Introduction to Statistics inPsychologyPSY 201Professor Greg FrancisLecture 08normal distributionpercentilesOld and new.NORMAL DISTRIBUTIONSwhen the distribution is a normaldistribution, we can describe thedistribution by just specifying• Mean: X• Standard deviation: s• Noting it is a normal distributionthat’s all we need!That’s part of our goal: describedistributions2STANDARD NORMALassume you have a standard normaldistribution (don’t worry about whereit came from)Y =1√2πe−z2/2-4 -2 0 2 4Score00.10.20.30.4if your distribution is normal, you can create astandard normal by converting to z-scores3USEsame as all other distributionsidentify key aspects of the datapercentilespercentile rankproportion of scores within a range...make it easier to interpret datasignificance!4AREA UNDER CURVEproportional to the frequency of scoreswithin the designated endpointssuppose you want to know theproportion of scores between the meanand another score (z-score)-4 -2 0 2 4Score00.10.20.30.45AREA UNDER CURVEsolving for the area requires calculusand numerical analysis(ack!)fortunately, we can also use tables (orcomputers, calculators)our textbook provides a normaldistribution calculator6CALCULATORthe calculator can compute:• Area between z1and z2• Area above z• Area outside z1and z2• Area below z7TABLEyou do not always have a calculatorlike this (e.g., on exams)you need to use a table to find thesame informationseveral different stylesTables•T-3Table entry for z is thearea under thestandard normal curveto the left of z.ProbabilityzTABL E AStandard normal probabilities(continued)z .00 .01 .02 .03 .04 .05 .06 .07 .08 .090.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .53590.1.5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .57530.2.5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .61410.3.6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .65170.4.6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .68790.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .72240.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .75490.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .78520.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .81330.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .83891.0.8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .86211.1.8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .88301.2.8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .90151.3.9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .91771.4.9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .93191.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .94411.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .95451.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .96331.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .97061.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .97672.0.9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .98172.1.9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .98572.2.9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .98902.3.9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .99162.4.9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .99362.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .99522.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .99642.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .99742.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .99812.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .99863.0.9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .99903.1.9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .99933.2.9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .99953.3.9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .99973.4.9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998Integre Technical Publishing Co., Inc. Moore/McCabe November 16, 2007 1:29p.m. moore page T-38TABLEtable only includes positive z-scores.Why?how would you find the area betweenz = −0.3 and z =2.4?Draw a picture!!!!!!!-4 -2 0 2 4Score00.10.20.30.49TABLEhow would you find the area betweenz = −0.3 and z =2.4?Look up area below −0.3 = 0.3821subtract from 0.5 = 0.1179Look up area below 2.4 = 0.9918subtract 0.5 from this area = 0.4918Add the areas = 0.610-4 -2 0 2 4Score00.10.20.30.410TABLEhow would you find the area beyondz =1.4?Look up area below 1.4 = 0.9192Recognize that the total area underthe curve is 1.0.Subtract the area from 1.01.0 - 0.9192 = 0.0808-4 -2 0 2 4Score00.10.20.30.411HINTSworking with these tables requiresexperiencehttp://psych-www.colorado.edu/∼mcclella/java/normal/tableNormal.htmlyou can learn short-cuts to answersalways draw a graph to sort out whatyou are looking foryou can never get a negative area!12PROPORTIONSsuppose you have 250 scores from atest that are normally distributedyou want to know how many scores arebetween 1.0 standard deviationsbelow the mean and 1.5 standarddeviations above the meantwo steps1. calculate the area under the stan-dard normal betweenz = −1.0 and z =1.5.2. convert the area under the curve tonumber of scores13PROPORTIONSLook up area below −1.0 = 0.1587subtract from 0.5 to get the area between 0and -1 = 0.3413Look up area below 1.5 = 0.9332subtract 0.5 to get area between 0 and 1.5 =0.4332Add the areas = 0.7745-4 -2 0 2 4Score00.10.20.30.414PROPORTIONSthis means that 77.45% of the scoreslie between one standard deviationbelow the mean and 1.5 standarddeviations above the meanso how many scores are in that range?multiply the total number of scores(250) with the percent in the range(decimal form)(0.7745) × (250) = 193.625 ≈ 19415PROPORTIONSsuppose you have 250 scores from atest that are normally distributedyou want to know how many scores arebelow 0.5 standard deviations abovethe mean, and how many scores arebeyond 2.5 standard deviationsabove the mean.two steps1. calculate the area under the stan-dard normal below z =0.5 and abovez =2.5.2. convert the area under the curve tonumber of scores16PROPORTIONSLook up area below 0.5 = 0.6915Look up area below −2.5 = 0.0062by symmetry, the area beyond 2.5 willalso be 0.0062Add the areas0.6915 + 0.0062 = 0.6977-4 -2 0 2 4Score00.10.20.30.417PROPORTIONSthis means that 69.77% of the scoreslie below 0.5 standard deviation abovethe mean or beyond 2.5 standarddeviations above the meanso how many scores are in that range?multiply the total number of scores(250) with the percent in the range(decimal form)(0.6997) × (250)