Normal Distribution Curve

What is the Normal Distribution?
(Bell Curve)

In many natural processes, random variation conforms to a particular probability distribution known as the normal distribution, which is the most commonly observed probability distribution. Mathematicians de Moivre and Laplace used this distribution in the 1700's. In the early 1800's, German mathematician and physicist Karl Gauss used it to analyze astronomical data, and it consequently became known as the Gaussian distribution among the scientific community.

The shape of the normal distribution resembles that of a bell, so it sometimes is referred to as the "bell curve", an example of which follows:

Normal Distribution

The above curve is for a data set having a mean of zero. In general, the normal distribution curve is described by the following probability density function:

Bell Curve Characteristics

The bells curve has the following characteristics:

• Symmetric
• Unimodal
• Extends to +/- infinity
• Area under the curve = 1

Completely Described by Two Parameters

The normal distribution can be completely specified by two parameters:

• 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.

Normal Distribution and the Central Limit Theorem

The normal distribution is a widely observed distribution. Furthermore, it frequently can be applied to situations in which the data is distributed very differently. This extended applicability is possible because of the central limit theorem, which states that regardless of the distribution of the population, the distribution of the means of random samples approaches a normal distribution for a large sample size.

Applications to Business Administration

The normal distribution has applications in many areas of business administration. For example:

• Modern portfolio theory commonly assumes that the returns of a diversified asset portfolio follow a normal distribution.
• In operations management, process variations often are normally distributed.
• In human resource management, employee performance sometimes is considered to be normally distributed.

The normal distribution often is used to describe random variables, especially those having symmetrical, unimodal distributions. In many cases however, the normal distribution is only a rough approximation of the actual distribution. For example, the physical length of a component cannot be negative, but the normal distribution extends indefinitely in both the positive and negative directions. Nonetheless, the resulting errors may be negligible or within acceptable limits, allowing one to solve problems with sufficient accuracy by assuming a normal distribution.

How to solve Z-score, raw scores, and areas in the Normal Distribution?

          Actually, we can solve the areas z-value ans raw scores using the

·         Empirical rule (usually for approximation)

·         Graphing Calculator

·         Solving by scientific Calculator and Z-table

We are going to focus for the Empirical rule and Solving

THE EMPIRICAL RULE

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 the normal distribution.

The empirical rule states that for a 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

Note that these values are approximations. For example, according to the normal curve probability density function, 95% of the data will fall within 1.96 standard deviations of the mean; 2 standard deviations is a convenient approximation.

SOLVING

Z-Score

Every unique pair of and defines a different normal distribution. This characteristic of the normal curve (actually a family of curves) could make analysis by the normal distribution tedious because volumes of normal curve tables – one for each different combinations of and - would be required.
Fortunately, all normal distributions can be converted into a single distribution, the standardized normal distribution or the z distribution, which has mean 0 and standard deviation 1. We write Z – N(0, 1).
The conversion formula for any x value of a given normal distribution is:

A z-score is the number of standard deviations that a value, x, is above or below the mean.
If the value of x is less than the mean, the z score is negative.
If the value of x is more than the mean, the z score is positive.
If the value of x equals the mean, the z score is zero.
This formula allows conversion of the distance of any x value form its mean into standard deviation units. A standard z score table can then be used to find probabilities for any normal distribution problem that has been converted to z scores.

Area under the Normal Curve

Whenever we will get the area, we will refer to the Z-table: here it is

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