## Presentation on theme: "Copyright © 2011 Pearson Education, Inc. Putting Statistics to Work."— Presentation transcript:

Copyright © 2011 Pearson Education, Inc. Slide 6-3 Unit 6C The Normal Distribution

6-C Copyright © 2011 Pearson Education, Inc. Slide 6-4 The Normal Distribution The normal distribution is a symmetric, bell- shaped distribution with a single peak. Its peak corresponds to the mean, median, and mode of the distribution.

6-C Copyright © 2011 Pearson Education, Inc. Slide 6-5 1.Most data values are clustered near the mean, giving the distribution a well-defined single peak. 2.Data values are spread evenly around the mean, making the distribution symmetric. 3.Larger deviations from the mean are increasingly rare, producing the tapering tails of the distribution. 4.Individual data values result from a combination of many different factors. A data set satisfying the following criteria is likely to have a nearly normal distribution. Conditions for a Normal Distribution

6-C The Empirical Rule (The 68-95-99.7 Rule) for a Normal Distribution Copyright © 2011 Pearson Education, Inc. Slide 6-6

6-C Copyright © 2011 Pearson Education, Inc. Slide 6-7 Standard Scores The number of standard deviations that a data value lies above or below the mean is called its standard score (or z-score), defined by Data Value above the mean below the mean Standard Score positive negative → →

6-C Copyright © 2011 Pearson Education, Inc. Slide 6-8 Standard Scores Example: If the mean were 21 with a standard deviation of 4.7 for scores on a nationwide test, find the z-score for a 30. What does this mean? This means that a test score of 30 would be about 1.91 standard deviations above the mean of 21.

6-C Using z-scores Copyright © 2011 Pearson Education, Inc. Slide 6-9 The mean height of males 20 years or older is 69.1 inches with a standard deviation of 2.8 inches. The mean height of females 20 years or older is 63.7 inches with a standard deviation of 2.7 inches. Data based on information obtained from National Health and Examination Survey. Who is relatively taller? Kevin Garnett whose height is 83 inches or Candace Parker whose height is 76 inches

6-C Using z-scores (cont.) Copyright © 2011 Pearson Education, Inc. Slide 6-10 Kevin Garnett Candace Parker Kevin Garnett’s height is 4.96 standard deviations above the mean. Candace Parker’s height is 4.56 standard deviations above the mean. Kevin Garnett is relatively taller.

6-C You try it! Copyright © 2011 Pearson Education, Inc. Slide 6-11 The mean commute time in the U.S. is 24.4 minutes with a standard deviation of 6.5 minutes. Find the z-score that corresponds to a commute time of 15 minutes. A. 1.45 B. –1.45 C. 11.25 D. –9.4

6-C You try it! (Answer) Copyright © 2011 Pearson Education, Inc. Slide 6-12 The mean commute time in the U.S. is 24.4 minutes with a standard deviation of 6.5 minutes. Find the z-score that corresponds to a commute time of 15 minutes. A. 1.45 B. –1.45 C. 11.25 D. –9.4

6-C Copyright © 2011 Pearson Education, Inc. Slide 6-13 The nth percentile of a data set is the smallest value in the set with the property that n% of the data are less than or equal to it. A data value that lies between two percentiles is said to lie in the lower percentile. Standard Scores and Percentiles

6-C Copyright © 2011 Pearson Education, Inc. Slide 6-14 Standard Scores and Percentiles

6-C Assignment P. 398 – 400 5-18, 20 – 28, 30, 37 Copyright © 2011 Pearson Education, Inc. Slide 6-15