The Normal Distribution

The 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

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.

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

z = Area =.05.05

Using the Tables Define “larger” versus “smaller” portionDefine “larger” versus “smaller” portion Distribution is symmetrical, so we don’t need negative values of zDistribution is symmetrical, so we don’t 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

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

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

Probable Limits X =  + z  X =  + 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.

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

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.