808 CHAPTER 12 Statistics mean, median, and mode for curves that are skewed to the right and left is given in Fig. 12.22. Skewed to the left Skewed to the right (a) (b) Mode Mean Median Mode Mean Median Figure 12.22 Normal Distributions Each of the distributions previously mentioned is useful in describing sets of data. However, the most important distribution is the normal or Gaussian distribution , named for German mathematician Carl Friedrich Gauss. (See the Profile in Mathematics on page 276). The histogram of a normal distribution is illustrated in Fig. 12.23. Normal distribution Figure 12.23 The normal distribution is important because many sets of data are normally distributed or closely resemble a normal distribution. Such distributions include intelligence quotients, heights and weights of males, heights and weights of females, blood pressure, birth weight, wearout mileage of automobile brakes, and life spans of refrigerators, to name just a few. The normal distribution is symmetric about the mean. If you were to fold the histogram of a normal distribution down the middle, the left side would fit the right side exactly. In a normal distribution, the mean, median, and mode all have the same value. When the histogram of a normal distribution is smoothed to form a curve, the curve is bell-shaped. The bell may be high and narrow or short and wide. Each of the three curves in Fig. 12.24 represents a normal curve. Curve 12.24(a) has the smallest standard deviation (spread from the mean); curve 12.24(c) has the largest. Did You Know? What Conclusions Can You Draw? City A City B Math scores 0 400 800 200 600 Based on the figure above, which shows the distribution of scores on the mathematics part of the SAT for two different cities, can we say that any given person selected at random from city B has outperformed any given person selected at random from city A? Both distributions appear normal, and the mean of city A is slightly less than the mean of city B. Consider, however, two randomly selected students who took this test: Sally from city A and Kendra from city B. The graph shows that many students in city A outperformed students from city B, so we cannot conclude that Sally scored higher than Kendra or that Kendra scored higher than Sally. (a) (b) (c) Figure 12.24 A Normal Distribution • The graph of a normal distribution is called a normal curve. • The normal curve is bell-shaped and symmetric about the mean. • The mean, median, and mode of a normal distribution all have the same value and all occur at the center of the distribution.
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