Dr. Lakey – Diekhoff (1996) – Chapter 4
PROBABILITY, STANDARD SCORES,
AND THE NORMAL CURVE
4.1 The Normal Distribution
Theoretical versus Empirical Distributions
Theoretical Normal or Gaussian Distribution
(1) Complete Continuous: Between Any Two Scores is Another
Score
(2) Infinite Sample or Population
(3) Goes to both Positive and Negative Infinity
Normal Distribution Approximated by Many
Variables?
(IQ and Personality Traits)
Required Assumption of Many Statistical Tests
but
“Robust Tests”
Normal Central Limits Phenomenon!
Example: One Die  Six Dice
(Rectangular  Triangular  Normal Distribution)
Shape of the Normal Distribution
Bell-Shaped ( Alexander G. Bell or Karl F. Gauss d. 1855)
Symmetrical (No Skew or No Asymmetry)
Mesokurtic (Kurtosis)
Figure 4.1 (p93)
Mean -- 34.13% --1-- 13.59% --2-- 2.14% --3-- 0.13% -4-- 0.01%
Usually Only 3 ( 34.13% -- 13.59% -- 2.27%)
Area Under Curve = Percentage of Cases = Probability of Cases
in a Random Draw from the Distribution
4.2 Rules of Probability
P(A) = f(A) / N
P(A) = Probability of A Outcomes
f(A) = Relative Frequency of A Outcomes
N = All Possible Outcomes
Example: 12 males and 18 females in a class of 30 students.
Example: 4 aces in a deck of 52 cards.
Venn Diagram
(1) List All Possible Outcomes (Usually the Trick!)
(2) Identify All A “Set” Outcomes
(3) Divide (2) by (1)
Normal Distribution = Venn Diagram
Example:
P(Drawing a Case between +1 and +2) = f(A) / N
= 2.14 / 100 = .0214
Converse Rule
P(NOT Drawing a Case between +1 and +2) =
1  P(Drawing a Case between +1 and +2)
= 1  .0214 = .9786
P(A) = 1  P(A) where letter overline = “NOT”
Probability Concepts
 Mutually Exclusive Outcomes:
Only One or the Other Event can Occur -- but not Both.
(1) Draw an Ace [P(A)] or a King [P(B)] from a Deck of Cards
(mutually exclusive)
(2) But Draw an Ace [P(A)] or a Spade [P(B)]
from a Deck of Cards
(not mutually exclusive: possible “joint” occurrence [P(AB)])
P(AB) = “the joint probability of A and B”
 Independent Outcomes:
The Occurrence of One Event does not change the Probability
of the Other Event.
Draw two successive Aces with replacement (independent)
Draw two successive Aces without replacement
(not independent: second draw is a “conditional” probability)
Addition Rule
P(A or B or C) = P(A) + P(B) + P(C)
when A, B and C are mutually exclusive.
P(A or B or C) = P(A) + P(B) + P(C)  P(AB)  P(AC) 
P(BC)  P(ABC)
when A, B or C are not mutually exclusive.
For Four Possible Outcomes?
P(A or B or C or D) = P(A) + P(B) + P(C) + P(D)  P(AB) 
P(AC)  P(AD)  P(BC)  P(BD)  P(CD)  P(ABC)  P(ABD) 
P(ACD)  P(BCD)  P(ABCD) !!! ???
For An Easier Two Possible Outcomes:
P(A or B) = P(A) + P(B)  P(AB)
where P(AB) = “the joint probability of A & B”
 Draw an Ace or a King:
P(A or B) = 4/52 + 4/52 – 0/52 = 2/52 = .154
 Draw an Ace or a Space:
P(A or B) = 4/52 + 13/52 – 1/52 = 16/52 = .308
Multiplication Rule
P(A and B and C) = P(A) x P(B) x P(C)
when A, B and C are independent.
P(A and B and C) = P(A) x P(BA) x P(CAB)
when A, B or C are not independent.
