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Notes: 3.1 Normal Density Curves Learning Target Use the properties of a normal curve to calculate areas and probabilities. ♦ I can use the properties of a normal density curve to sketch a normal distribution♦ Density Curve: Any curve (graph) where the area under the curve = 1 (100%). We use density curves to simplify histograms. Example 1: Simplify the following histograms by drawing density curves. Normal Density Curve: A special density curve that has the following properties: Density Curve (total area under curve = 1) Symmetrical (mean = median) Bell shaped Most of the data is within + 3 standard deviations of the mean (99.7%) The distribution of data follows the Empirical Rule (68-95-99.7) The inflection point on the curve corresponds to + 1 standard deviation Empirical Rule: Also called the 68-95-99.7 Rule. It is used to estimate the area (determine probabilities) under a normal density curve within +1, +2, and +3 standard deviations from the mean. In a normal distribution: + 1 standard deviation away from the mean contains 68% of the data + 2 standard deviations away from the mean contains 95% of the data + 3 standard deviations away from the mean contains 99.7% of the data Intro to Stats Page 1 Notes: 3.1 Normal Density Curves ♦ I can use the Empirical Rule to calculate the probability of an event based on a normal distribution ♦ Example 1: Use the curve below to show the percent of data within 1, 2, and 3 standard deviations away from the mean Example 2: Use the Empirical Rule (68-95-99.7) to determine the area under the curve within each standard deviation. The mean = median so half of the data is above the mean and half is below. 68% of data is within 1 standard deviation, so 34% (half) will be below the mean and 34% will be above. 95% of data is within 2 standard deviations, 47.5% on each side of the mean. Since 34% is within 1 standard deviation, then 47.5 – 34 = 13.5. The area between 1 and 2 standard deviations is 13.5% 99.7% is within 3 standard deviations, 49.85% on each side. Since 47.5 is within 2 standard deviations, then 49.85 – 47.5 = 2.35. The area between 2 and 3 standard deviations is 2.35% 99.7% of the data is within 3 standard deviations (not quite 100%). That leaves 0.3% outside of 3 standard deviations or 0.15% on either side Intro to Stats Page 2