# Chapter 6 Lessons 1 and 2

## Graphs of Normal Probability Distributions Lesson 1

This distribution was studied by the French mathematician Abraham de Moivre (1667–1754) and later by the German mathematician Carl Friedrich Gauss (1777–1855), whose work is so important that the normal distribution is sometimes called Gaussian.

## Application

Applications of a normal probability distribution are so numerous that some mathematicians refer to it as “a veritable Boy Scout knife of statistics.”

A rather complicated formula, presented later in this section, defines a normal distribution in terms of m and s, the mean and standard deviation of the population distribution.

## Bell Shape Curve

We see that a general normal curve is smooth and symmetrical about the vertical line extending upward from the mean m.

Notice that the highest point of the curve occurs over m. If the distribution were graphed on a piece of sheet metal, cut out, and placed on a knife edge, the balance point would be at m.

## Properties of a Normal Curve

1. The curve is bell shaped, with the highest point over the mean

2. The curve is symmetrical about a vertical line through

## Empirical Rule

For a distribution that is symmetrical and bell shaped:

1 standard deviation=68%

2 standard deviation=95%

3 standard deviation=99.7%

## Solution to Problems

Since m = 600 and m + s = 600 + 100 = 700, we see that the shaded area is simply the area between m and m + s.

The area from m to m + s is 34% of the total area.

This tells us that the probability a Sunshine radio will last between 600 and 700 playing hours is about 0.34.