16112 Words65 Pages

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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*…show more content…*

No variable ﬁts a normal distribution perfectly, since a normal distribution is a theoretical distribution. However, a normal distribution can be used to describe many variables, because the deviations from a normal distribution are very small. This concept will be explained further in Section 6–1.

When the data values are evenly distributed about the mean, a distribution is said to be a symmetric distribution. (A normal distribution is symmetric.) Figure 6–2(a) shows a symmetric distribution. When the majority of the data values fall to the left or right of the mean, the distribution is said to be skewed. When the majority of the data values fall to the right of the mean, the distribution is said to be a negatively or left-skewed distribution. The mean is to the left of the median, and the mean and the median are to the left of the mode. See Figure 6–2(b). When the majority of the data values fall to the left of the mean, a distribution is said to be a positively or right-skewed distribution. The mean falls to the right of the median, and both the mean and the median fall to the right of the mode. See Figure 6–2(c).

6–3

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Chapter 6 The Normal Distribution

The “tail” of the curve indicates the direction of skewness (right is positive, left is negative). These distributions can be compared with the ones shown in Figure 3–1 in

Chapter 3. Both types

8/13/08

4:39 PM

Page 299

Confirming Pages

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

No variable ﬁts a normal distribution perfectly, since a normal distribution is a theoretical distribution. However, a normal distribution can be used to describe many variables, because the deviations from a normal distribution are very small. This concept will be explained further in Section 6–1.

When the data values are evenly distributed about the mean, a distribution is said to be a symmetric distribution. (A normal distribution is symmetric.) Figure 6–2(a) shows a symmetric distribution. When the majority of the data values fall to the left or right of the mean, the distribution is said to be skewed. When the majority of the data values fall to the right of the mean, the distribution is said to be a negatively or left-skewed distribution. The mean is to the left of the median, and the mean and the median are to the left of the mode. See Figure 6–2(b). When the majority of the data values fall to the left of the mean, a distribution is said to be a positively or right-skewed distribution. The mean falls to the right of the median, and both the mean and the median fall to the right of the mode. See Figure 6–2(c).

6–3

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302

Chapter 6 The Normal Distribution

The “tail” of the curve indicates the direction of skewness (right is positive, left is negative). These distributions can be compared with the ones shown in Figure 3–1 in

Chapter 3. Both types

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