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1、Exam Performance,Grade Distribution,Homework and Exam Performance,Exam Rules,Review today in classReturn at the end of classYou may come see them in my office at your leisure,Our Friend,the Normal Distribution,Frequency Distributions,How do we find the number of students who score between 4 and 6,in
2、clusive?Add!Score 4=5 StudentsScore 5=3 StudentsScore 6=2 Students=10 Students,Score,Freq.,What do we do with Continuous Scores?,Frequencies get messyDistribution is not clearWe need something Else,Probability Distributions,How do we find the probability of a student scoring between 4 and 6,inclusiv
3、e?Add!P(4)=.2P(5)=.1P(6)=.05=.35 Students,Score,Freq.,Continuous Probability Distributions,What 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 Curve,The normal distribution can also be a Probab
4、ility Distribution.,Family of Normal Curves,All 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 Deviations,Whatever the mean and std dev:If the distribution is nor
5、mally 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 ma
6、ny standard deviations a certain number is away from the mean,Example,SAT 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%,Practice,SAT scores are normally distributed with a mean of 500 and a std.dev.of 100.What perce
7、ntage 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 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 b
8、etween 85 and 115?,It works with different and,68%,Practice,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 100?,34%,Practice,IQ scores are normally distributed with a mean of 100 and a std.dev.of 15.What percentage of peo
9、ple 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 un
10、its of the standard Deviation.Example:Distance between 100 and 105 is.333 Standard Deviations,Think of the Std.Dev.As a Unit,How 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 deviatio
11、n for SAT scores?100,How 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 problem,Party-time employee salaries in a company are normally distributed with mean$20,000 and Standard Dev.$1,000 Ho
12、w 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 problem,How many Std.Devs.Is$19,371 away from the mean?,Intuitively,we cant do thisUsing the formula,we get=-.269 Std.devs.away,
13、X=19,371,?,Z Scores,We 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
14、 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
15、 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
16、 web site,It Looks like this,Suppose you find a Z-score of.12Column B says that 4.78%of cases lie between the mean and your Z-score,It Looks like this,Suppose you find a Z-score of.12Column C says that 45.22%of cases lie beyond your Z-score,Column C,IQ is normally distributed with a mean of 100 and
17、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 table,Look at Column C for.60 Only 27.43%of p
18、eople 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 SCORE,Describe scores relative to other scores in a single distribution when we divide the deviation by the stan
19、dard 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.,
