Part III
Taking Chances for Fun and Profit
Chapter 8   
Are Your Curves Normal? Probability and
Why it Counts
0900 Quiz #3 N=26
 2|1389
 3|01112333335669
 4|00012334
 X-bar=34.62; Median=13th and 14th dp=33
Mode=33;
 S=6.03;
1030 Quiz #3 N=33
 2|0355678899
 3|033334668899
 4|00111223455
 X-bar=34.73; Median=33+1/2=17th dp=36;
 Mode=33;
s= 7.02;
Frequency distribution: 900 quiz
scores
Freq
CF
RF
CRF
21 – 24
2
2
.077
.077
25 – 28
1
3
.038
.115
29 – 32
6
9
.231
.346
33 – 36
8
17
.308
.654
37 – 40
4
21
.154
.808
41 – 44
5
26
.182
1.00
What you will learn in Chapter 7
 Understanding probability is basic to
understanding statistics
 Characteristics of the “normal” curve

i.e. the bell-shaped curve
 All about z scores


Computing them
Interpreting them
Why Probability?
 Basis for the normal curve

Provides basis for understanding probability of
a possible outcome
 Basis for determining the degree of
confidence that an outcome is “true”

Example:

Are changes in student scores due to a particular
intervention that took place or by chance along?
The Normal Curve
(a.k.a. the Bell-Shaped Curve)
 Visual representation of a distribution of
scores
 Three characteristics…



Mean, median, and mode are equal to one
another
Perfectly symmetrical about the mean
Tails are asymptotic (get closer to horizontal
axis but never touch)
The Normal Curve
Hey, That’s Not Normal!
 In general, many events occur right in the
middle of a distribution with few on each end.
More Normal Curve 101
More Normal Curve 101
 For all normal distributions…

almost 100% of scores will fit between -3 and
+3 standard deviations from the mean.

So…distributions can be compared

Between different points on the X-axis, a
certain percentage of cases will occur.
What’s Under the Curve?
The z Score
 A standard score that is the result of dividing
the amount that a raw score differs from the
mean of the distribution by the standard
deviation.
(X  X )
z
,
s
 What about those symbols?
The z Score
 Scores below the mean are negative (left of
the mean) and those above are positive (right
of the mean)
A
z score is the number of standard
deviations from the mean
 z scores across different distributions are
comparable
What z Scores Represent
 The areas of the curve that are covered by
different z scores also represent the
probability of a certain score occurring.
 So try this one…

In a distribution with a mean of 50 and a
standard deviation of 10, what is the
probability that one score will be 70 or above?
Why Use Z scores?
• Percentages can be used to compare
different scores, but don’t convey as much
information
• Z scores also called standardized scores,
making scores from different distributions
comparable; Ex: You get two different scores
in two different subjects(e.g Statistics 28 and
English 76). They are not yet comparable, so
lets turn them into percentages( e.g
28/35=80% and 76/100, 76%). Relatively you
did better in statistics.
Percentages Verse Z scores
• How do you compare to others? From
percentages alone, you have no way of
knowing. Say µ on English exam was =70
with ó of 8 pts, your 76 gives you a z-score of
.75, three-fourths of one stand deviation
above the mean; Mean on statistics test is 21,
with ó of 5 pts; your score of 28 gives a z
score of 1.40 standard deviations above
mean; Although English and statistics scores
were similar, comparing z scores shows you
did much better in statistics
Using z scores to find percentiles
• Prof Oh So Wise, scores 142 on an evaluation. What
is Wise’s percentile ranking? Assume profs’ scores
are normally distributed with µ of 100 and ó of 25.
X-µ
142-100
z= 1.68
ó
25
Area under curve ‘Small Part’ = .0465, equals those
who scored above the prof;
1–.0465= 95.35th percentile. Oh so wise is in top 5% of
all professors. Not bad at all.
Never use from ‘mean to z’ to find percentile!! We’re
only concerned with scores above or below a certain
rank
Starting with An Area Under Curve
and Finding Z and then X…
 Using the previous parameters of µ of 100
and ó of 25, what score would place a
professor in top 10% of this distribution? After
some algebra, we have X=µ+z (ó)
 100(µ) + 1.28(z)(25)(ó)=132 (X). A score of
132 would place a professor in top 10 %;
 What scores place a professor in most
extreme 5% of all instructors?
What does ‘most extreme’ mean?
 It is not just one end of the distribution, but
both ends, or 2.5% at either end;
 X= µ + z(ó)= 100+ 1.96(25)= 149
 100 +-1.96(25)=51; 51 and 149 place a
professor at the most extreme 5 % of the
distribution;
The Difference between z scores
What z Scores Really Represent
 Knowing the probability that a
z score will
occur can help you determine how extreme a
z score you can expect before determining
that a factor other than chance produced the
outcome
 Keep in mind…
z scores are typically
reserved for populations. 
Hypothesis Testing & z Scores
 Any event can have a probability associated
with it.


Probability values help determine how
“unlikely” the even might be
The key --- less than 5% chance of occurring
and you have a significant result
Some rules regarding normal
distribution
 Percentiles – if raw score is below the mean
use’ small part’ to find percentile ; if raw score
is above the mean, use’ big part’ to find
percentile; check to see that you’re right by
constructing a frequency distribution and
identifying cumulative percentage
 If raw scores are on opposite sides of the
mean, add the areas/percentages. If raw
scores are on same side of mean, subtract
areas/percentages
Using the Computer
 Calculating z Scores
Glossary Terms to Know
 Probability
 Normal curve

Asymptotic
 Standard Scores

z scores