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Normal Distributions & z score Dec. 9 Normal Distribution • • • • Tend to cluster around the mean Symmetrical bell-shaped frequency polygon Mode, median, mean all occur at the highest point, in the center of distribution • 3 units of SD to the left of the mean and 3 units of SD to the right of the mean • 99.74% of polygon area lies btw –3SD and +3SD Skewed Distribution • A large percentage the values is on either left or right side • Positively skewed – Has a tail to the right • Negatively skewed – Has a tail to the left z scores • Used to compare values of interval or ratio –level variables – Rita’s score of 21 out of 24 vs. Mary’s score 85 out of 100 • Variables should be normally distributed • z score equals raw score minus mean divided by SD • Can be converted to percentiles • Known as standard scores Areas of Normal Curve Table Examples of z Scores & corresponding Areas and percentiles z score .12 1.78 -2.90 1.15 -1.15 row 0.1 1.7 2.9 1.1 1.1 column .02 .08 .00 .05 .05 area 04.78 46.25 49.81 37.49 37.49 percentile 54.78 96.25 .19 87.49 12.51 Exercise 1. On an IQ test with a mean of 100, SD of 16, what is the percentile of IQ 104? • Debora leads a treatment group of college students diagnosed as chronic anxiety. Group members have been selected for treatment on the basis their scores on Anxiety Scale A with a mean of 70 and SD of 10 or Scale B with mean of 50 and SD of 12. Gina had score of 78 on the Scale A and Tom received 66 on the Scale B. There is only one vacancy in the group. Who should be in the treatment group? Deriving Raw Scores from Percentiles • Lauren wishes to form a treatment group of college students with high anxiety levels using the results of testing with Scale B (mean=50, SD=12). She wants to include the top 10% of all students measured by Scale B. What is the cutoff point? • Subtract the percentile of mean (50) from 90 to determine the area of a normal curve that would fall between the 90th percentile and the mean • Find the z score corresponding to the area • Use the z score formula, add 50 if the z score is positive Exercise One hundred students took mid-term exam designed to measure knowledge of statistics. Their scores on it formed a normal distribution with a mean of 80 and a standard deviation of 6. Instructor decided to assign A to the top 10% of the scores. What score does John have to get to earn? Exercise • 89,56,45,78,98,45,55,77,88,99,98,97,54,34,94 • 77,88,87,67,98,87,55,77,45,44,88,99,69,67,98 • Calculate the mode, median, mean, range, variance, and standard deviation for both sections. Which section did better overall on the exam?