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STA 291 Fall 2009

STA 291 Fall 2009. Lecture 6 Dustin Lueker. Measures of Variation. Statistics that describe variability Two distributions may have the same mean and/or median but different variability Mean and Median only describe a typical value, but not the spread of the data Range Variance

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STA 291 Fall 2009

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  1. STA 291Fall 2009 Lecture 6 Dustin Lueker

  2. Measures of Variation • Statistics that describe variability • Two distributions may have the same mean and/or median but different variability • Mean and Median only describe a typical value, but not the spread of the data • Range • Variance • Standard Deviation • Interquartile Range • All of these can be computed for the sample or population STA 291 Fall 2009 Lecture 6

  3. Range • Difference between the largest and smallest observation • Very much affected by outliers • A misrecorded observation may lead to an outlier, and affect the range • The range does not always reveal different variation about the mean STA 291 Fall 2009 Lecture 6

  4. Example • Sample 1 • Smallest Observation: 112 • Largest Observation: 797 • Range = • Sample 2 • Smallest Observation: 15033 • Largest Observation: 16125 • Range = STA 291 Fall 2009 Lecture 6

  5. Deviations • The deviation of the ith observation xi from the sample mean is the difference between them, • Sum of all deviations is zero • Therefore, we use either the sum of the absolute deviations or the sum of the squared deviations as a measure of variation STA 291 Fall 2009 Lecture 6

  6. Sample Variance • Variance of nobservations is the sum of the squared deviations, divided by n-1 STA 291 Fall 2009 Lecture 6

  7. Example STA 291 Fall 2009 Lecture 6

  8. Interpreting Variance • About the average of the squared deviations • “average squared distance from the mean” • Unit • Square of the unit for the original data • Difficult to interpret • Solution • Take the square root of the variance, and the unit is the same as for the original data • Standard Deviation STA 291 Fall 2009 Lecture 6

  9. Properties of Standard Deviation • s ≥ 0 • s = 0 only when all observations are the same • If data is collected for the whole population instead of a sample, then n-1 is replaced by n • s is sensitive to outliers STA 291 Fall 2009 Lecture 6

  10. Variance and Standard Deviation • Sample • Variance • Standard Deviation • Population • Variance • Standard Deviation STA 291 Fall 2009 Lecture 6

  11. Population Parameters and Sample Statistics • Population mean and population standard deviation are denoted by the Greek letters μ (mu) and σ (sigma) • They are unknown constants that we would like to estimate • Sample mean and sample standard deviation are denoted by and s • They are random variables, because their values vary according to the random sample that has been selected STA 291 Fall 2009 Lecture 6

  12. Empirical Rule • If the data is approximately symmetric and bell-shaped then • About 68% of the observations are within one standard deviation from the mean • About 95% of the observations are within two standard deviations from the mean • About 99.7% of the observations are within three standard deviations from the mean STA 291 Fall 2009 Lecture 6

  13. Example • SAT scores are scaled so that they have an approximate bell-shaped distribution with a mean of 500 and standard deviation of 100 • About 68% of the scores are between • About 95% of the scores are between • If you have a score above 700, you are in the top % • What percentile would this be? STA 291 Fall 2009 Lecture 6

  14. Example • According to the National Association of Home Builders, the U.S. nationwide median selling price of homes sold in 1995 was $118,000 • Would you expect the mean to be larger, smaller, or equal to $118,000? • Which of the following is the most plausible value for the standard deviation? (a) –15,000, (b) 1,000, (c) 45,000, (d) 1,000,000 STA 291 Fall 2009 Lecture 6

  15. Coefficient of Variation • Standardized measure of variation • Idea • A standard deviation of 10 may indicate great variability or small variability, depending on the magnitude of the observations in the data set • CV = Ratio of standard deviation divided by mean • Population and sample version STA 291 Fall 2009 Lecture 6

  16. Example • Which sample has higher relative variability? (a higher coefficient of variation) • Sample A • mean = 62 • standard deviation = 12 • CV = • Sample B • mean = 31 • standard deviation = 7 • CV = STA 291 Fall 2009 Lecture 6

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