1、Exam PerformanceGrade DistributionMean83.2Median84.5Std.Dev.9.5Mean Dev.7.4Min61Max9605101520Percent60708090100var6Homework and Exam Performance60708090100Exam Performance0.2.4.6.81Average Homework PercentFitted valuesvar6Exam Rules Review today in class Return at the end of class You may come see t
2、hem in my office at your leisureOur Friend,the Normal DistributionFrequency Distributions How do we find the number of students who score between 4 and 6,inclusive?Add!Score 4=5 Students Score 5=3 Students Score 6=2 Students =10 Students ScoreFreq.What do we do with Continuous Scores?Frequencies get
3、 messy Distribution is not clear We need something ElseProbability Distributions How do we find the probability of a student scoring between 4 and 6,inclusive?Add!P(4)=.2 P(5)=.1 P(6)=.05 =.35 Students ScoreFreq.Continuous Probability Distributions What is the probability of scoring 3.141592654?Virt
4、ually 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
5、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
6、+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 Rule We just have to know how many standard deviations a certain number is away from the meanExample SAT scores are normally distributed with a mean of 500 and a std.dev.of 100.What
7、 percentage of students score between 400 and 600?68%Practice SAT 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%Practice SAT scores are normally distributed with a mean of 500 and a std.dev.of 100.What percentage of s
8、tudents 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 X68%Practice IQ scores are normally distributed with a mean of 100 and a std.dev.of 15.What percentage of peop
9、le have an IQ between 85 and 100?34%Practice IQ 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 happ
10、ens when we want to know the probability of scoring IQ between 100 and 105 We 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 Unit How many inches are in a foot?12 How
11、many cups are in a pint?2 How many IQ points are there in a standard deviation for IQ?15 How 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 st
12、d.devs=xConsider this problem Party-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)x18,50
13、0 20,0001,5001,0001,000?Consider this problem How many Std.Devs.Is$19,371 away from the mean?Intuitively,we cant do this Using the formula,we get=-.269 Std.devs.away19,371 20,0002691,0001,000X=19,371?Z Scores We call these standard deviation values “Z-scores”Z score is defined as the number of stand
14、ard 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
15、.16%are within+1.13 s.d.in the normal distribution 1.14 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 you The probability of getting a score between your Z-score and the
16、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 this Suppose you find a Z-score of.12 Column B says that 4.78%of cases lie between the me
17、an and your Z-scoreIt Looks like this Suppose you find a Z-score of.12 Column 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 109 1009.61515xzWhat does this Z-score.60 mean?Does not m
18、ean 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 table Look 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 alm
19、ost 75%of people in the world!72.57%of people have an IQ less than this.USEFULNESS OF Z SCORE Describe 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.
