The Normal Distribution Review Questions --cont. How do we convert X to z?How do we convert X to z? How do we use the tables of z?How do we use the tables of z? Of what use are probable limits?Of what use are probable limits? If we know your test score, how do we calculate your percentile?If we know your test score, how do we calculate your percentile? What is a T score and why do we care?What is a T score and why do we care?