Post on 06-Apr-2017
Submitted By: 2014-UETR-CS-01,03,05,15,21Submitted To: Sir Ihsan-Ul-Ghafoor
Normal distribution
NORMAL PROBABILITY DISTRIBUTIONS
The Most Important Probability Distribution in Statistics
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
A normal distribution is a continuous, symmetric, bell-shaped distribution of a variable.The normal distribution is defined as
2
2
2)(
)(2
1)(
X
eXf
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.
INCHESF
70.569.5
68.567.5
66.565.5
64.563.5
62.561.5
60.559.5
58.557.5
56.555.5
Cases weighted by PCTF
20
18
16
14
12
10
8
6
4
20
Std. Dev = 2.48 Mean = 63.7
N = 99.68
INCHESM
76.575.5
74.573.5
72.571.5
70.569.5
68.567.5
66.565.5
64.563.5
62.561.5
60.559.5
Cases weighted by PCTM
20
10
0
Std. Dev = 2.61 Mean = 69.1
N = 99.23
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(~),(~ NYZNY
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.
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
• 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 52.25.2
7.630.70
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.
1
2
3