Normal Distribution • Many common statistics (such as human height, weight, or blood pressure) gathered from samples in the natural world tend to have a normal distribution about their mean. – A normal distribution has data that vary randomly from the mean. Characteristics of a Normal Distribution • The graph of the curve is continuous, bellshaped, and symmetric with respect to the mean. • The mean, median, and mode are equal and located at the center. • The curve approaches, but never touches, the 𝑥-axis. • The total area under the curve is equal to 1 or 100%. In a normal distribution, • Approximately 68% of the population is within 1 standard deviation of the mean. • Approximately 95% of the population is within 2 standard deviations of the mean. • Approximately 99.7% of the population is within 3 standard deviations of the mean. • This is the Empirical or 68-95-99.7 Rule. Empirical Rule Example 1 A normal distribution has a mean of 21 and a standard deviation of 4. a. Find the range of values that represent the middle 68% of the distribution. b. What percent of the data will be greater than 29? Example 2 The heights of 1800 adults are normally distributed with a mean of 70 inches and a standard deviation of 2 inches. a. How many adults are between 66 and 74 inches? b. What is the probability that a random adult is more than 72 inches tall? Example 3 IQ scores are known to be normally distributed with a mean of 100 and a standard deviation of 15. What percent of the population has a score above 130? What score cuts off the lowest 16% of the population? • The Empirical Rule is only useful for evaluating specific values, such as 𝜇 + 𝜎 . • Therefore, we standardize the data set by converting the data to 𝑧-values. z X Standard Normal Distribution • Has the following properties: 1. Its graph is bell-shaped and the total area under the curve is 1 or 100%. 2. Almost all the area is between 𝑧 = −3 and 𝑧 = 3. 3. Its mean is equal to 0 (𝜇 = 0). 4. Its standard deviation is equal to 1 (𝜎 = 1) Example 4 Find 𝑧 if 𝑋 = 18, 𝜇 = 22, and 𝜎 = 3.1. Indicate the position of 𝑋 in the distribution. X Example 5 The U.S. Air Force requires that that pilots have heights between 64 in. and 77 in. Given that men have normally distributed heights with a mean of 69.5 in. and a standard deviation of 2.4 in., find the percentage of men who satisfy that height requirement. Use the z-score table!