The Normal Distribution &
Standard Normal Distribution
I The Normal Distribution
A
What is it?
B
Why is it everywhere?
Probability Theory is why
C
The Skewed Normal Distribution
D
Kurtosis
II The Standard Normal Distribution
A
Standardizing a Normal Distribution
B
Computing Proportions using Table B.1
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A Normal Distribution:
Chest Sizes of Scottish Militia Men
1200
1000
800
600
400
200
0
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
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A Normal Distribution:
Histogram of Human Gestation
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The Normal Distribution: Height
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A Normal Distribution:
Age At Retirement
2000
1800
1600
1400
1200
1000
800
600
400
200
0
31-35 36-40 41-45 46-50 51-55 56-60 61-65 66-70 71-75 76-80 81-85 86-90 91-95 96-100
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Normally Distributed Variables
• The most common continuous (interval/ratio) variable type
• Occurs predominantly in nature (biology, psychology, etc.)
• Determined by the principles of Probability
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Probability and the Normal
Distribution
Probability is the Underlying Cause of the
Normal Distribution
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Possible outcomes
for four coin tosses
HHHH
HTHH
THHH
TTHH
HHHT
HTHT
THHT
TTHT
HHTH
HTTH
THTH
TTTH
HHTT
HTTT
THTT
TTTT
There are 16 possibilities because there are 2
possible outcomes for each toss and 4 tosses: 24
In general the possible outcomes are mn where m is
the number of outcomes per event and n is the
number of events
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Probability distribution of the number
of heads obtained in 4 coin tosses
No. of Heads
x
0
Probability
P(X=x)
0.0625
=
1/16
1
0.2500
=
4/16
2
0.3750
=
6/16
3
0.2500
=
4/16
4
0.0625
=
1/16
1.0000
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9
Probability distribution of the
number of heads obtained in 4
coin tosses
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
1
2
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4
10
Frequencies for the numbers of
heads obtained in 4 tosses for
1000 observations
No. of Heads
x
0
Probability
P(X=x)
0.0625
Observed
Frequency
64
1
0.2500
248
2
0.3750
392
3
0.2500
268
4
0.0625
28
1.0000
1
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(a) Probability for 4 coin flips vs.
(b) 1000 observations
0.40
400
0.35
350
0.30
300
0.25
250
0.20
200
0.15
150
0.10
100
0.05
50
0.00
0
0
1
2
3
4
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0
1
2
3
4
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Interpretation of a Normal
Distribution in terms of Probability
Consider what would happen if there were only 4 genes for height
(there are more), each of which has only 2 possible states (like heads
versus tails for a coin), call the states T for tall and S for short. The
distributions would be identical to that for the coin tosses (see left
below) with the possibility of 0, 1, 2, 3, and 4 T’s. In reality height
is controlled by many genes so that more than 5 outcomes are
possible (see right below).
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
1
2
3
4
And for 6 coins instead of 4?
0.40
0.35
0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
0.00
0.00
0
1
2
3
4
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0
1
2
3
4
5
6
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Another Example
2 Dice
Possible outcomes:
1,1 1,2 1,3 1,4
2,1 2,2 2,3 2,4
3,1 3,2 3,3 3,4
4,1 4,2 4,3 4,4
5,1 5,2 5,3 5,4
6,1 6,2 6,3 6,4
1,5
2,5
3,5
4,5
5,5
6,5
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1,6
2,6
3,6
4,6
5,6
6,6
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Another Example
x
f (x)
2
1
3
2
7
6
4
3
5
4
4
6
5
3
7
6
1
8
5
0
9
4
10
3
11
2
12
1
5
2
2
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4
5
6
7
8
9
10
11
12
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Examples of the Normal
Distribution
•
•
•
•
•
•
•
Age
Height
Weight
I.Q.
Sick Days per Year
Hours Sleep per Night
Words Read per
Minute
• Calories Eaten per
Day
• Hours of Work Done
per Day
• Eyeblinks per Hour
• Insulting Remarks per
Week
• Number of Pairs of
Socks Owned
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The Skewed Normal Distribution
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Examples of Skewed Normal
Distributions
• Income
• Number of Empty
Soda Cans in Car
• Drug Use per Week
• Car Accidents per Year
• Lifetime
Hospitalizations
• Number of Guitars
Owned
• Consecutive Days
Unemployed
• Hand-Washings per
Day
• Number of Languages
Spoken Fluently
• Hours of T.V. per Day
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Graph of a Normal Distribution
34.13% 34.13%
13.59%
13.59%
2.28%
2.28%
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Shapes of the Normal
Distribution
Kurtosis
• Leptokurtic
• Platokurtic
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Frequency and relative-frequency
distributions for heights
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What do we do with Normal
Distributions?
1. Determine the position of a given score
relative to all other scores.
2. Compare distributions.
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Relative-frequency
histogram for heights
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Comparing Two Distributions
Two distributions of exam scores. For both distributions, µ = 70,
but for one distribution, σ = 12. The position of X = 76 is very
different for these two distributions.
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Data Transformations are
Reversible and Do not Alter the
Relations Among Items
1) Add or Subtract a Constant From Each
Score
2) Multiply Each Score By a Constant
•
e.g., if you wanted to convert a group of
Fahrenheit temperatures to Centigrade you
would subtract 32 from each score then multiply
by 5/9ths
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Transforming a distribution does not change the
shape of the distribution, only its units
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0.04
0.02
0.02
0
0
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202
0.04
192
0.06
182
0.06
172
0.08
162
0.08
152
0.1
80
0.1
76
0.12
72
0.12
68
0.14
64
0.14
60
0.16
56
0.16
142
Height a) in inches b) in centimeters
inches X 2.54 = centimeters
28
Transformations
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Standard Normal Distribution
A normally distributed variable having mean 0 and standard
deviation 1 is said to have the standard normal distribution.
Its associated normal curve is called the standard normal
curve.
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Transformation to Standard Units
The idea is to transform (reversibly) any normal distribution into
a STANDARD NORMAL distribution with μ = 0 and σ = 1
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Standardized Normally Distributed
Variable
A normally distributed variable, x, is converted to a standard
normal distribution, z, with the following formula
z
x

