**stix**

**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

Data: Types, Sources, Validity

Presentation Statistics: Tables, Charts, Graphs and Graphics, Distribution Displays; Use of computerized displays and graphics

Central Tendency

Dispersion / Deviation

Confidence Levels

Relationships: Correlation and Regression; Time Series

Probability

*Marketing, Accounting, Business Management, Financial Management, Materials Management (Inventory & Production Control; Logistics), Personnel Management

**Industrial Mid-Management, Supervisory-Management, Quality Control (Quality Management)

***Industrial Engineering, Maintenance and Plant Engineering

**Contact Us at:
FICOA,
5928 W. Michigan St., Wauwatosa, WI 53213-4248**

phone: (414) 258 - 6492 ... ask for Hank

3Jul05

"The race may not always go to the swift,

nor the contest to the strong but,

that's the way to lay your bets." - Ring Lardner