Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
2.1 Density Curves & The Normal Distribution Density Curves • Sometimes it is easier to work with a smooth curve than a histogram. • Curves describe proportions versus the counts that histograms describe. • Areas under the curve represent proportions of observations. Density Curves (continued) • Total area under ANY density curve is 1 (100%) • Density curves ALWAYS lie on or above the horizontal axis 1 Density Curves (continued) 12% of the observations take values between 7 and 8 Mean & Median of a Density Curve • Median is equal-areas point, the point with half the area to the left and half to the right – Same is true for the quartiles • Mean is point at which the curve would balance if it were made of solid material – Mean of skewed distribution is pulled toward the direction of the long tail Mean & Median of a Density Curve The mean is the balance point of a density curve. 2 Density Curves (continued) A symmetric density curve median = mean A skewed right density curve median < mean Notable Notations Sample Population Actual observations Idealized distribution Histogram Density curve x s Try This! 1. Sketch a density curve that is symmetric but has a different shape from the actual symmetric density curve. 2. Sketch a density curve that is strongly skewed to the left and estimate the median and mean. 3 Try This! 3. This is the density curve for a uniform distribution. (a) What percent of the observations lie above 0.8? (b) What percent of the observations lie below 0.6? (c) What percent of the observations lies between 0.25 and 0.75? Normal Curves • Symmetric, unimodal, bell-shaped curves • Described by the mean and standard deviation • Mean and median are always located at the center of distribution • The larger the standard deviation, the more spread out the curve N (, ) Normal Curves • Symmetric, unimodal, bell-shaped curves • Described by the mean and standard deviation • Mean and median are always located at the center of distribution • The larger the standard deviation, the more spread out the curve N (, ) 4 The 68-95-99.7 Rule The Empirical Rule Approx. 68% of the data will fall within 1 standard deviation of the mean ( 1 , 1 ) The 68-95-99.7 Rule The Empirical Rule Approx. 95% of the data will fall within 2 standard deviation of the mean ( 2, 2) The 68-95-99.7 Rule The Empirical Rule Approx. 99.7% of the data will fall within 3 standard deviation of the mean ( 3 , 3 ) 5 Try This! 4. Suppose it takes you 20 minutes, on average, to drive to school, with a standard deviation of 2 minutes. Assume the driving time distribution is approximately normal. (a) What percent of the time will you arrive to school in less than 22 minutes? Try This! 4. Suppose it takes you 20 minutes, on average, to drive to school, with a standard deviation of 2 minutes. Assume the driving time distribution is approximately normal. (b) What percent of the time will it take you more than 24 minutes? Try This! 5. The distribution of heights of adult American men is approx. normal with a mean of 69 inches and a standard deviation of 2.5 inches. (a) What percent are taller than 74 in.? (b) Between what heights do the middle 95% of men fall? (c) What percent are shorter than 66.5 in.? (d) A height of 71.5 inches corresponds to what percentile of adult male American heights? 6