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Standardizing normal distributions
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Standard Normal Distribution
• For a variable x, the variable (z-score)
z
x

• is called the standardized version of x or
the standardized variable corresponding to
the variable x.
• This transformation is standard for any
variable and preserves the exact
relationships among the scores
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Standard Normal Distributions
• The z-score transformation is entirely
reversible but allows any distribution to be
compared (e.g., I.Q. and SAT score; does a
top I.Q. score correspond to a top SAT
score?)
• z-scores all have a mean of zero and a
standard deviation of 1, which gives them
the simplest possible mathematical
properties.
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Standard Normal Distributions
An example of a z transformation from a
variable (x) with mean 3 and standard
deviation 2
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Understanding x and z-scores
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Basic Properties of the Standard
Normal Curve
Property 1: The total area under the standard normal curve
is equal to 1.
Property 2: The standard normal curve extends indefinitely
in both directions, approaching, but never touching, the
horizontal axis as it does so.
Property 3: The standard normal curve is symmetric about
0; that is, the left side of the curve should be a mirror image
of the right side of the curve.
Property 4: Most of the area under the standard normal
curve lies between –3 and 3.
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Finding percentages for a normally
distributed variable from areas
under the standard normal curve
Because the standard normal distribution is the same for all
variables, it is an easy way to determine what proportion of scores
is less than a, what proportion lies between a and b, and what
proportion is greater than b (for
any distribution and any desired39
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points a and b).
The relationship between z-score values
and locations in a population
distribution.
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The X-axis is relabeled in z-score units. The distance that is
equivalent to σ corresponds to 1 point on the z-score scale.
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APPENDIX B
T AB L E B.1
Statistical Tables
THE UNIT NORMAL TABLE*
—————————————————————
Table
B.1
p. 687
*Column A lists z-score values. A vertical line drawn through a normal distribution at a z-score location divides the distribution into two sections.
Column B identifies the proportion in the larger section, called the body.
Column C identifies the proportion in the smaller section, called the tail.
Column D identifies the proportion between the mean and the z-score.
Note: Because the normal distribution is symmetrical, the proportions for negative z-scores are the same as those for positive z-scores.
(A)
z
(B)
PROPORTION
IN BODY
(C)
PROPORTION
IN TAIL
(D)
PROPORTION
BETWEEN MEAN AND z
(A)
z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.24
.5000
.5040
.5080
.5120
.5160
.5199
.5239
.5279
.5319
.5359
.5398
.5438
.5478
.5517
.5557
.5596
.5636
.5675
.5714
.5753
.5793
.5832
.5871
.5910
.5948
.5000
.4960
.4920
.4880
.4840
.4801
.4761
.4721
.4681
.4641
.4602
.4562
.4522
.4483
.4443
.4404
.4364
.4325
.4286
.4247
.4207
.4168
.4129
.4090
.4052
.0000
.0040
.0080
.0120
.0160
.0199
.0239
.0279
.0319
.0359
.0398
.0438
.0478
.0517
.0557
.0596
.0636
.0675
.0714
.0753
.0793
.0832
.0871
.0910
.0948
0.25
0.26
0.27
0.28
0.29
0.30
0.31
0.32
0.33
0.34
0.35
0.36
0.37
0.38
0.39
0.40
0.41
0.42
0.43
0.44
0.45
0.46
0.47
0.48
0.49
(B)
(C)
(D)
PROPORTION PROPORTIONIN
PROPORTION
IN BODY
TAIL
BETWEENMEAN AND z
.5987
.6026
.6064
.6103
.6141
.6179
.6217
.6255
.6293
.6331
.6368
.6406
.6443
.6480
.6517
.6554
.6591
.6628
.6664
.6700
.6736
.6772
.6808
.6844
.6879
.4013
.3974
.3936
.3897
.3859
.3821
.3783
.3745
.3707
.3669
.3632
.3594
.3557
.3520
.3483
.3446
.3409
.3372
.3336
.3300
.3264
.3228
.3192
.3156
.3121
.0987
.1026
.1064
.1103
.1141
.1179
.1217
.1255
.1293
.1331
.1368
.1406
.1443
.1480
.1517
.1554
.1591
.1628
.1664
.1700
.1736
.1772
.1808
.1844
.1879
Table
B.1
A
Closer
Look
(A)
z
(B)
(C)
PROPORTION PROPORTION
IN BODY
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.24
.5675
.5714
.5753
.5793
.5832
.5871
.5910
.5948
IN TAIL
.4325
.4286
.4247
.4207
.4168
.4129
.4090
.4052
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(D)
PROPORTION
BETWEEN MEAN AND z
.0675
.0714
.0753
.0793
.0832
.0871
.0910
.0948
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The Normal Distribution:
why use a table?
P
x2

