The name of the statistic: Normal Distribution
Alternative name: Gaussian distribution in honor of Carl Friedrich Gauss


I. Overview of Normal Distribution
A. Definition
B. Characteristics
C. Examples
D. Importance
E. Applications
F. Assumptions

II. Test strengths and weaknesses

III. Statistical formula

IV. Examples

V. Additional resources

VI. References
Appendices

I. Overview
Definition (http://www.childrens-mercy.org/stats/definitions/norm_dist.htm)

What is normal distribution?
Normal distributions are a family of distributions that have the same general shape. They are symmetric with scores more concentrated in the middle than in the tails. Normal distributions are sometimes described as bell shaped.
The simplest case of normal distribution, known as the Standard Normal Distribution, has expected value zero and variance one. This is written as N (0, 1).

Characteristics (http://www.onlinemathlearning.com/normal-distribution.html)
• It is a continuous distribution
• It is symmetrical about the mean. Each half of the distribution is a mirror image of the other half.
• It is asymptotic to the horizontal axis.
• It is unimodal.
• The area under the curve is 1.

Examples
The normal distribution has application in many areas of business administration as examples:
• Modern portfolio theory commonly assumes that the returns of a diversified asset portfolio follow a normal distribution.
• In human resource management, employee performance sometimes is considered to be normally distributed.

Importance and Application

Why is it important? , and when is it used? (http://www.stat.wvu.edu/SRS/Modules/Normal/normal.html)
• Many things are normally distributed, or very close to it. For example, height and intelligence are approximately normally distributed; measurement errors also often have a normal distribution.
• The normal distribution is easy to work with mathematically. In many practical cases, the methods developed using normal theory work quite well even when the distribution is not normal.
• There is a very strong connection between the size of a sample N and the extent to which a sampling distribution approaches the normal form. Many sampling distributions based on large N can be approximated by the normal distribution even though the population distribution itself is not normal.
• The normal distribution is the most used statistical distribution. The principal reasons are: normality arises naturally in many physical, biological, and social measurement situations.
In addition, normality is important in statistical inference.

Assumptions
What are the rules for? (http://www.netmba.com/statistics/distribution/normal/)
The normal distribution can be completely specified by two parameters:
This is written as N (0, 1).
• Mean
• Standard deviation
If the mean and standard deviation are known, then one essentially knows as much as if one had access to every point in the data set.
The empirical rule is a handy quick estimate of the spread of the data given the mean and standard deviation of a data set that follows normal distribution.
• 68% of the data will fall within 1 standard deviation of the mean
• 95% of the data will fall within 2 standard deviations of the mean
• Almost all (99.7% ) of the data will fall within 3 standard deviations of the mean

II. Test strengths and weaknesses
The strengths of normal distribution are:
• probably the most widely known and used of all distributions.
• infinitely divisible probability distributions.
• strictly stable probability distributions.
The weakness of normal distributions is for reliability calculations. In this case, using the normal distribution starts at negative infinity. This case is able to result in negative values for some of the results (http://www.weibull.com/LifeDataWeb/characteristics_of_the_normal_distribution.htm) .

III. Statistical formula
How do you use this formula?
We usually work with the standardized normal distribution, where μ = 0 and σ = 1, N (0, 1).
1. We first convert the problem into an equivalent one dealing with a normal variable measured in standardized deviation units, called a standardized normal variable. To do this, if X ∼ N (μ, σ5), then Z = X- μ/ σ ~N (0, 1)
2. A table of standardized normal values (from the table in the first appendix of this paper) can then be used to obtain an answer in terms of the converted problem.
3. If necessary, we can then convert back to the original units of measurement. To do this, simply note that, if we take the formula for Z, multiply both sides by σ, and then add μ to both sides, we get X= Z σ+ μ
4. The interpretation of Z values is straightforward. Since σ = 1, if Z = 2, the corresponding X value is exactly 2 standard deviations above the mean. If Z = -1, the corresponding X value is one standard deviation below the mean. If Z = 0, X = the mean, i.e. μ.

IV. Examples
Couple Husband age (X) Wife age (Y)
1 33 30
2 32 29
3 26 28
4 25 23
5 27 26
6 30 25
7 27 26
8 28 25
9 29 29
10 39 36
11 27 28
12 35 36
13 20 23
14 28 25
15 34 35
16 31 26
17 36 32
18 32 31
19 28 25
20 32 32
Source: data adapted from homework 5 of CEP 932
As a result, the research can learn about the shape of normal distribution though SPSS program.

Figure_1..jpg

figure_2..jpg

V. Additional resources
http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalLesson.htm
http://www.coventry.ac.uk/ec/~styrrell/pages/norex.htm
http://www.oswego.edu/~srp/stats/z.htm

VI. References
http://www.onlinemathlearning.com/normal-distribution.html
http://www.netmba.com/statistics/distribution/normal/
http://www.childrens-mercy.org/stats/definitions/norm_dist.htm
http://www.weibull.com/LifeDataWeb/characteristics_of_the_normal_distribution.htm

Appendices

Area between 0 and z

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549
0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015
1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177
1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441
1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857
2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890
2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916
2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936
2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964
2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990

Normal Distribution with a curb
figure3.jpg