I.
The Normal Curve
A.
Important in behavioral sciences.
1.
Many variables of interest are approximately normally distributed.
2.
Statistical inference tests have sampling distributions which become
normally distributed as sample size increases.
3.
Many statistical inference tests require sampling distributions that are
normally distributed.
B.
Characteristics
1.
Symmetrical, bellshaped curve.
2.
Equation:
Shows that the curve is asymptotic to the abscissa; i.e., it approaches the
X
axis and gets closer and closer but never touches it.
3.
Area contained under the normal curve:
a.
Area under the curve represents the percentage of scores contained
within the area.
b.
34.13% of scores between mean (
μ
) and +1
σ
; 13.59% of area
contained between a score equal to
μ
+ 1
σ
and a score of
μ
+ 2
σ
;
2.15% of area is between
μ
+ 2
σ
and
μ
+ 3
σ
; and 0.13% falls beyond
μ
+ 3
σ
.
c.
Since the curve is symmetrical, the same percentages hold for scores
below the mean.
II.
Standard Scores (
z
Scores)
A.
Converts raw scores into standard scores symbolized as
z
.
1.
Definition. A standard score is a transformed score which designates how
many standard deviation units the corresponding raw score is above or
below the mean.
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2.
Equation:
B.
Comparisons between different distributions.
1.
Allows comparisons even when the units of the distributions are different.
2.
Percentile ranks are possible.
C.
Characteristics of
z
scores.
1.
z
scores have the same shape as the set of raw scores from which they
were transformed.
2.
μ
z
= 0. The mean of
z
scores equals zero.
3.
σ
z
= 1.00. The standard deviation of
z
scores equals 1.00.
D.
Using
z
scores.
1.
Finding the percentage or frequency (area) corresponding to any raw
score.
z
= (
X

μ
)
/ σ
Use above formula to calculate z score. Then use table to determine the
area under the normal curve for the various values of z.
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 Summer '09
 ShahramGhiasinejad
 Normal Distribution, Standard score

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