SECTION 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 235 A normal distribution can have any mean and any positive standard deviation. These two parameters, m and s, determine the shape of the normal curve. The mean gives the location of the line of symmetry, and the standard deviation describes how much the data are spread out. For instance, in the figures below, curves A and B have the same mean, and curves B and C have the same standard deviation. The total area under each curve is 1. Also, in each graph, one of the inflection points occurs one standard deviation to the left of the mean, and the other occurs one standard deviation to the right of the mean. 0 1 2 3 4 5 6 7 x Inflection points A 0 1 2 3 4 5 6 7 x Inflection points B 0 1 2 3 4 5 6 7 x Inflection points C Mean: m = 3.5 Mean: m = 3.5 Mean: m = 1.5 Standard deviation: Standard deviation: Standard deviation: s = 1.5 s = 0.7 s = 0.7 Understanding Mean and Standard Deviation 1. Which normal curve has a greater mean? 2. Which normal curve has a greater standard deviation? 6 9 12 15 18 21 x A B SOLUTION 1. The line of symmetry of curve A occurs at x = 15. The line of symmetry of curve B occurs at x = 12. So, curve A has a greater mean. 2. Curve B is more spread out than curve A. So, curve B has a greater standard deviation. TRY IT YOURSELF 1 1. Which normal curve has the greatest mean? 2. Which normal curve has the greatest standard deviation? C A B 40 50 60 30 70 x Answer: Page A39 EXAMPLE 1 Picturing the World According to the National Center for Health Statistics, the number of births in the United States in a recent year was 3,747,540.The weights of the newborns can be approximated by a normal distribution, as shown in the figure. (Adapted from National Center for Health Statistics) Weight (in grams) Weights of Newborns 2100 1500 2700 3300 3900 4500 5100 x What is the mean weight of the newborns? Estimate the standard deviation of this normal distribution.
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