Normal Distribution PPT
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Transcript Normal Distribution PPT
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