Exam PerformanceGrade DistributionMean83.2Median84.5Std. Dev.9.5Mean Dev.7.4Min61Max96Homework and Exam PerformanceExam RulesReview today in classReturn at the end of classYou may come see them in my office at your leisureOur Friend, the Normal DistributionFrequency DistributionsHow do we find the number of students who score between 4 and 6, inclusive?Add! Score 4 = 5 StudentsScore 5 = 3 StudentsScore 6 = 2 Students = 10 Students ScoreFreq.What do we do with Continuous Scores?Frequencies get messyDistribution is not clearWe need something ElseProbability DistributionsHow do we find the probability of a student scoring between 4 and 6, inclusive?Add! P(4) = .2P(5) = .1P(6) = .05 = .35 Students ScoreFreq.Continuous Probability DistributionsWhat is the probability of scoring 3.141592654?Virtually zero!When it is continuous, we need to find the probability of scoring in a range.Normal Distribution CurveThe normal distribution can also be a Probability Distribution.Family of Normal CurvesAll in family are “frequency distributions” which conform to the 68-95-99.7 rule. The means and standard deviations of different distributions differ but the symmetry holds.Changing Standard DeviationsWhatever the mean and std dev: If the distribution is normally distributed the 68-95-99.7 rule applies. At 68%, two-thirds of all the cases fall within + 1 standard deviation of the mean, 95% of the cases within + 2 standard deviation of the mean, and 99.7% of the cases within + 3 sd of the mean.How do we do it?We have our 68-95-99.7 RuleWe just have to know how many standard deviations a certain number is away from the meanExampleSAT scores are normally distributed with a mean of 500 and a std. dev. of 100. What percentage of students score between 400 and 600?68%PracticeSAT scores are normally distributed with a mean of 500 and a std. dev. of 100. What percentage of students score between 400 and 500?34%PracticeSAT scores are normally distributed with a mean of 500 and a std. dev. of 100. What percentage of students score less than 300?2.5%IQ scores are normally distributed with a mean of 100 and a std. dev. of 15. What percentage of people have an IQ between 85 and 115?It works with different and 68%PracticeIQ scores are normally distributed with a mean of 100 and a std. dev. of 15. What percentage of people have an IQ between 85 and 100?34%PracticeIQ scores are normally distributed with a mean of 100 and a std. dev. of 15. What percentage of people have an IQ less than 70?2.5%Problem:By manipulating probabilities, we can only handle situations where we are 1, 2, or 3 Std. Devs. Away from the mean.What happens when we want to know the probability of scoring IQ between 100 and 105We need to convert the IQ score (or SAT score or whatever) into units of the standard Deviation.Example: Distance between 100 and 105 is .333 Standard DeviationsThink of the Std. Dev. As a UnitHow many inches are in a foot?12How many cups are in a pint?2How many IQ points are there in a standard deviation for IQ?15How many SAT points are there in a standard deviation for SAT scores?100How do you convert inches to feet? Distance in feet = Distance in inches 12Distance in IQ std devs = Distance in IQ points 100Distance in IQ std. devs = Consider this problemParty-time employee salaries in a company are normally distributed with mean $20,000 and Standard Dev. $1,000 How many Std. Devs. Is $18,500 away from the mean?Intuitively, we see that 1,500 is 1.5 Std. Devs. from Using the formula, we get-1.5 (negative specifies direction)?Consider this problemHow many Std. Devs. Is $19,371 away from the mean?Intuitively, we cant do thisUsing the formula, we get= -.269 Std. devs. awayX =19,371?Z ScoresWe call these standard deviation values “Z-scores”Z score is defined as the number of standard units any score or value is from the mean.Z score states how many standard deviations the observation X falls away from the mean and in which direction plus or minus.What Good does this do?Someone figured out that 68% are within + 1 s.d. and about 95% are within + 2 s.d.Someone did this to show that 74.16% are within + 1.13 s.d. in the normal distribution1.14 s.d = 74.58%1.15 s.d = 74.98%1.16 s.d = 75.4%It goes on and on and on.These results appear in a “Z-table”You calculate a Z score, find that score in column A and the Z-table will tell youThe probability of getting a score between your Z-score and the mean (column B)The probability of getting a score greater than your Z-score, that is, from your Z-score out to the end of the normal distribution (column C)This Table can be downloaded from my web siteIt Looks like thisSuppose you find a Z-score of .12Column B says that 4.78% of cases lie between the mean and your Z-scoreIt Looks like thisSuppose you find a Z-score of .12Column C says that 45.22% of cases lie beyond your Z-scoreColumn CIQ is normally distributed with a mean of 100 and sd of 15.How do you interpret a score of 109?Use Z score What does this Z-score .60 mean?Does not mean 60 percent of cases below this score BUT rather that this Z score is .60 standard units above the mean, We need the Z-table to interpret this!Using the Z tableLook at Column C for .60 Only 27.43% of people have an IQ higher than this.If your IQ is 109 (.6 s.d. above the mean), you are smarter than almost 75% of people in the world!72.57% of people have an IQ less than this.USEFULNESS OF Z SCOREDescribe scores relative to other scores in a single distribution when we divide the deviation by the standard deviation. The Z score is the probability of getting a particular value in any normal distribution.Can make comparisons across different normal distributions, across different samples of individuals or different groups. The Z score standardizes all NDCs, makes all NDCs comparable even when the means are different and standard deviations are different.