Section 12.6 The Normal Distribution Math in Our World

Learning Objectives Identify characteristics of a data set that is normally distributed. Apply the empirical rule. Compute z scores. Use a table and z scores to find areas under the standard normal distribution.

Normal Distributions A wide variety of quantities in the real world, like sizes of individuals in a population, IQ scores, and many others, tend to exhibit the same phenomenon, in which we see that the largest number have values somewhere in the middle of the range, and the classes further away from the center have smaller values. In fact, it’s so common that frequency distributions of this type came to be known as normal distributions.

Normal Distributions A normal distribution is a continuous, symmetric, bell-shaped distribution.

Normal Distributions Some Properties of a Normal Distribution 1. It is bell-shaped. 2. The mean, median, and mode are equal and located at the center of the distribution. 3. It is unimodal (i.e., it has only one mode). 4. It is symmetrical 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.

Normal Distributions Some Properties of a Normal Distribution 5. It is continuous—i.e., there are no gaps or holes. 6. The area under a portion of a normal curve is the percentage (in decimal form) of the data that falls between the data values that begin and end that region. 7. The total area under a normal curve is exactly 1. This makes sense based on property 6, since the area under the whole curve encompasses all data values.

The Empirical Rule When data are normally distributed, approximately 68% of the values are within 1 standard deviation of the mean, approximately 95% are within 2 standard deviations of the mean, and approximately 99.7% are within 3 standard deviations of the mean.

EXAMPLE 1 Using the Empirical Rule According to the website answerbag.com, the mean height for male humans is 5 feet 9.3 inches, with a standard deviation of 2.8 inches. If this is accurate, out of 1,000 randomly selected men, how many would you expect to be between 5 feet 6.5 inches and 6 feet 0.1 inch? SOLUTION The given range of heights corresponds to those within 1 standard deviation of the mean, so we would expect about 68% of men to fall in that range. In this case, we expect about 680 men to be between 5 feet 6.5 inches and 6 feet 0.1 inch.

The Standard Normal Distribution The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. The values under the curve shown indicate the proportion of area in each section.

Z Score For a given data value from a data set that is normally distributed, we define that value’s z score to be The z score is a measure of position: it describes how many standard deviations a data value lies above (if positive) or below (if negative) the mean.

EXAMPLE 2 Computing a z Score Based on the information in Example 1, find the z score for a man who is 6 feet 4 inches tall. SOLUTION Using the formula for z scores with mean 5 feet 9.3 inches and standard deviation 2.8 inches:

Z Score The value of z scores is that they will allow us to find areas under a normal curve using only areas under a standard normal curve, which can be read from a table, like this one. An area table with more values can be found in Appendix A of the text.

The Standard Normal Distribution Two Important Facts about the Standard Normal Curve 1. The area under any normal curve is divided into two equal halves at the mean. Each of the halves has area 0.500. 2. The area between z = 0 and a positive z score is the same as the area between z = 0 and the negative of that z score.

Find the area under the standard normal distribution (a) Between z = 1.55 and z = 2.25. (b) Between z = – 0.60 and z = – 1.35. (c) Between z = 1.50 and z = – 1.75. EXAMPLE 3 Finding the Area between Two z Scores

EXAMPLE 3 Finding the Area between Two z Scores (a) Draw the picture, label z scores, and shade the requested area. SOLUTION Using Table 12-3, the area between z = 0 and z = 2.25 is 0.488, and the area between z = 0 and z = 1.55 is 0.439. The area we are looking for is the larger area minus the smaller: 0.488 – 0.439 = 0.049.

EXAMPLE 3 Finding the Area between Two z Scores (b) Draw the picture, label z scores, and shade the requested area. SOLUTION Using key fact 2, we can find the area between z = 0 and the two given negative z scores by finding the corresponding positive z scores in the table. The area we are looking for is the larger area minus the smaller: 0.412 – 0.226 = 0.186.

EXAMPLE 3 Finding the Area between Two z Scores (c) Draw the picture, label z scores, and shade the requested area. SOLUTION Next , find the corresponding areas in the table. Since the z values are on opposite sides of the mean, the areas corresponding to the z values are added. 0.433 + 0.460 = 0.893.

Find the area under the standard normal distribution To the right of z = 1.70. To the right of z = – 0.95. EXAMPLE 4 Finding the Area to the Right of a z Score

EXAMPLE 4 Finding the Area to the Right of a z Score (a) Draw the picture, label z scores, and shade the requested area. SOLUTION The area of the entire portion to the right of z = 0 is 0.500. According to the table, the area of the portion between z = 0 and z = 1.70 is 0.455. So the shaded portion is the difference between 0.500 and 0.455: 0.500 – 0.455 = 0.045.

EXAMPLE 4 Finding the Area to the Right of a z Score (b) Draw the picture, label z scores, and shade the requested area. SOLUTION The area of the entire portion to the right of z = 0 is 0.500. According to the table, the area of the portion between z = 0 and z = – 0.95 is 0.329. This time, the area is the sum of 0.500 and 0.329: 0.500 + 0.329 = 0.829.

Find the area under the standard normal distribution To the left of z = – 2.20. To the left of z = 1.95. EXAMPLE 5 Finding the Area to the Left of a z Score

EXAMPLE 5 Finding the Area to the Left of a z Score (a) Draw the picture, label z scores, and shade the requested area. SOLUTION The area of the entire portion to the left of z = 0 is 0.500. According to the table, the area of the portion between z = 0 and z = – 2.20 is 0.486. So the shaded portion is the difference between 0.500 and 0.486: 0.500 – 0.486 = 0.014.

EXAMPLE 5 Finding the Area to the Left of a z Score (b) Draw the picture, label z scores, and shade the requested area. SOLUTION The area of the entire portion to the left of z = 0 is 0.500. According to the table, the area of the portion between z = 0 and z = 1.95 is 0.474. This time, the area is the sum of 0.500 and 0.474: 0.500 + 0.474 = 0.974.