1. Standardized Distributions Statistics 2126

2. Introduction Last time we talked about measures of spread Specifically the variance and the standard deviation s and s 2 You might ask yourself “Why is this useful?”

3. So, what did you get? Say you are comparing your quiz marks with other people in the class Let’s say you got 8 And the class average was 7 That is a population mean, we are considering the class to be a population so  = 7

4. What did you get, in relation to others By how much are you better than the class average By 1…. If everyone got say below you, you rock This is where the population standard deviation or  comes into play Let’s say  = 1.5

5. So compare How many standard deviations are you from the mean? We call this a z score

6. x=8  =7  =1.5

7. So what does that mean? It means you are.67 standard deviations away from the mean. We now have a measure of how far away you are from a mean We call this a standard score Let’s say you get 8 on the next quiz But now the class mean is 7.5

8. Change it up a little Now let’s say the standard deviation is.5 So now on this quiz the scores were packed much more tightly Did you do relatively better on the first quiz or on the second one?

9. x=8  =7.5  =.5

10. So compare the two You did better on the second quiz than you did on the first one You are 1 standard deviation from the mean You are simply comparing the two z scores

11. Properties of z It can be negative or positive If you are off to the left of the mean you will get a negative score If you are off to the right, your z score will be positive What is the shape? What is the average z score? What is the standard deviation?

12. You can answer these questions by looking at the formula

13. An example IQ has a mean of 100 and a standard deviation of 15 N(100,15) That just means it is normal with a mean of 100 and a sd of 15 So what is the z score of someone with an IQ of 118

14. x = 118  = 100  = 15

15. You could go the other way too So say someone had a z score of 1.62 What is their IQ? Well again just list what you know z = 1.62  = 100  = 15 x = ?

16. Now just sub into the formula and cross multiply

17. Well this must all have a point Using a z table Or this VERY cool website: http://davidmlane.com/hyperstat/z_table.htmlhttp://davidmlane.com/hyperstat/z_table.html So if you know the z, you can find out what the probability of getting a z score at a certain level is.

18. So it looks like this