Section 7.1 The Standard Normal Curve 289 Properties of Normal Distributions 1. Normal distributions have one mode. 2. Normal distributions are symmetric around the mode. 3. The mean and median of a normal distribution are both equal to the mode. In other words, the mean, median, and mode of a normal distribution are all the same. 4. The normal distribution follows the Empirical Rule (see Figure 7.5): ∙ Approximately 68% of the population is within one standard deviation of the mean. In other words, approximately 68% of the population is in the interval �� − �� to �� + ��. ∙ Approximately 95% of the population is within two standard deviations of the mean. In other words, approximately 95% of the population is in the interval �� − 2�� to �� + 2��. ∙ Approximately 99.7% of the population is within three standard deviations of the mean. In other words, approximately 99.7% of the population is in the interval �� − 3�� to �� + 3��. Recall: The Empirical Rule holds for most unimodal symmetric distributions. See Section 3.2. ≈ 99.7% ≈ 95% ≈ 68% μ − 3σ μ − 2σ μ − σ μ μ + σ μ + 2σ μ + 3σ Figure 7.5 Normal curve with mean �� and standard deviation �� Objective 3 Find areas under the standard normal curve Finding Areas Under the Standard Normal Curve A normal distribution can have any mean and any positive standard deviation, but it is only necessary to work with the normal distribution that has mean 0 and standard deviation 1, which is called the standard normal distribution. The probability density function for the standard normal distribution is called the standard normal curve. For any interval, the area under a normal curve over the interval represents the proportion of the population that is contained within the interval. Finding an area under a normal curve is a crucial step in many statistical procedures. When finding an area under the standard normal curve, we use the letter z to indicate a value on the horizontal axis (see Figure 7.6). We refer to such a value as a z-score. Since the mean of the standard normal distribution, which is located at the mode, is 0, the z-score at the mode of the curve is 0. Points on the horizontal axis to the left of the mode have negative z-scores, and points to the right of the mode have positive z-scores. z 0 Figure 7.6
navidi_monk_elementary_statistics_2e_ch7-9
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