For Four Possible Outcomes?
P(A and B and C and D) = P(A) x P(BA) x P(CAB) x
P(CABD )
For An Easier Two Possible Outcomes:
P(A and B) = P(AB) = P(A) x P(BA)
where BA = B “given” A has occurred.
 Draw two successive aces with replacement:
P(AB) = 4/52 x 4/52 = .0059
 Draw two successive aces without replacement:
P(AB) = 4/52 x 3/51 = .0045
4.3 Standard Scores
Important: The Stock & Trade of Psychology
Ordinal Scale (Percentile Ranks)  Interval Scale (z-Scores)
See Figure 4.1 Again (p93)
z = (X  X)  s
The score’s difference from the mean in SD units.
Examples:
Given IQ Test with Mean = 100 and s = 15
 Find z-Score for an IQ of 115?
z = (115  100)  15 = +1.00
 Find z-Score for an IQ of 90?
z = (90  100)  15 = 0.67
Characteristics of Standard z-Scores
(1) z-Score transformation distribution SAME SHAPE
as raw score distribution.
(2) z-Score mean always zero (Xz = 0.00).
(3) z-Score standard deviation always one (sz = 1.00).
Some Uses of Standard Scores
 Locating Scores in a Distribution
Ordinal Percentile Ranks  Interval z-Scores
See Figure 4.1 Again (p93) Note Percentile “Rubber Ruler”
ACTs and SATs push Percentile Ranks
for ease of understanding
 Comparing Scores on Different Variables
Is Dirty Harry dirtier or hairier?
Given Male Hairiness: Mean = 8.3 hairs/square inch, SD =
2.6 hairs/square inch.
Given Male Dirtiness: Mean = 1.7 g dirt, SD = 0.5 g dirt.
Dirty Hairy: 11.0 hairs/square inch; 3.1 g dirt.
z = (11.0  8.3)  2.6 = +1.04 for Hairiness
z = ( 3.1  1.7)  0.5 = +2.08 for Dirtiness
Harry is dirtier than hairier than the comparison group.
Is Harry twice as dirtier as hairier?
(Need Ratio Scale data to answer this question)
4.4 Standard Normal Distribution
See Figure 4.1 Again (p93)
Mean = 0.00 and SD = 1.00
See Table 1 in Appendix B (p399-402)
Three Columns
(A) z-Score
(B) Area Between Mean and z-Score
(C) Area Beyond z-Score
Study Figure 4.2 (p105) and Figure 4.3 (p107)
Examples:
 For z = 1.00: What is the area above the mean?
What is the percentile rank?
What percentage of cases are higher?
 For z = 1.96: What is the area above the mean? What is
area below the mean?
What is the total area?
What percentage of cases is outside this central area?
 For z = 2.58: What is the area above the mean?
What is area below the mean?
What is the total area?
What percentage of cases is outside this central area?
 What percentage of cases fall between z-scores of +0.20 and
+.40? [Fig. 4.3c]
 What percentage of cases fall between z-scores of 0.50 and
0.30?

ESTIMATING PERCENTILES RANKS:
Percentile X  z-score  PR
 Most competitive Graduate Programs have student bodies
that average 1400+ on the GRE: What is Percentile Rank of
their average student?
X = 700+
?.?? = ( 700 – 500 ) / 100
z = +?.??  PR = 97.72+
ESTIMATING PERCENTILES (SCORES):
PR  z-score  Percentile X
 Most Graduate Programs require GRE scores at the 85
percentile rank or higher: How high do you have to score on
each test (what is the percentile score)?
PR = 85.00  z-score
?.?? = ( X – 500 ) / 100
X = 604
4.5 Data Distributions That Are Not Normal?
Table 4.1 p114
Cheating Justification Questionnaire
Empirical %c versus Theoretical Normal (Est. PR)
Very close “goodness of fit”!
What if “significantly different”?
Large samples help (next chapter).