x1
1
2
2
e
 ( X   ) 2 / 2 2
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d
dx
44
From x or z to P
To determine a percentage or
probability for a normally
distributed variable
Step 1 Sketch the normal curve associated with the variable
Step 2 Shade the region of interest and mark the delimiting xvalues
Step 3 Compute the z-scores for the delimiting x-values found
in Step 2
Step 4 Use Table B.1 to obtain the area under the standard
normal curve delimited by the z-scores found in Step 3
Use Geometry and remember that the total area under
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the curve is always 1.00.
45
From x or z to P
Finding percentages for a normally
distributed variable from areas under
the standard normal curve
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Finding percentages for a normally
distributed variable from areas under the
standard normal curve
1.
,  are given.
2. a and b are any two values of the variable x.
3. Compute z-scores for a and b.
4. Consult table B-1
5. Use geometry to find desired area.
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Given that a quiz has a mean score of 14
and an s.d. of 3, what proportion of the
class will score between 9 & 16?
1.
 = 14 and  = 3.
2. a = 9 and b = 16.
3. za = -5/3 = -1.67, zb = 2/3 = 0.67.
4. In table B.1, we see that the area to the left of a is 0.0475
and that the area to the right of b is 0.2514.
5. The area between a and b is therefore
1 – (0.0475 + 0.2514) = 0.701 or 70.01%
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Finding the area under the standard
normal curve to the left of z = 1.23
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What if you start with x
instead of z?
z = 1.50: Use Column C; Anthony
P = 0.0668
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What is the
probability of
selecting a
random
student who
scored above
650 on the
SAT?
50
Finding the area under the standard
normal curve to the right of z = 0.76
The easiest way would be to use Column C, but lets use
Column B instead
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Finding the area under the standard
normal curve that lies between
z = –0.68 and z = 1.82
P = 1 – 0.0344 – 0.2483
= 0.7173
One Strategy: Start with the area to the left of 1.82, then
subtract the area to the right of -0.68.
Second Strategy: Start with 1.00 and subtract off the two
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tails
52
Determination of the percentage of
people having IQs between 115 and
140
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From x or z to P
Review of Table B.1 thus far
Using Table B.1 to find the area under the standard normal
curve that lies
(a) to the left of a specified z-score,
(b) to the right of a specified z-score,
(c) between two specified z-scores
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Then if x is asked for, convert
from
z to x
54
From P to z or x
Now the other way around
To determine the observations
corresponding to a specified
percentage or probability for a
normally distributed variable
Step 1 Sketch the normal curve associated the the variable
Step 2 Shade the region of interest (given as a probability or area
Step 3 Use Table B.1 to obtain the z-scores delimiting the region
in Step 2
Step 4 Obtain the x-values having the z-scores found in Step 3
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From P to z or x
Finding z- or x-scores corresponding
x
to a given region.
z
Finding the z-score having area 0.04 to its left
x=σ×z+μ

