# Normal Distribution

**Topics:**Normal distribution, Standard deviation, Probability theory

**Pages:**129 (16101 words)

**Published:**February 17, 2014

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C H A P T E

R

6

The Normal

Distribution

Objectives

Outline

After completing this chapter, you should be able to

1

2

3

Identify distributions as symmetric or skewed.

4

Find probabilities for a normally distributed

variable by transforming it into a standard

normal variable.

Introduction

6–1

Normal Distributions

Identify the properties of a normal distribution.

Find the area under the standard normal

distribution, given various z values.

5

Find speciﬁc data values for given

percentages, using the standard normal

distribution.

6

6–3 The Central Limit Theorem

6–4 The Normal Approximation to the Binomial

Distribution

Use the central limit theorem to solve

problems involving sample means for large

samples.

7

6–2 Applications of the Normal Distribution

Use the normal approximation to compute

probabilities for a binomial variable.

Summary

6–1

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Chapter 6 The Normal Distribution

Statistics

Today

What Is Normal?

Medical researchers have determined so-called normal intervals for a person’s blood pressure, cholesterol, triglycerides, and the like. For example, the normal range of systolic blood pressure is 110 to 140. The normal interval for a person’s triglycerides is from 30 to 200 milligrams per deciliter (mg/dl). By measuring these variables, a physician can determine if a patient’s vital statistics are within the normal interval or if some type of treatment is needed to correct a condition and avoid future illnesses. The question then is, How does one determine the so-called normal intervals? See Statistics Today—Revisited at the end of the chapter.

In this chapter, you will learn how researchers determine normal intervals for speciﬁc medical tests by using a normal distribution. You will see how the same methods are used to determine the lifetimes of batteries, the strength of ropes, and many other traits.

Introduction

Random variables can be either discrete or continuous. Discrete variables and their distributions were explained in Chapter 5. Recall that a discrete variable cannot assume all values between any two given values of the variables. On the other hand, a continuous variable can assume all values between any two given values of the variables. Examples of continuous variables are the heights of adult men, body temperatures of rats, and cholesterol levels of adults. Many continuous variables, such as the examples just mentioned, have distributions that are bell-shaped, and these are called approximately normally distributed variables. For example, if a researcher selects a random sample of 100 adult women, measures their heights, and constructs a histogram, the researcher gets a graph similar to the one shown in Figure 6–1(a). Now, if the researcher increases the sample size and decreases the width of the classes, the histograms will look like the ones shown in Figure 6–1(b) and (c). Finally, if it were possible to measure exactly the heights of all adult females in the United States and plot them, the histogram would approach what is called a normal distribution, shown in Figure 6–1(d). This distribution is also known as 6–2

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Chapter 6 The Normal Distribution

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Figure 6–1

Histograms for the

Distribution of Heights

of Adult Women

(a) Random sample of 100 women

(b) Sample size increased and class width decreased

(c) Sample size increased and class width

decreased further

(d) Normal distribution for the population

Figure 6–2

Normal and Skewed

Distributions

Mean

Median

Mode

(a) Normal

Mean Median Mode

(b) Negatively skewed

Objective

1

Identify distributions

as symmetric or

skewed.

Mode Median Mean

(c) Positively skewed

a bell curve or a Gaussian...

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