THE NORMAL DISTRIBUTION AND Z-SCORESAreas Under the Curve

Let’s Practice! x: { 3, 8, 1}

Find s: x381

m444

(x- )m-14-3

(x- )m 2

1169

(S x- )m 2 = SS(S x- )m 2 = 26

(S x- )m 2

N√ =√(26/3)

= 2.94OR

SS = Sx2

(Sx)2

N__ __

x381

x2

9641 74 - (144/3) = 26

Then √(26/3) = 2.94

Sx = 12Sx2 = 74

The Philosophy of Statistics & Standard Deviation

N=50

The Philosophy of Statistics & Standard Deviation

.24

.20

.16

.12

.08

.04

Proportion

The Philosophy of Statistics & Standard Deviation

.24

.20

.16

.12

.08

.04

Proportion

Standard Deviation and Distribution Shape

IQ

Example: IQs of Sample of Psychologists

ID IQ

1 128

2 155

3 135

4 134

5 144

6 101

7 167

8 198

9 94

10 128

11 155

12 145

= +0.13, “normal”

= +2.07, abnormally high

= -1.66, low side of normal

z(144) = [ 144 – 140.33]/ 27.91

z(198) = [ 198 – 140.33]/ 27.91

z(94) = [ 94 – 140.33]/ 27.91

x - x z =

s x = 140.33

s = 27.91

With some simple calculation we find:

Forward and reverse transforms

Example: If population μ = 120 and σ =20Find the raw score associated with a z-score of 2.5

x = 120 + 2.5(20)x = 120 + 50x = 170

“forward”

x - mz = s

x - xz = s

population

sample

Raw score Z-score

“reverse”

x = m + z s

x = x + z s

Z- score Raw Score

Why are z-scores important?• z-scores can be used to describe how normal/abnormal scores

within a distribution are

• With a normal distribution, there are certain relationships between z-scores and the proportion of scores contained in the distribution that are ALWAYS true.

1. The entire distribution contains 100% of the scores2. 68% of the scores are contained within 1 standard deviation below and above the mean3. 95% of the scores are contained within 2 standard deviations below and above the mean

The Normal Curve

0

0.005

0.01

0.015

0 32 64 96 128 160 192 224 256

Rel

ativ

e Fr

eque

ncy

Raw Score

Z- Score -4 -3 -2 -1 0 1 2 3 4

• What percentage of scores are contained between 96 and 160?• What percentage of scores are between 128 and 160?• If I have a total of 200 scores, how many of them are less than 128?

Z-score -4 -3 -2 -1 0 1 2 3 4

m= 128s = 32

68%95%

The Normal Curve

0

0.005

0.01

0.015

0 32 64 96 128 160 192 224 256

Rel

ativ

e

Fre

qu

ency

Raw Score

Z- Score -4 -3 -2 -1 0 1 2 3 4 Z-score -4 -3 -2 -1 0 1 2 3 4

m= 128s = 32

Table A in appendix D contains the areas under the normal curve indexed by Z-score.

From these tables you can determine the numberof individuals on either side of any z-score.

But how do we find areas associated with z-scores that are not simply0, 1, or 2?

What proportion of people got a z score of 1.5 or higher?

z-score

1.5

Examples of AREA C

2.3 -1.7

What percentage of people have a z-score of 0 or greater? What percentage of people have a z-score of 1 or greater? What percentage of people have a z-score of -2.5 or less? What percentage of people have a z-score of 2.3 or greater? What percentage of people have a z-score of -1.7 or less?

50%15.87%.62%

1.07%4.46%

Examples of AREA B

What percentage of people have a z-score between 0 and 1?What percentage of people have a z-score between 0 and 2.3? What percentage of people have a z-score between 0 and -2.4? What percentage of people have a z-score between 0 and 1.27? What percentage of people have a z-score between 0 and 1.79? What percentage of people have a z-score between 0 and -3.24?

34.13%48.93%49.18%39.80%46.33%49.94%

What percentage of people have a z-score of 1 or less? 84.13%

Areas which require a COMBINATION of z-scores

What percentage of people have a z-score between -1 and 2.3?84.13%

Areas which require a COMBINATION of z-scores

What percentage of people have a z-score of 1 or less? 84.13%

Areas which require a COMBINATION of z-scores

What percentage of people have a z-score of 1 or less? 84.13%

Areas which require a COMBINATION of z-scores

What percentage of people have a z-score of 1 or less? 84.13%

Areas which require a COMBINATION of z-scores

What percentage of people have a z-score of 1 or less? 84.13%

Areas which require a COMBINATION of z-scores

What percentage of people have a z-score of -1.7 or less? 4.46%

What z-score is required for someone to be in the bottom 4.46%? -1.7

What percentage of people have a score of 73.6 or less? 4.46%?

128 + (-1.7)32

m= 128s = 32

128 - 54.473.6 or below

Raw Score 0 32 64 96 128 160 192 224 256

What score is required for someone to be in the bottom 4.46%?

What z-score is required for someone to be in the top 25%?What z-score is required for someone to be in the top 5%? What z-score is required for someone to be in the bottom 10%? What z-score is required for someone to be in the bottom 70%? What z-score is required for someone to be in the top 50%? What z-score is required for someone to be in the bottom 30%?

.681.65

-1.29.52

0-.53

What percentage of scores fall between the mean and a score of 132?

Here, we must first convert this raw score to a z-score in order to be able to use what we know about the normal distribution. (132-128)/32 = 0.125, or rounded, 0.13.

Area B in the z-table indicates that the area contained between the mean and a z-score of .13 is .0517, which is 5.17%

m= 128s = 32

What percentage of scores fall between a z-score of -1 and 1.5?

If we refer to the illustration above, it will require two separate areas added together in order to obtain the total area:

Area B for a z-score of -1: .3413Area B for a z-score of 1.5: .4332

Added together, we get .7745, or 77.45%

m= 128s = 32

What percentage of scores fall between a z-score of 1.2 and 2.4?

Notice that this area is not directly defined in the z-table. Again, we must use two different areas to come up with the area we need. This time, however, we will use subtraction.

Area B for a z-score of 2.4: .4918Area B for a z-score of 1.2: .3849When we subtract, we get .1069, which is 10.69%

m= 128s = 32

If my population has 200 people in it, how many people have an IQbelow a 65?

First, we must convert 65 into a z-score: (65-128)/32 = -1.96875, rounded = -1.97Since we want the proportion BELOW -1.97, we are looking for Area C of a z-score of 1.97 (remember, the distribution is symmetrical!) : .0244 = 2.44%

Last step: What is 2.44% of 200?200(.0244) = 4.88

m= 128s = 32

What IQ score would I need to have in order to make it to the top 5%?

Since we’re interested in the ‘top’ or the high end of the distribution, we want to find an Area C that is closest to .0500, then find the z-score associated with it.

The closest we can come is .0495 (always better to go under). The z-score

associated with this area is 1.65.Let’s turn this z-score into a raw score: 128 + 1.65(32) = 180.8

m= 128s = 32

A possible type of test question:A class of 30 students takes a difficult statistics exam. The average grade turns out to be 65. Michael is a student in this class. His grade on the exam is 80. The following is known:

SS = 2883.2 Assuming that these 30 students make up the population of interest, what is the approximate number of people that did better than Michael on the exam?

SS= 2883.2m = 65

N = 30

x

z80 - 65

s = √SS N

= √2883.2 30 = √96.11 = 9.80

9.80

z(80) = (80-65)/9.80 = 1.53Area C for a z score of 1.53 = .0630, so about 6.3%, or 1.89 people