x  z  
If μ is 242 σ is 100, then
x = 100 × -1.75 + 242
x = 67
Use Column C:
The z corresponding to 0.04 Anthony
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left tail is -1.75
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The z
Notation
The symbol zα is used to denote the
z-score having area α (alpha) to its
right under the standard normal
curve. We read “zα” as “z sub α” or
more simply as “z α.”
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The z notation : P(X>x) = α
P(X>x)= α
This is the z-score that
demarks an area under the
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curve with P(X>x)= α
58
The z notation : P(X<x) = α
P(X<x)= α
Z
This is the z-score that
demarks an area under the
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curve with P(X<x)= α
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The z notation : P(|X|>|x|) = α
P(|X|>|x|)= α
α/2
1- α
This is the z-score that
demarks an area under the
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curve with P(|X|>|x|)= α
α/2
60
Finding z 0.025
Use Column C:
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The z corresponding to 0.025Anthony
in the
right tail is 1.96
61
Finding z 0.05
Use Column C:
The z corresponding to 0.05 Anthony
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right tail is 1.64
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Finding the two z-scores dividing the
area under the standard normal curve
into a middle 0.95 area and two outside
0.025 areas
Use Column C:
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The z corresponding to 0.025
in both
63
Finding the 90th percentile for IQs
z0.10 = 1.28
z = (x-μ)/σ
1.28 = (x – 100)/16
120.48 = x
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What you should be able to do
1. Start with z-or x-scores and compute regions
2. Start with regions and compute z- or x-scores
z
x

x  z  
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DESCRIPTIVES
EXERCISE & REVIEW
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Descriptives
1. Non-Parametric Statistics:
a) Frequency & percentile
b) Median, Range, Interquartile Range, SemiInterquartile Range
2. Parametric Statistics:
a) Mean, Variance, Standard Deviation
b) z-score & proportion
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NonParametric
Analysis
Weekly Income
540
275
680
8275
425
380
2370
4185
155
0
490
380
265
145
755
125
430
675
125
155
185
505
425
785
NonParametric
Analysis
Weekly Income Sorted Scores
540
0
275
125
680
125
8275
145
425
155
380
155
2370
185
4185
265
155
275
0
380
490
380
380
425
265
425
145
430
755
490
125
505
430
540
675
675
125
680
155
755
185
785
505
2370
425
4185
785
8275
NonParametric
Analysis
Range = H-L+1
= 8276
-or= URL-LRL
= 8275.5-(-0.5)
= 8276
Weekly Income Sorted Scores
540
0
275
125
680
125
8275
145
425
155
380
155
2370
185
4185
265
155
275
0
380
490
380
380
425
265
425
145
430
755
490
125
505
430
540
675
675
125
680
155
755
185
785
505
2370
425
4185
785
8275
NonParametric
Analysis
Q1: 25/4 or 6 ¼
Q1: ¼ of the distance
between 155 and 185
Q1 = 162.5
Q2 = 425 = median
Q3: 75/4 or 18 ¼
Q3: ¼ of the distance
between 675 and 680
Q3 = 676.25
Weekly Income Sorted Scores 25%, 50%, 75%
540
0
275
125
680
125
8275
145
425
155
380
155
155
2370
185
185
4185
265
155
275
0
380
490
380
380
425
425
265
425
425
145
430
755
490
125
505
430
540
675
675
675
125
680
680
155
755
185
785
505
2370
425
4185
785
8275
NonParametric
Analysis
Q1 = 162.5
Q2 = 425 = median
Q3 = 676.25
IR = 513.75
Weekly Income Sorted Scores 25%, 50%, 75%
540
0
275
125
680
125
8275
145
425
155
380
155
155
2370
185
185
4185
265
155
275
0
380
490
380
380
425
425
265
425
425
145
430
755
490
125
505
430
540
675
675
675
125
680
680
155
755
185
785
505
2370
425
4185
785
8275
NonParametric
Analysis
Weekly Income Sorted Scores 25%, 50%, 75% Proportion
540
0
0.00
275
125
0.08
680
125
0.08
8275
145
0.13
425
155
0.21
380
155
155
0.21
2370
185
185
0.25
4185
265
0.29
155
275
0.33
0
380
0.42
490
380
0.42
380
425
425
0.50
265
425
425
0.50
145
430
0.54
755
490
0.58
125
505
0.63
430
540
0.67
675
675
675
0.71
125
680
680
0.75
155
755
0.79
185
785
0.83
505
2370
0.88
425
4185
0.92
785
8275
0.96
NonParametric
Analysis
Weekly Income Sorted Scores 25%, 50%, 75% Proportion
540
0
0.00
275
125
0.08
680
125
0.08
8275
145
0.13
425
155
0.21
380
155
155
0.21
2370
185
185
0.25
4185
265
0.29
155
275
0.33
0
380
0.42
490
380
0.42
380
425
425
0.50
265
425
425
0.50
145
430
0.54
755
490
0.58
125
505
0.63
430
540
0.67
675
675
675
0.71
125
680
680
0.75
155
755
0.79
185
785
0.83
505
2370
0.88
425
4185
0.92
785
8275
0.96
Parametric
Analysis
(sample)
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
Parametric
Analysis

Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
823.00
Parametric
Analysis

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
823.00
41.15
Parametric
Analysis
M

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
823.00
41.15
Parametric
Analysis

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
823.00
41.15
x-M
6.85
-5.15
30.85
-37.15
-1.15
-5.15
-11.15
-7.15
-1.15
0.85
3.85
18.85
19.85
-16.15
-12.15
-0.15
3.85
13.85
-10.15
7.85
Parametric
Analysis

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
823.00
41.15
2
x-M
(x-M)
6.85
46.9225
-5.15
26.5225
30.85 951.7225
-37.15 1380.1225
-1.15
1.3225
-5.15
26.5225
-11.15 124.3225
-7.15
51.1225
-1.15
1.3225
0.85
0.7225
3.85
14.8225
18.85 355.3225
19.85 394.0225
-16.15 260.8225
-12.15 147.6225
-0.15
0.0225
3.85
14.8225
13.85 191.8225
-10.15 103.0225
7.85
61.6225
Parametric
Analysis

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
823.00
41.15
x-M
(x-M) 2
6.85
46.9225
-5.15
26.5225
30.85 951.7225
-37.15 1380.1225
-1.15
1.3225
-5.15
26.5225
-11.15 124.3225
-7.15
51.1225
-1.15
1.3225
0.85
0.7225
3.85
14.8225
18.85 355.3225
19.85 394.0225
-16.15 260.8225
-12.15 147.6225
-0.15
0.0225
3.85
14.8225
13.85 191.8225
-10.15 103.0225
7.85
61.6225
 4154.55
/(n-1)
Parametric
Analysis
SS

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
x-M
(x-M)
6.85
46.9225
-5.15
26.5225
30.85 951.7225
-37.15 1380.1225
-1.15
1.3225
-5.15
26.5225
-11.15 124.3225
-7.15
51.1225
-1.15
1.3225
0.85
0.7225
3.85
14.8225
18.85 355.3225
19.85 394.0225
-16.15 260.8225
-12.15 147.6225
-0.15
0.0225
3.85
14.8225
13.85 191.8225
-10.15 103.0225
7.85
61.6225
823.00
41.15
 4154.55
/(n-1)
2
Parametric
Analysis

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
x-M
(x-M)
6.85
46.9225
-5.15
26.5225
30.85 951.7225
-37.15 1380.1225
-1.15
1.3225
-5.15
26.5225
-11.15 124.3225
-7.15
51.1225
-1.15
1.3225
0.85
0.7225
3.85
14.8225
18.85 355.3225
19.85 394.0225
-16.15 260.8225
-12.15 147.6225
-0.15
0.0225
3.85
14.8225
13.85 191.8225
-10.15 103.0225
7.85
61.6225
823.00
41.15
 4154.55
/(n-1) 206.73
2
Parametric
Analysis
Variance

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
823.00
41.15
x-M
(x-M) 2
6.85
46.9225
-5.15
26.5225
30.85 951.7225
-37.15 1380.1225
-1.15
1.3225
-5.15
26.5225
-11.15 124.3225
-7.15
51.1225
-1.15
1.3225
0.85
0.7225
3.85
14.8225
18.85 355.3225
19.85 394.0225
-16.15 260.8225
-12.15 147.6225
-0.15
0.0225
3.85
14.8225
13.85 191.8225
-10.15 103.0225
7.85
61.6225

/(n-1)
4154.55
206.73
Parametric
Analysis

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
x-M
(x-M)
6.85
46.9225
-5.15
26.5225
30.85 951.7225
-37.15 1380.1225
-1.15
1.3225
-5.15
26.5225
-11.15 124.3225
-7.15
51.1225
-1.15
1.3225
0.85
0.7225
3.85
14.8225
18.85 355.3225
19.85 394.0225
-16.15 260.8225
-12.15 147.6225
-0.15
0.0225
3.85
14.8225
13.85 191.8225
-10.15 103.0225
7.85
61.6225
823.00
41.15
 4154.55
/(n-1) 206.73
sqrt 14.38
2
Parametric
Analysis
s

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
x-M
(x-M)
6.85
46.9225
-5.15
26.5225
30.85 951.7225
-37.15 1380.1225
-1.15
1.3225
-5.15
26.5225
-11.15 124.3225
-7.15
51.1225
-1.15
1.3225
0.85
0.7225
3.85
14.8225
18.85 355.3225
19.85 394.0225
-16.15 260.8225
-12.15 147.6225
-0.15
0.0225
3.85
14.8225
13.85 191.8225
-10.15 103.0225
7.85
61.6225
823.00
41.15
 4154.55
/(n-1) 206.73
sqrt 14.38
2
Parametric
Analysis
z

