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1.2 Describing Distributions with Numbers

AP Statistics. 1.2 Describing Distributions with Numbers. Fuel Economy for 2004 car models. Mean and Median. Mean: The Average Median: The Middle. Mean Highway mileage for two seaters. Caution!!!

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1.2 Describing Distributions with Numbers

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  1. AP Statistics 1.2 Describing Distributions with Numbers

  2. Fuel Economy for 2004 car models

  3. Mean and Median • Mean: The Average • Median: The Middle

  4. Mean Highway mileage for two seaters Caution!!! The mean is sensitive to the influence of a few extreme observations. It is not a resistant measure of center.

  5. Measuring Spread • The simplest useful numerical description of a distribution includes a description of both the center and the spread. • Range: simplest measure of spread. The highest value minus the lowest value. • Pth percentile of a distribution is the percent of observations that fall below it.

  6. The Quartiles

  7. Calculating Quartiles The highway mileage of 20 2-seaters arranged in numerical order are: 13 15 16 16 17 19 20 22 23 23 |23 24 25 25 26 28 28 28 29 32 The median is marked by |. The 1st Quartile (Q1) is the median of the 10 numbers to the left of the median. The 3rd Quartile (Q1) is the median of the 10 numbers to the right of the median. If there arean odd number of observations, the median is the middle number and it is excluded from calculations of Q1 and Q3.

  8. The Five–Number Summary and Boxplots

  9. The Five–Number Summary and Boxplots

  10. Boxplots of the highway and city gas mileages for cars classified as two-seaters and minicompacts.

  11. Outliers • Lower Bound: Q1 – 1.5(IQR) • If an observation falls below the lower bound it is considered an outlier. • Upper Bound: Q3 + 1.5(IQR) • If an observation falls above the upper bound it is considered an outlier.

  12. Measuring Spread: Variance and Standard Deviation

  13. Some FAQs • Why do we square the deviations? • If not, they would all add to zero. • Why do we emphasize the standard deviation rather than the variance? • The standard deviation measures spread about the mean in the original scale. • Why do we average by dividing by n-1 rather than n in calculating the variance? • Because the sum of the deviations is always zero, the last deviation can be found once we know the other n-1. The number n-1 is called the degrees of freedom.

  14. Choosing Measures of Center and Spread

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