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Chapter 13, Part 1

Chapter 13, Part 1. STA 200 Summer I 2011. At this point…. we have a couple of methods for graphing a data set (histogram, stem-and-leaf plot) we have a general idea of what to look for in a graph (shape, outliers)

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Chapter 13, Part 1

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  1. Chapter 13, Part 1 STA 200 Summer I 2011

  2. At this point… • we have a couple of methods for graphing a data set (histogram, stem-and-leaf plot) • we have a general idea of what to look for in a graph (shape, outliers) • we have a couple of ways to describe a distribution numerically (mean and standard deviation, five number summary)

  3. Density Curves • Sometimes, the pattern of a large data set is so regular that we can describe it by a smooth curve (a density curve). • A density curve can be obtained by drawing a curve through the tops of the bars in a histogram. • Density curves always show percentages or proportions (rather than counts), and the total percentage under the curve is always 100%.

  4. Density Curves and Shape • We use the same terminology to describe the shape of a density curve as we would to describe the shape of a histogram (symmetric, left-skewed, right-skewed). • Regarding measures of center: • if a distribution is exactly symmetric: mean = median • if a distribution is left-skewed: mean < median • if a distribution is right-skewed: mean > median

  5. Normal Distribution • Normal curves a symmetric, single-peaked, and bell-shaped. • A specific normal curve is completely described by its mean and standard deviation. • The mean is located at the center of the curve. • The standard deviation is the distance from the mean to one of the change-of-curvature points on either side.

  6. 68-95-99.7 Rule • For a normal distribution, • approximately 68% of the data will be within 1 standard deviation of the mean • approximately 95% of the data will be within 2 standard deviations of the mean • approximately 99.7% of the data will be within 3 standard deviations of the mean • This is also known as the Empirical Rule.

  7. 68-95-99.7 Rule (cont.)

  8. Example • An IQ test is normally distributed with mean 100 and standard deviation 15. • Between what two values do approximately 68% of IQ scores lie? • Between what two values do approximately 95% of IQ scores lie? • Between what two values do approximately 99.7% of IQ scores lie?

  9. Example (cont.) • More questions: • What percentage of IQ scores are less than 115? • What percentage of IQ scores are greater than 130? • What percentage of IQ scores are less than 85? • What percentage of IQ scores are between 70 and 115%?

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