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The Normal Distribution Section 11.10 Discrete Probability Distribution - a finite number of possible events, or values Continuous Probability Distribution - events for this can be any value in an interval of real numbers Normal Distribution Has data that vary randomly from the mean. The graph of a normal distribution is a normal curve Normal Distribution For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics: • About 68% of the data lie within one standard deviation of the mean. • About 95% of the data lie within two standard deviations of the mean. • About 99.7% of the data lie within three standard deviations of the mean. Normal Distribution 3 standard deviations 2 standard deviations 1 standard deviation 34% 2.35% x 3s 34% 13.5% x 2s 13.5% x s x xs 2.35% x 2s x 3s A normal distribution has a symmetric bell shape centered on the mean. The normal distribution and standard deviations In a normal distribution: Approximately 68% of scores will fall within one standard deviation of the mean The normal distribution and standard deviations In a normal distribution: Approximately 95% of scores will fall within two standard deviations of the mean The normal distribution and standard deviations In a normal distribution: Approximately 99.7% of scores will fall within three standard deviations of the mean The Shape of Distributions Sometimes data are not normally distributed. They can be Skewed, an asymmetric curve where one end stretches out further than the other end. Normal distributions (bell shaped) are a family of distributions that have the same general shape. They are symmetric (the left side is an exact mirror of the right side) with scores more concentrated in the middle than in the tails. Examples of normal distributions are shown to the right. Notice that they differ in how spread out they are. The area under each curve is the same. The normal distribution and standard deviations 34% 2.35% 34% 13.5% In a normal distribution: The total area under the curve is 1. 13.5% 2.35% Analyze Normally Distributed Data The bar graph gives the weights of a population of female brown bears. The red curve shows how the weights are normally distributed about the mean, 115kg. Approximately what percent of female brown bears weigh between 100 and 129 kg? Example For a population of male European eels, the mean body length is shown below. Sketch a normal curve showing the eel lengths at one, two, and three standard deviations from the mean. For a population of male European eels, the mean body length is shown below. Sketch a normal curve showing the eel lengths at one, two, and three standard deviations from the mean. Example The height of adult American males are approximately normally distributed with mean of 69.5 in and standard deviation 2.5 in. What percent of adult males are between 67 in and 74.5 in tall? Example In a group of 2000 adults American males. About how many would you expect to be taller than 6 ft? Exit Ticket The heights of men in a survey are normally distributed about their mean. 1. About what percent of men aged 25 to 34 are 69 – 71 inches tall? 2. About what percent of men aged 25 to 34 are less than 70 in tall? 3. Suppose the survey included data on 100 men. About how many would you expect to be 69 – 71 in tall? 4. The scores on the Algebra 2 final are approximately normally distributed with a mean of 150 and a standard deviation of 15. a. What percentage of students who took the test scored above 180? b. If 250 students took the final exam, approximately how many scored above 135? 5. For a population of female European eels, the mean body length is 21.1 in. The standard deviation is 4.7 in. Sketch a normal curve showing eel lengths at one, two, and three standard deviations from the mean.