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
x-M
(x-M)
6.85
46.9225
-5.15
26.5225
30.85 951.7225
-37.15 1380.1225
-1.15
1.3225
-5.15
26.5225
-11.15 124.3225
-7.15
51.1225
-1.15
1.3225
0.85
0.7225
3.85
14.8225
18.85 355.3225
19.85 394.0225
-16.15 260.8225
-12.15 147.6225
-0.15
0.0225
3.85
14.8225
13.85 191.8225
-10.15 103.0225
7.85
61.6225
823.00
41.15
 4154.55
/(n-1) 206.73
sqrt 14.38
2
(x-M)/s
0.476
-0.358
2.146
-2.584
-0.080
-0.358
-0.775
-0.497
-0.080
0.059
0.268
1.311
1.381
-1.123
-0.845
-0.010
0.268
0.963
-0.706
0.546
Parametric
Analysis

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
x-M
(x-M)
6.85
46.9225
-5.15
26.5225
30.85 951.7225
-37.15 1380.1225
-1.15
1.3225
-5.15
26.5225
-11.15 124.3225
-7.15
51.1225
-1.15
1.3225
0.85
0.7225
3.85
14.8225
18.85 355.3225
19.85 394.0225
-16.15 260.8225
-12.15 147.6225
-0.15
0.0225
3.85
14.8225
13.85 191.8225
-10.15 103.0225
7.85
61.6225
823.00
41.15
 4154.55
/(n-1) 206.73
sqrt 14.38
2
(x-M)/s
0.476
-0.358
2.146
-2.584
-0.080
-0.358
-0.775
-0.497
-0.080
0.059 0.524
0.268
1.311
1.381
-1.123
-0.845
-0.010
0.268
0.963
-0.706
0.546
Parametric
Analysis

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
x-M
(x-M)
6.85
46.9225
-5.15
26.5225
30.85 951.7225
-37.15 1380.1225
-1.15
1.3225
-5.15
26.5225
-11.15 124.3225
-7.15
51.1225
-1.15
1.3225
0.85
0.7225
3.85
14.8225
18.85 355.3225
19.85 394.0225
-16.15 260.8225
-12.15 147.6225
-0.15
0.0225
3.85
14.8225
13.85 191.8225
-10.15 103.0225
7.85
61.6225
823.00
41.15
 4154.55
/(n-1) 206.73
sqrt 14.38
2
(x-M)/s
0.476
-0.358
2.146
-2.584 0.005
-0.080
-0.358
-0.775
-0.497
-0.080
0.059
0.268
1.311
1.381
-1.123
-0.845
-0.010
0.268
0.963
-0.706
0.546
Parametric
Analysis

/n
Hours Work
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
x-M
(x-M)
6.85
46.9225
-5.15
26.5225
30.85 951.7225
-37.15 1380.1225
-1.15
1.3225
-5.15
26.5225
-11.15 124.3225
-7.15
51.1225
-1.15
1.3225
0.85
0.7225
3.85
14.8225
18.85 355.3225
19.85 394.0225
-16.15 260.8225
-12.15 147.6225
-0.15
0.0225
3.85
14.8225
13.85 191.8225
-10.15 103.0225
7.85
61.6225
823.00
41.15
 4154.55
/(n-1) 206.73
sqrt 14.38
2
(x-M)/s
0.476
-0.358
2.146 0. 984
-2.584
-0.080
-0.358
-0.775
-0.497
-0.080
0.059
0.268
1.311
1.381
-1.123
-0.845
-0.010
0.268
0.963
-0.706
0.546
Parametric
Analysis
(population)
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
Parametric
Analysis

Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
1638.00
Parametric
Analysis

/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
1638.00
81.90
Parametric
Analysis


/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
1638.00
81.90
Parametric
Analysis

/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
1638.00
81.90
X-
-5.90
1.10
-0.90
8.10
11.10
6.10
3.10
-29.90
8.10
9.10
6.10
13.10
-20.90
8.10
18.10
11.10
-36.90
-1.90
1.10
-7.90
Parametric
Analysis

/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
1638.00
81.90
X-
(X-)2
-5.90
34.8100
1.10
1.2100
-0.90
0.8100
8.10
65.6100
11.10
123.2100
6.10
37.2100
3.10
9.6100
-29.90
894.0100
8.10
65.6100
9.10
82.8100
6.10
37.2100
13.10
171.6100
-20.90
436.8100
8.10
65.6100
18.10
327.6100
11.10
123.2100
-36.90 1361.6100
-1.90
3.6100
1.10
1.2100
-7.90
62.4100
Parametric
Analysis

