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)21
169
S(x-m)2 = SSS(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
Rela
tive
Freq
uenc
y
Raw ScoreZ- 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
Rela
tive
Freq
uenc
y
Raw ScoreZ- 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 = 65N = 30
sm-
=xz 80 - 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