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The Standard Deviation as a Ruler

The Standard Deviation as a Ruler. Chapter 6: Part I. 68-95-99.7 Rule. For unimodal, symmetric distributions: About 68% of all observations are within one standard deviation of the mean About 95% of all observations are within two standard deviations of the mean

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The Standard Deviation as a Ruler

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  1. The Standard Deviation as a Ruler Chapter 6: Part I

  2. 68-95-99.7 Rule • For unimodal, symmetric distributions: • About 68% of all observations are within one standard deviation of the mean • About 95% of all observations are within two standard deviations of the mean • Almost all (99.7%) of the observations are within three standard deviations of the mean

  3. 68-95-99.7 Rule Continued

  4. Application of the 68-95-99.7 Rule • IQ scores for a group of students are unimodal and symmetric with a mean ( ) of 100 and a standard deviation (s) of 16. • Draw the model for these IQ scores. Label it clearly showing what the 68-95-99.7 model predicts about the scores. • In what interval would you expect the central 95% of IQ scores to be found?

  5. Application Continued • About what percent of students should have IQ scores above 116? • About what percent of students should have IQ scores between 68 and 84? • About what percent of students should have IQ scores above 132?

  6. Z-Scores • A z-score measures the number of standard deviation that a data value is from the mean. • Z = Distance between the data value and the mean Standard Deviation • Z = • A positive z-score indicates that the data value is above the mean, whereas a negative z-score indicate that the data value is below the mean.

  7. Application Using Z-scores • Adult female Dalmatians weigh an average of 50 pounds with a standard deviation of 3.3 pounds. Adult female Boxers weigh an average of 57.5 pounds with a standard deviation of 1.7 pounds. One ISU professor owns an underweight Dalmatian and an underweight Boxer. The Dalmatian weighs 45 pounds, and the Boxer weighs 52 pounds. Which dog is more underweight? Explain

  8. Using the 68-95-99.7 Rule and Z-scores Together

  9. Application #2 Using Z-Scores • The middle 50% of a college’s students have SAT scores between 1030 and 1150. Students in the top quartile are eligible for full ride scholarships. If a student plans to take the ACT test as an alternative to the SAT, how well does she need to perform to be eligible for the scholarship? Additional information that might come in handy: Average SAT score of college students = 1010 and standard deviation is 150 points. Average ACT score of college students is 20.5 with a standard deviation of 3.5.

  10. Assignment • Read Chapter 6 • Try the following exercises: • 1, 5, 7, 9, 15, 17 • Quiz 2 next Wednesday, covers Chapters 5 and 6.

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