/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
1638.00
81.90
X-
(X-)2
-5.90
34.8100
1.10
1.2100
-0.90
0.8100
8.10
65.6100
11.10
123.2100
6.10
37.2100
3.10
9.6100
-29.90
894.0100
8.10
65.6100
9.10
82.8100
6.10
37.2100
13.10
171.6100
-20.90
436.8100
8.10
65.6100
18.10
327.6100
11.10
123.2100
-36.90 1361.6100
-1.90
3.6100
1.10
1.2100
-7.90
62.4100
3905.80
Parametric
Analysis
SS

/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
1638.00
81.90
X-
(X-)2
-5.90
34.8100
1.10
1.2100
-0.90
0.8100
8.10
65.6100
11.10
123.2100
6.10
37.2100
3.10
9.6100
-29.90
894.0100
8.10
65.6100
9.10
82.8100
6.10
37.2100
13.10
171.6100
-20.90
436.8100
8.10
65.6100
18.10
327.6100
11.10
123.2100
-36.90 1361.6100
-1.90
3.6100
1.10
1.2100
-7.90
62.4100
3905.80
Parametric
Analysis

/N
(X-)2
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
-5.90
34.8100
1.10
1.2100
-0.90
0.8100
8.10
65.6100
11.10
123.2100
6.10
37.2100
3.10
9.6100
-29.90
894.0100
8.10
65.6100
9.10
82.8100
6.10
37.2100
13.10
171.6100
-20.90
436.8100
8.10
65.6100
18.10
327.6100
11.10
123.2100
-36.90 1361.6100
-1.90
3.6100
1.10
1.2100
-7.90
62.4100
1638.00
81.90
3905.80
195.29
X-
Parametric
Analysis
Variance

/N
(X-)2
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
-5.90
34.8100
1.10
1.2100
-0.90
0.8100
8.10
65.6100
11.10
123.2100
6.10
37.2100
3.10
9.6100
-29.90
894.0100
8.10
65.6100
9.10
82.8100
6.10
37.2100
13.10
171.6100
-20.90
436.8100
8.10
65.6100
18.10
327.6100
11.10
123.2100
-36.90 1361.6100
-1.90
3.6100
1.10
1.2100
-7.90
62.4100
1638.00
81.90
3905.80
195.29
X-
Parametric
Analysis

/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
X-
(X-)2
-5.90
34.8100
1.10
1.2100
-0.90
0.8100
8.10
65.6100
11.10
123.2100
6.10
37.2100
3.10
9.6100
-29.90
894.0100
8.10
65.6100
9.10
82.8100
6.10
37.2100
13.10
171.6100
-20.90
436.8100
8.10
65.6100
18.10
327.6100
11.10
123.2100
-36.90 1361.6100
-1.90
3.6100
1.10
1.2100
-7.90
62.4100
1638.00
81.90
sqrt
3905.80
195.29
13.97
Parametric
Analysis


/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
X-
(X-)2
-5.90
34.8100
1.10
1.2100
-0.90
0.8100
8.10
65.6100
11.10
123.2100
6.10
37.2100
3.10
9.6100
-29.90
894.0100
8.10
65.6100
9.10
82.8100
6.10
37.2100
13.10
171.6100
-20.90
436.8100
8.10
65.6100
18.10
327.6100
11.10
123.2100
-36.90 1361.6100
-1.90
3.6100
1.10
1.2100
-7.90
62.4100
1638.00
81.90
sqrt
3905.80
195.29
13.97
(X-)/ 
Parametric
Analysis
z

/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
X-
(X-)2
-5.90
34.8100
1.10
1.2100
-0.90
0.8100
8.10
65.6100
11.10
123.2100
6.10
37.2100
3.10
9.6100
-29.90
894.0100
8.10
65.6100
9.10
82.8100
6.10
37.2100
13.10
171.6100
-20.90
436.8100
8.10
65.6100
18.10
327.6100
11.10
123.2100
-36.90 1361.6100
-1.90
3.6100
1.10
1.2100
-7.90
62.4100
1638.00
81.90
sqrt
3905.80
195.29
13.97
(X-)/ 
-0.422
0.079
-0.064
0.580
0.794
0.437
0.222
-2.140
0.580
0.651
0.437
0.937
-1.496
0.580
1.295
0.794
-2.641
-0.136
0.079
-0.565
Parametric
Analysis

