# Chapter 6: Normal Distribution Curves

**Basic**Created by: CK-12

## Introduction

What a way to start math class! Imagine your math teacher allowing you to bounce a basketball around the classroom.

Prior to the beginning of class, go to the physical education department of your school and borrow some basketballs. These will be used in this chapter to introduce the concept of a normal distribution. You may find your class to be somewhat noisy, but you're sure to enjoy the activity. Begin the class by reviewing the concept of a circle and the terms associated with the measurements of a circle. You and your classmates should be able to provide these facts based on your previous learning. To ensure that everyone understands the concepts of center, radius, diameter, and circumference, work in small groups to create posters to demonstrate your understanding. These posters can then be displayed around the classroom. The following is a sample of the type of poster you may create:

Once you and your classmates have completed this and you are confident that you all understand the measurements associated with a circle, distribute the basketballs to everyone in the class. While your classmates are playing with the balls, set up a table with tape, rulers, string, scissors, markers, and any other materials that you think you may find useful to answer the following questions:

- What is the circumference of the basketball?
- What is the diameter of the basketball?
- What is the radius of the basketball?
- How did you determine these measurements?
- What tools did you use to find your answers?

When playtime is over, use the tools provided to answer all of the above questions. It is the job of you and your classmates to determine a method of calculating these measurements. For the questions that require numerical measurements as answers, take 2 measurements for each. You must plot your results for the diameter of the basketball, so remember to record your data.

Oops! Don't forget that the ruler cannot go through the basketball!

When you have answered the above questions, plot your 2 results for the diameter of the basketball on a large sheet of grid paper, and have your classmates do the same.

- 6.1.
## Standard Deviation of a Normal Distribution

- 6.2.
## Variance of a Data Set

- 6.3.
## Standard Deviation of a Data Set

- 6.4.
## Applications of Variance and Standard Deviation

- 6.5.
## Empirical Rule

### Chapter Summary

## Summary

This chapter covers describing in detail the spread of a normal (bell-shape) distribution. The *standard deviation* is a measure of how spread out the data is. It is represented as \begin{align*}\sigma\end{align*} for a population or \begin{align*}s\end{align*} for a sample. If \begin{align*}x\end{align*} is a data value, \begin{align*}\mu\end{align*} and \begin{align*}\overline{x}\end{align*} are the mean, and \begin{align*}n\end{align*} is number of data values, then the standard deviation is:

\begin{align*}\sigma^2 = \frac{\sum(x-\mu)^2}{n}\end{align*} (population) **or** \begin{align*}s^2 = \frac{\sum(x-\overline{x})^2}{n-1}\end{align*} (sample)

The only difference in the formulas is the number by which the sum is divided. The standard deviation found by this method defines the spread of the curve. The *Empirical Rule* or *68-95-99.7 Rule* defines how much of the distribution is within 1, 2, and 3 standard deviations of the mean.