Introduction to Statistics in Psychology PSY 201 Professor Greg Francis Lecture 08 normal distribution percentiles NORMAL DISTRIBUTIONS STANDARD NORMAL when the distribution is a normal distribution we can describe the distribution by just specifying assume you have a standard normal distribution don t worry about where it came from 2 1 Y e z 2 2 Mean X Standard deviation s Noting it is a normal distribution Old and new 0 4 0 3 that s all we need 0 2 That s part of our goal describe distributions 0 1 0 4 2 0 2 4 Score if your distribution is normal you can create a standard normal by converting to z scores 2 3 USE AREA UNDER CURVE AREA UNDER CURVE same as all other distributions proportional to the frequency of scores within the designated endpoints solving for the area requires calculus and numerical analysis ack identify key aspects of the data percentiles percentile rank suppose you want to know the proportion of scores between the mean and another score z score proportion of scores within a range 0 4 0 3 make it easier to interpret data significance fortunately we can also use tables or computers calculators our textbook provides a normal distribution calculator 0 2 0 1 0 4 2 0 2 4 Score 4 5 6 CALCULATOR TABLE TABLE the calculator can compute you do not always have a calculator like this e g on exams table only includes positive z scores Why you need to use a table to find the same information how would you find the area between z 0 3 and z 2 4 Area between z1 and z2 Area above z Area outside z1 and z2 several different styles Integre Technical Publishing Co Inc Area below z Moore McCabe November 16 2007 1 29 p m moore Draw a picture page T 3 Tables T 3 0 4 Probability 0 3 Table entry for z is the area under the standard normal curve to the left of z 0 2 z TABLE A Standard normal probabilities continued 7 z 00 01 02 03 04 05 06 07 08 09 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 3 1 3 2 3 3 3 4 5000 5398 5793 6179 6554 6915 7257 7580 7881 8159 8413 8643 8849 9032 9192 9332 9452 9554 9641 9713 9772 9821 9861 9893 9918 9938 9953 9965 9974 9981 9987 9990 9993 9995 9997 5040 5438 5832 6217 6591 6950 7291 7611 7910 8186 8438 8665 8869 9049 9207 9345 9463 9564 9649 9719 9778 9826 9864 9896 9920 9940 9955 9966 9975 9982 9987 9991 9993 9995 9997 5080 5478 5871 6255 6628 6985 7324 7642 7939 8212 8461 8686 8888 9066 9222 9357 9474 9573 9656 9726 9783 9830 9868 9898 9922 9941 9956 9967 9976 9982 9987 9991 9994 9995 9997 5120 5517 5910 6293 6664 7019 7357 7673 7967 8238 8485 8708 8907 9082 9236 9370 9484 9582 9664 9732 9788 9834 9871 9901 9925 9943 9957 9968 9977 9983 9988 9991 9994 9996 9997 5160 5557 5948 6331 6700 7054 7389 7704 7995 8264 8508 8729 8925 9099 9251 9382 9495 9591 9671 9738 9793 9838 9875 9904 9927 9945 9959 9969 9977 9984 9988 9992 9994 9996 9997 5199 5596 5987 6368 6736 7088 7422 7734 8023 8289 8531 8749 8944 9115 9265 9394 9505 9599 9678 9744 9798 9842 9878 9906 9929 9946 9960 9970 9978 9984 9989 9992 9994 9996 9997 5239 5636 6026 6406 6772 7123 7454 7764 8051 8315 8554 8770 8962 9131 9279 9406 9515 9608 9686 9750 9803 9846 9881 9909 9931 9948 9961 9971 9979 9985 9989 9992 9994 9996 9997 5279 5675 6064 6443 6808 7157 7486 7794 8078 8340 8577 8790 8980 9147 9292 9418 9525 9616 9693 9756 9808 9850 9884 9911 9932 9949 9962 9972 9979 9985 9989 9992 9995 9996 9997 5319 5714 6103 6480 6844 7190 7517 7823 8106 8365 8599 8810 8997 9162 9306 9429 9535 9625 9699 9761 9812 9854 9887 9913 9934 9951 9963 9973 9980 9986 9990 9993 9995 9996 9997 5359 5753 6141 6517 6879 7224 7549 7852 8133 8389 8621 8830 9015 9177 9319 9441 9545 9633 9706 9767 9817 9857 9890 9916 9936 9952 9964 9974 9981 9986 9990 9993 9995 9997 9998 8 0 1 0 4 2 0 2 4 Score 9 TABLE TABLE HINTS how would you find the area between z 0 3 and z 2 4 how would you find the area beyond z 1 4 working with these tables requires experience Look up area below 0 3 0 3821 Look up area below 1 4 0 9192 http psych www colorado edu mcclella java normal tableNormal html Recognize that the total area under the curve is 1 0 you can learn short cuts to answers Subtract the area from 1 0 1 0 0 9192 0 0808 always draw a graph to sort out what you are looking for subtract from 0 5 0 1179 Look up area below 2 4 0 9918 subtract 0 5 from this area 0 4918 Add the areas 0 610 0 4 you can never get a negative area 0 4 0 3 0 3 0 2 0 2 0 1 0 1 0 0 4 4 2 0 2 2 0 4 10 2 4 Score Score 11 12 PROPORTIONS PROPORTIONS PROPORTIONS suppose you have 250 scores from a test that are normally distributed Look up area below 1 0 0 1587 this means that 77 45 of the scores lie between one standard deviation below the mean and 1 5 standard deviations above the mean subtract from 0 5 to get the area between 0 and 1 0 3413 you want to know how many scores are between 1 0 standard deviations below the mean and 1 5 standard deviations above the mean subtract 0 5 to get area between 0 and 1 5 0 4332 two steps Add the areas 0 7745 Look up area below 1 5 0 9332 0 4 1 calculate the area under the standard normal between z 1 0 and z 1 5 2 convert the area under the curve to number of scores so how many scores are in that range multiply the total number of scores 250 with the percent in the range decimal form 0 3 0 7745 250 193 625 194 0 2 0 1 0 4 2 0 2 4 Score 13 14 15 PROPORTIONS PROPORTIONS PROPORTIONS suppose you have 250 scores from a test that are normally distributed Look up area below 0 5 0 6915 this means that 69 77 of the scores lie below 0 5 standard deviation above the mean or beyond 2 5 standard deviations above the mean you want to know how many …