The Normal Probability Distribution

Carl Friedrich (Johann) Gauss (1777-1855); German mathematician considered to be one of the three greatest mathematicians; the other two being Archimedes and Newton. Gauss originated the Normal Probability formula which generates the bell-shaped curve and, which is generally accepted as describing naturally distributed populations. The Normal Curve and its attendant Standard Deviation form the principle measures of dispersion and statistical significance.

The basic normal probability curve is centered on zero (0) and extends, in a bell-shape to positive (+) infinity (to the right) and negative (-) infinity (to the left). An examination of a bell-shaped curve reveals a rounded middle (at the mean) which is concave downward and gently slopes to a concave upward shape further from the mean. Where the curve changes direction, from concave downward to concave upward, is the point of deflection. That point of deflection is at  x = 1  or  x = -1 and is the distance of the standard deviation.

Since the mathematical expression provides a fixed shape, the areas which are enclosed by the standard deviation(s) are also fixed and can be considered to be probability spaces.

Since the normal curve is symmetrical about the mean, half (50%) of the probability space is above the mean and the other half is below the mean.

The probability space under the normal curve can be determined by using integral calculus but this is seldom used since extensive tables of the probabilities are widely published. The probability spaces are expressed in terms of the distance from the mean to some point in the population and in terms of the standard deviation.

Probability Areas*** under the Normal Curve

x/s*    area**               x/s    area

.10    .0398                1.50    .4332

.20    .0793                1.645  .4500 

.30    .1179                1.96    .4750

.40    .1554                 2.00    .4772

.50    .1915                2.33    .4901

1.00    .3413               3.00    .4987

*distance from the mean in terms of a standard deviation

**area of probability, which is bounded by the mean and a point, "x";   "s" is the calculated value of the estimate of the standard deviation of a given population

***abridged; more complete table may be found in most statistics texts

A proposed curriculum for an introductory Statistics course:

[Target audience: Business*, Industrial**, and Engineering*** and Technical*** Students at the technical and community college level.]

Introduction; What is "Statistics" - a definition

How, Where, When, and Why we use Statistics

The Branches of Statistics

Central Tendency

Dispersion / Deviation

Relationships: Correlation and Regression; Time Series

Probability

***Industrial Engineering, Maintenance and Plant Engineering

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