Standardized Distributions

Statistics 2126

Introduction

• Last time we talked about measures of spread

• Specifically the variance and the standard deviation

• s and s2

• You might ask yourself “Why is this useful?”

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

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

So compare

• How many standard deviations are you from the mean?

• We call this a z score

€

z =x −μ

σ

x=8 =7 =1.5

€

z =x −μ

σ

z =8 − 7

1.5

z =1

1.5z = .67

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

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?

x=8 =7.5 =.5

€

z =x −μ

σ

z =8 − 7.5

.5

z =.5

.5z =1.00

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

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?

You can answer these questions by looking at the

formula

€

z =x −μ

σ

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

x = 118 = 100 = 15

€

z =x −μ

σ

z =118 −100

15

z =18

15z =1.2

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 = ?

Now just sub into the formula and cross multiply

€

z =x −μ

σ

1.62 =x −100

15x −100 =15(1.62)

x −100 = 24.3

x =124.3

Well this must all have a point

• Using a z table• Or this VERY cool website:• http://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.

So it looks like this