/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
X-
(X-)2
-5.90
34.8100
1.10
1.2100
-0.90
0.8100
8.10
65.6100
11.10
123.2100
6.10
37.2100
3.10
9.6100
-29.90
894.0100
8.10
65.6100
9.10
82.8100
6.10
37.2100
13.10
171.6100
-20.90
436.8100
8.10
65.6100
18.10
327.6100
11.10
123.2100
-36.90 1361.6100
-1.90
3.6100
1.10
1.2100
-7.90
62.4100
1638.00
81.90
sqrt
3905.80
195.29
13.97
(X-)/ 
-0.422
0.079
-0.064
0.580
0.794
0.437
0.222
-2.140
0.580
0.651
0.437
0.937
-1.496
0.580
1.295
0.794
-2.641
-0.136
0.079
-0.565
0.476
Parametric
Analysis
What proportion
of scores is
below 45?
0.004
Above?
0.996

/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
X-
(X-)2
(X-)/ 
-5.90
34.8100
1.10
1.2100
-0.90
0.8100
8.10
65.6100
11.10
123.2100
6.10
37.2100
3.10
9.6100
-29.90
894.0100
8.10
65.6100
9.10
82.8100
6.10
37.2100
13.10
171.6100
-20.90
436.8100
8.10
65.6100
18.10
327.6100
11.10
123.2100
-36.90 1361.6100
-1.90
3.6100
1.10
1.2100
-7.90
62.4100
-0.422
0.079
-0.064
0.580
0.794
0.437
0.222
-2.140
0.580
0.651
0.437
0.937
-1.496
0.580
1.295
0.794
-2.641
-0.136
0.079
-0.565
3905.80
195.29
13.97
0.00
1638.00
81.90
sqrt
0.004
Parametric
Analysis

/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
X-
(X-)2
(X-)/ 
-5.90
34.8100
1.10
1.2100
-0.90
0.8100
8.10
65.6100
11.10
123.2100
6.10
37.2100
3.10
9.6100
-29.90
894.0100
8.10
65.6100
9.10
82.8100
6.10
37.2100
13.10
171.6100
-20.90
436.8100
8.10
65.6100
18.10
327.6100
11.10
123.2100
-36.90 1361.6100
-1.90
3.6100
1.10
1.2100
-7.90
62.4100
-0.422
0.079
-0.064
0.580
0.794
0.437
0.222
-2.140
0.580
0.651
0.437
0.937
-1.496
0.580
1.295
0.794
-2.641
-0.136
0.079
-0.565
3905.80
195.29
13.97
0.00
1638.00
81.90
sqrt
0.902
Parametric
Analysis
What proportion
of scores is
between 100 and
45?
0.902 – 0.004
= 0.898

/N
Exam Score
76
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
80
83
74
X-
(X-)2
(X-)/ 
-5.90
34.8100
1.10
1.2100
-0.90
0.8100
8.10
65.6100
11.10
123.2100
6.10
37.2100
3.10
9.6100
-29.90
894.0100
8.10
65.6100
9.10
82.8100
6.10
37.2100
13.10
171.6100
-20.90
436.8100
8.10
65.6100
18.10
327.6100
11.10
123.2100
-36.90 1361.6100
-1.90
3.6100
1.10
1.2100
-7.90
62.4100
-0.422
0.079
-0.064
0.580
0.794
0.437
0.222
-2.140
0.580
0.651
0.437
0.937
-1.496
0.580
1.295
0.794
-2.641
-0.136
0.079
-0.565
3905.80
195.29
13.97
0.00
1638.00
81.90
sqrt
0.902
0.004
What z-score corresponds to the Top 10%?
1.28
What z-scores correspond to the Middle 60%?
±0.84
Anthony J Greene
108
Given a mean of 58 and a st. dev. Of 10, what is the
likelihood of randomly being between 55 and 65?
Can use column D: 0.1179
+0.2580
Anthony
J Greene = 0.3759
109
Given a mean of 58 and a st. dev. Of 10, what is the
likelihood of randomly being between 65 and 75?
Anthony J Greene
Can use column C: 0.2420
-0.0446 = 0.1974
110
What Scores are in the Top 15%?
Use column C
z = 1.04
x = z ×σ + μ
Anthony J Greene
x = 604
111
What Scores are the middle 80%?
Can Use Column D: z = ±1.28
x
z

x  z   
x  1.28100 500
x  128 500
x  628
x  628, 372
Anthony J Greene
x  128 500
x  372
112
What is the percent of the population that lies below 114?
Anthony
Use Column
B: JzGreene
= 1.40; P = 0.9192
113
What is the percentile rank of x = 92?
Anthony
Use Column
C: JzGreene
= -0.80; P = 0.2119
114
What x score corresponds to the bottom 34%?
Use Column B: z = - .41
x=z×σ+μ
x = -0.41 × 5 + 60 = 57.95
Anthony J Greene
115