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15.3 THE NORMAL DISTRIBUTION Learn to recognize and analyze normal distributions BELL-SHAPED CURVES data is clumped in the middle and tapered off at the extremes Other examples: heights of all 11th grade girls in a large city levels of soil samples from all farms in a rural county achievement test scores for all high school students in a state STANDARD NORMAL DISTRIBUTION mean = 0 standard deviation = 1 The Normal Distribution has: • mean = median = mode • symmetry about the center • 50% of values less than the mean and 50% greater than the mean The spread of a normal distribution is controlled by the standard deviation; the smaller the standard deviation the more concentrated the data. STANDARD NORMAL CURVE bell-shaped curve that represents the standard normal distribution properties of the standard normal curve 1. symmetric with respect to the y-axis 2. approaches the x-axis asymptotically as |x| increases 3. total area under the curve and above the x-axis =1 AREA UNDER THE STANDARD NORMAL CURVE can be used to estimate the fraction of data falling between two values tables have been developed to help find this area AREA UNDER THE STANDARD NORMAL CURVE FOR 0 x 4.0 IN A STANDARD NORMAL DISTRIBUTION… what fraction of the data is between 2.6 and 2.8? what percent of data is within two standard deviations from the mean? what fraction of the data is more than 2.0? what percent of the data is more than two standard deviations from the mean? STANDARDIZED VALUE CORRESPONDING TO Z gives number of standard deviations that “x” is from the mean of the distribution z x The heights of a certain group of adults are normally distributed with a mean of 180 cm and a standard deviation of 8cm. Find the percent of the group having…. a. a height greater than 196 cm b. heights between 172 and 180