# Normal distribution

CHAPTER 6 |

The Normal Distribution |

**Objectives**

After completing this chapter, you should be able to

**1** __Identify distributions as symmetric or skewed.__

**2** __Identify the properties of a normal distribution.__

**3** __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** __Use the central limit theorem to solve problems involving sample means for large samples.__

**7** __Use the normal approximation to compute probabilities for a binomial variable.__

**Outline**

__6–2Applications of the Normal Distribution__

__6–4The Normal Approximation to the Binomial Distribution__

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**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 a

*bell curve*or a

*Gaussian distribution*, named for the German mathematician Carl Friedrich Gauss (1777–1855), who derived its equation.

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Figure 6–1

Histograms for the Distribution of Heights of Adult Women

Figure 6–2

Normal and Skewed Distributions

Objective 1

Identify distributions as symmetric or skewed.

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

**The mean is to the left of the median, and the mean and the median are to the left of the mode. See**

__negatively or left-skewed distribution.____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

**The mean falls to the right of the median, and both the mean and the median fall to the right of the mode. See**

__positively or right-skewed distribution.____Figure 6–2(c)__.

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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 follow the same principles.

This chapter will present the properties of a normal distribution and discuss its applications. Then a very important fact about a normal distribution called the * central limit theorem *will be explained. Finally, the chapter will explain how a normal distribution curve can be used as an approximation to other distributions, such as the binomial distribution. Since a binomial distribution is a discrete distribution, a correction for continuity may be employed when a normal distribution is used for its approximation.

Objective 2

Identify the properties of a normal distribution.

**6–1Normal Distributions**

In mathematics, curves can be represented by equations. For example, the equation of the circle shown in __Figure 6–3__ is *x*2 + *y*2 = *r*2, where *r* is the radius. A circle can be used to represent many physical objects, such as a wheel or a gear. Even though it is not possible to manufacture a wheel that is perfectly round, the equation and the properties of a circle can be used to study many aspects of the wheel, such as area, velocity, and acceleration. In a similar manner, the theoretical curve, called a *normal distribution curve*, can be used to study many variables that are not perfectly normally distributed but are nevertheless approximately normal.

Figure 6–3

Graph of a Circle and an Application

The mathematical equation for a normal distribution is

where

*e* ≈ 2.718 (≈ means “is approximately equal to”)

*π* ≈ 3.14

*μ* = population mean

*σ* = population standard deviation

This equation may look formidable, but in applied statistics, tables or technology is used for specific problems instead of the equation.

Another important consideration in applied statistics is that the area under a normal distribution curve is used more often than the values on the *y* axis. Therefore, when a normal distribution is pictured, the *y* axis is sometimes omitted.

Circles can be different sizes, depending on their diameters (or radii), and can be used to represent wheels of different sizes. Likewise, normal curves have different shapes and can be used to represent different variables.

The shape and position of a normal distribution curve depend on two parameters, the *mean* and the *standard deviation*. Each normally distributed variable has its own normal distribution curve, which depends on the values of the variable’s mean and standard deviation. __Figure 6–4(a)__ shows two normal distributions with the same mean values but different standard deviations. The larger the standard deviation, the more dispersed, or spread out, the distribution is. __Figure 6–4(b)__ shows two normal distributions with the same standard deviation but with different means. These curves have the same shapes but are located at different positions on the *x* axis. __Figure 6–4(c)__ shows two normal distributions with different means and different standard deviations.

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Figure 6–4

Shapes of Normal Distributions

Historical Note

The discovery of the equation for a normal distribution can be traced to three mathematicians. In 1733, the French mathematician Abraham DeMoivre derived an equation for a normal distribution based on the random variation of the number of heads appearing when a large number of coins were tossed. Not realizing any connection with the naturally occurring variables, he showed this formula to only a few friends. About 100 years later, two mathematicians, Pierre Laplace in France and Carl Gauss in Germany, derived the equation of the normal curve independently and without any knowledge of DeMoivre’s work. In 1924, Karl Pearson found that DeMoivre had discovered the formula before Laplace or Gauss.

A **normal distribution** is a continuous, symmetric, bell-shaped distribution of a variable.

The properties of a normal distribution, including those mentioned in the definition, are explained next.

**Summary of the Properties of the Theoretical Normal Distribution**

1.A normal distribution curve is bell-shaped.

2.The mean, median, and mode are equal and are located at the center of the distribution.

3.A normal distribution curve is unimodal (i.e., it has only one mode).

4.The curve is symmetric about the mean, which is equivalent to saying that its shape is the same on both sides of a vertical line passing through the center.

5.The curve is continuous; that is, there are no gaps or holes. For each value of *X*, there is a corresponding value of *Y*.

6.The curve never touches the *x* axis. Theoretically, no matter how far in either direction the curve extends, it never meets the *x* axis—but it gets increasingly closer.

7.The total area under a normal distribution curve is equal to 1.00, or 100%. This fact may seem unusual, since the curve never touches the *x* axis, but one can prove it mathematically by using calculus. (The proof is beyond the scope of this textbook.)

8.The area under the part of a normal curve that lies within 1 standard deviation of the mean is approximately 0.68, or 68%; within 2 standard deviations, about 0.95, or 95%; and within 3 standard deviations, about 0.997, or 99.7%. See __Figure 6–5__ , which also shows the area in each region.

The values given in item 8 of the summary follow the *empirical rule* for data given in __Section 3–2__ .

You must know these properties in order to solve problems involving distributions that are approximately normal.

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Figure 6–5

Areas Under a Normal Distribution Curve

Objective 3

Find the area under the standard normal distribution, given various *z* values.

**The Standard Normal Distribution**

Since each normally distributed variable has its own mean and standard deviation, as stated earlier, the shape and location of these curves will vary. In practical applications, then, you would have to have a table of areas under the curve for each variable. To simplify this situation, statisticians use what is called the * standard normal distribution *.

The **standard normal distribution** is a normal distribution with a mean of 0 and a standard deviation of 1.

The standard normal distribution is shown in __Figure 6–6__ .

The values under the curve indicate the proportion of area in each section. For example, the area between the mean and 1 standard deviation above or below the mean is about 0.3413, or 34.13%.

The formula for the standard normal distribution is

All normally distributed variables can be transformed into the standard normally distributed variable by using the formula for the standard score:

This is the same formula used in __Section 3–3__ . The use of this formula will be explained in __Section 6–3__ .

As stated earlier, the area under a normal distribution curve is used to solve practical application problems, such as finding the percentage of adult women whose height is between 5 feet 4 inches and 5 feet 7 inches, or finding the probability that a new battery will last longer than 4 years. Hence, the major emphasis of this section will be to show the procedure for finding the area under the standard normal distribution curve for any *z* value. The applications will be shown in __Section 6–2__ . Once the *X* values are transformed by using the preceding formula, they are called *z* values. The **z value **is actually the number of standard deviations that a particular

*X*value is away from the mean.

__Table E__in

__Appendix C__gives the area (to four decimal places) under the standard normal curve for any

*z*value from –3.49 to 3.49.

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