Statistics

1. Submitted By:
2014-UETR-CS-01,03,05,15,21
Submitted To:
Sir Ihsan-Ul-Ghafoor
Normal distribution

3. NORMAL PROBABILITY
DISTRIBUTIONS
The Most Important Probability
Distribution in Statistics

4.  The normal probability
distribution ,which is
considered the cornestone of the
mode statistical theory , was
discover by
 Abraham de moivre (1667-1754)
As the limiting form of the binomial
distribution
 It is also called Gaussian
distribution in honor of the great
German mathemation
 Carl F.Gauss (1777-1855)
Abraham de moivre

5. A normal distribution is a continuous,
symmetric, bell-shaped distribution of
a variable.
The normal distribution is defined as
2
2
2
)(
)(
2
1
)( σ
µ
πσ
−
−
=
X
eXf

7.  The normal curve is bell-shaped and has a single
peak at the exact center of the distribution.
 The normal distribution is symmetrical about
its mean.
 The normal distribution is asymptotic - the
curve gets closer and closer to the x-axis but
never actually touches it.
 The arithmetic mean, median, and mode of the
distribution are equal and located at the peak.

8. INCHESF
70.5
69.5
68.5
67.5
66.5
65.5
64.5
63.5
62.5
61.5
60.5
59.5
58.5
57.5
56.5
55.5
Cases weighted by PCTF
20
18
16
14
12
10
8
6
4
2
0
Std. Dev = 2.48
Mean = 63.7
N = 99.68
INCHESM
76.5
75.5
74.5
73.5
72.5
71.5
70.5
69.5
68.5
67.5
66.5
65.5
64.5
63.5
62.5
61.5
60.5
59.5
Cases weighted by PCTM
20
10
0
Std. Dev = 2.61
Mean = 69.1
N = 99.23

9.  A normal distribution with a mean of 0 and a
standard deviation of 1 is called the standard
normal distribution.
 The distance between a selected value is X, and
the population mean , divided by the
population standard deviation,
)1,0(~),(~ N
Y
ZNY
σ
µ
σµ
−
=⇒

10.  The monthly incomes of recent MBA
graduates in a large corporation are
normally distributed with a mean of
$2000 and a standard deviation of $200.
What is the Z value for an income of
$2200? An income of $1700?
A Z value of 1 indicates that the value of
$2200 is 1 standard deviation above the
mean of $2000, while a Z value of $1700 is
1.5 standard deviation below the mean of
$2000.

11.  Professor Mann has determined that the final averages
in his statistics course is normally distributed with a
mean of 72 and a standard deviation of 5. He decides
to assign his grades for his current course such that the
top 15% of the students receive an A. What is the
lowest average a student can receive to earn an A?
 Let X be the lowest average. Find X such that P(X
>X)=.15. The corresponding Z value is 1.04. Thus we
have (X-72)/5=1.04, or X=77.2

12. • What is the probability arandomly selected female is 5’10” or
taller (70 inches)?
• Step 1 - Y ~ N(63.7 , 2.5)
• Step 2 - YL = 70.0 YU = ∞
• Step 3 -
∞==
−
= UL ZZ 52.2
5.2
7.630.70

13.  About 68 percent of the area under the normal
curve is within one standard deviation of the
mean.
 About 95 percent is within two standard
deviations of the mean.
 99.74 percent is within three standard
deviations of the mean.
σµ 1±
µ σ±2
µ σ± 3