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

https://en.wikipedia.org/wiki/File:Normal_Distribution_PDF.svg

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.

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