Download The Normal Distribution

The Normal Distribution
Data can be "distributed" (spread out)
in different ways.
It can be spread out
more on the left
Or more on the right
Or it can be all jumbled up
The Normal Distribution
A normal distribution is a very important statistical
data distribution pattern occurring in many natural
and social phenomena.
The Normal Distribution
A normal distribution is characterized by its mean (μ) and
variance (σ2)
f(X)
Changing μ shifts the
distribution left or right.
σ
µ
Changing σ increases
or decreases the
spread.
X
A normal distribution is characterized by its mean (μ) and
variance (σ2)
- Changing μ shifts the distribution left or right.
- Changing σ increases or decreases the spread.
f(X)
σ
µ
X
The Normal Distribution
- Mean
- Variance
-  Standard
deviation
-  Standard
error of
the mean
1 n
µ = ! xi
n i
n
1
2
var(x) = ! =
(xi " µ )
!
n "1 i
n
1
2
std(x) = var(x) = ! =
(x
"
µ
)
!
i
i
n "1
!
SEM =
n
2
The Normal Distribution:
as mathematical function (pdf)
f ( x) =
1
σ 2π
Note constants:
π=3.14159
e=2.71828
1 x−µ 2
− (
)
⋅e 2 σ
This is a bell shaped
curve with different
centers and spreads
depending onσand
onμ
The Normal PDF
It s a probability density function (PDF), so no matter
what the values of σand μ, must integrate to 1!
+∞
∫σ
−∞
1
2π
1 x−µ 2
− (
)
⋅ e 2 σ dx
=1
The beauty of the normal curve:
The area between µ-σ and µ+σ is about 68%; the area between
µ-2σ and µ+2σ is about 95%; and the area between µ-3σ and µ+3σ
is about 99.7%. Almost all values fall within 3 standard deviations.
The beauty of the normal curve:
The area between µ-σ and µ+σ is about 68%; the area between
µ-2σ and µ+2σ is about 95%; and the area between µ-3σ and µ+3σ
is about 99.7%. Almost all values fall within 3 standard deviations.
The beauty of the normal curve:
The area between µ-σ and µ+σ is about 68%; the area between
µ-2σ and µ+2σ is about 95%; and the area between µ-3σ and µ+3σ
is about 99.7%. Almost all values fall within 3 standard deviations.
The Standard Normal Distribution (Z)
All normal distributions can be converted into
the standard normal curve by subtracting the
mean and dividing by the standard deviation:
Z=
X −µ
σ
Somebody calculated all the integrals for the standard
normal and put them in a table! So we never have to
integrate!
Even better, computers now do all the integration.
The Normal Distribution
1 n
µ = ! i xi
n
n
1
2
!=
(x
!
µ
)
"
i
n !1 i
μ-3σμ-2σ μ-σ
μ+3σ
μ+2σ
μ+σ
μ
Z=
-3
-2
-1
0
1
2
3
X −µ
Z- Score
σ