This preview shows pages 1–3. Sign up to view the full content.
Normal Distributions
Page 1 of
8
What makes a
normal distribution
normal? This is just another way of saying we
should see something that looks like a
bellshaped continuous curve
.
Each of the curves to the right has that characteristic.
If we are talking about heights of men in a certain group, say
the Bambutu pygmy tribe in Africa, we are not surprised that
the average height of an adult male is about 51 inches. Most
adults have a height near that size while a few are
somewhat taller or shorter. By contrast, in the Tutsi tribe,
also called the
Watusi
, heights average near a spectacular
7 feet tall!
Neither of these populations is
normal
in height to our way of
thinking, but each tribe’s
distribution of heights is normal
.
C
Mathematically a curve is normal when it demonstrates a symmetry with scores more
concentrated in the middle than in the tails.
C
Normal curves are defined by two parameters: the
mean
(
μ
) and the
standard deviation
(
σ
).
C
The more normal a curve is the closer the
median value
1
is to the
mean
(or vice versa). They
also get very close to the absolute maximum of the normal curve.
C
Many kinds of behavioral data are approximated well by a normal distribution. Many statistical
tests assume a normal distribution. Most of those tests work well even if the distribution is only
approximately normal as long as the distribution does not deviate greatly from normality.
We should be able to find a
mean
(
μ
) and a
standard deviation
(
σ
) so that a normally distributed
population can be modeled by or fitted to
.
2
2
()
2
2
1
2
x
fx
e
That function has all the characteristics mentioned before and has become the standard equation for
a
normal curve
. After that, everything we have ever said about a
continuous probability
distribution
holds true.
C
The area under a normal curve is one.
Hence,
.
2
2
2
2
1
1
2
x
e
C
No probability of an event can exceed one and all probabilities must be positive values.
Hence,
where
are both
2
2
2
2
1
0(
)
(
)
1
2
x
bb
aa
Pa x b
f x
e
ab
reasonable outcomes in the experiment.
C
Since
is continuous, the probability of any specific result, say a height of exactly 6 feet,
f
x
doesn’t really come from the function directly. The value
describes one of infinitely many
6
x
1
This is the value that splits the data into to halves with 50% above and 50% below.
Copyright 2010  ASU School of Mathematical and Statistical Sciences (Terry Turner)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Normal Distributions
Page 2 of
8
Normal Distribution
z 
Table
Z
0.00
0.01
0.0 0.5000
0.5040
0.1 0.5398
0.5438
0.2 0.5793
0.5832
0.3 0.6179
0.6217
0.4 0.6554
0.6591
0.5
0.6915
0.6950
0.6 0.7257
0.7291
0.7 0.7580
0.7611
0.8 0.7881
0.7910
0.9 0.8159
0.8186
1.0 0.8413
0.8438
1.1 0.8643
0.8665
1.2 0.8849
0.8869
1.3 0.9032
0.9049
1.4 0.9192
0.9207
1.5
0.9332
0.9345
1.6 0.9452
0.9463
1.7 0.9554
0.9564
1.8 0.9641
0.9649
1.9 0.9713
0.9719
2.0
0.9772
0.9778
points.
So
technically and is infinitesimally small realistically. We should more correctly
(6
)
0
Px
ask what is the probability of a height between say 5.999 feet and 6.001 feet. Then we can use
integration
to find that probability.
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 06/16/2011 for the course MAT 211 taught by Professor Seal during the Summer '08 term at ASU.
 Summer '08
 SEAL
 Normal Distribution

Click to edit the document details