Introduction

This photo shows many different pairs of shoes in various colors. The shoes appear to be hanging from a wall by cords.
Figure 6.1 If you ask enough people about their shoe size, you will find that your graphed data is shaped like a bell curve and can be described as normally distributed. (credit: Ömer Ünlϋ)

Chapter Objectives

By the end of this chapter, the student should be able to do the following:

  • Recognize the normal probability distribution and apply it appropriately
  • Recognize the standard normal probability distribution and apply it appropriately
  • Compare normal probabilities by converting to the standard normal distribution

The normal, a continuous distribution, is the most important of all the distributions. It is widely used and even more widely abused. Its graph is bell-shaped. You see the bell curve in almost all disciplines, including psychology, business, economics, the sciences, nursing, and, of course, mathematics. Some of your instructors may use the normal distribution to help determine your grade. Most IQ scores are normally distributed. Often, real-estate prices fit a normal distribution. The normal distribution is extremely important, but it cannot be applied to everything in the real world.

In this chapter, you will study the normal distribution, the standard normal distribution, and applications associated with them.

The normal distribution has two parameters—two numerical descriptive measures—the mean (μ) and the standard deviation (σ). If X is a quantity to be measured that has a normal distribution with mean (μ) and standard deviation (σ), we designate this by writing

This shows the graph of a normal distribution bell curve labeled NORMAL: X ~ N (μ, σ).
Figure 6.2

The curve is symmetric about a vertical line drawn through the mean, μ. In theory, the mean is the same as the median, because the graph is symmetric about μ. With a normal distribution, the mean, median, and mode all lie at the same point. The normal distribution depends only on the mean and the standard deviation. The location of the mean simply indicates the location of the line of symmetry, in a normal distribution. Since the area under the curve must equal one, a change in the standard deviation, σ, causes a change in the shape of the curve; the curve becomes fatter or skinnier depending on σ. A change in μ causes the graph to shift to the left or right. The location of the mean simply indicates the location of the line of symmetry, in a normal distribution. This means there are an infinite number of normal probability distributions. One distribution of special interest is called the standard normal distribution.

Collaborative Exercise

Your instructor will record the heights of both men and women in your class, separately. Draw histograms of your data. Then draw a smooth curve through each histogram. Is each curve somewhat bell-shaped? Do you think that if you had recorded 200 data values for men and 200 for women that the curves would look bell-shaped? Calculate the mean for each data set. Write the means on the x-axis of the appropriate graph below the peak. Shade the approximate area that represents the probability that one randomly chosen male is taller than 72 inches. Shade the approximate area that represents the probability that one randomly chosen female is shorter than 60 inches. If the total area under each curve is 1, does either probability appear to be more than 0.5?