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3-2: Measures of Variation

Learn how to calculate and interpret measures of variation and analyze what they can tell you about a set of data. Includes examples and explanations of range, variance, and standard deviation.

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3-2: Measures of Variation

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  1. 3-2: Measures of Variation SWBAT: calculate and interpret measures of variation and analyze what these measures can tell them about a set of data. Warm-up/quiz HW?s Notes: Measures of variation Assignment…

  2. Warm-up/quiz

  3. HW?s

  4. 3-2 Measures of Variation How Can We Measure Variability? Range Variance Standard Deviation Coefficient of Variation Chebyshev’s Theorem Empirical Rule (Normal) Bluman, Chapter 3 4

  5. Measures of Variation: Range The range is the difference between the highest and lowest values in a data set. Bluman, Chapter 3 5

  6. Chapter 3Data Description Section 3-2 Example 3-18/19 Page #123 Bluman, Chapter 3 6

  7. Example 3-18/19: Outdoor Paint Two experimental brands of outdoor paint are tested to see how long each will last before fading. Six cans of each brand constitute a small population. The results (in months) are shown. Find the mean and range of each group. Bluman, Chapter 3 7

  8. Example 3-18: Outdoor Paint The average for both brands is the same, but the range for Brand A is much greater than the range for Brand B. Which brand would you buy? Bluman, Chapter 3 8

  9. Measures of Variation: Variance & Standard Deviation The variance is the average of the squares of the distance each value is from the mean. The standard deviation is the square root of the variance. The standard deviation is a measure of how spread out your data are. Bluman, Chapter 3 9

  10. Uses of the Variance and Standard Deviation To determine the spread of the data. To determine the consistency of a variable. To determine the number of data values that fall within a specified interval in a distribution (Chebyshev’s Theorem). Used in inferential statistics. Bluman, Chapter 3 10

  11. Measures of Variation: Variance & Standard Deviation (Population Theoretical Model) The population variance is The population standard deviation is Bluman, Chapter 3 11

  12. Chapter 3Data Description Section 3-2 Example 3-21 Page #125 Bluman, Chapter 3 12

  13. Example 3-21: Outdoor Paint Find the variance and standard deviation for the data set for Brand A paint. 10, 60, 50, 30, 40, 20 -25 25 15 -5 5 -15 625 625 225 25 25 225 35 35 35 35 35 35 1750 Bluman, Chapter 3 13

  14. Measures of Variation: Variance & Standard Deviation(Sample Theoretical Model) The sample variance is The sample standard deviation is Bluman, Chapter 3 14

  15. Why n – 1? • We use the sample variance to estimate the population variance • When the sample is small (< 30), it may underestimate the population variance • n – 1 makes the sample variance larger, likely giving us a better estimate for the population

  16. Measures of Variation: Variance & Standard Deviation(Sample Computational Model) Is mathematically equivalent to the theoretical formula. Saves time when calculating by hand Does not use the mean Is more accurate when the mean has been rounded. Bluman, Chapter 3 16

  17. Measures of Variation: Variance & Standard Deviation(Sample Computational Model) The sample variance is The sample standard deviation is Bluman, Chapter 3 17

  18. Chapter 3Data Description Section 3-2 Example 3-23 Page #129 Bluman, Chapter 3 18

  19. Example 3-23: European Auto Sales Find the variance and standard deviation for the amount of European auto sales for a sample of 6 years. The data are in millions of dollars. 11.2, 11.9, 12.0, 12.8, 13.4, 14.3 125.44 141.61 166.41 163.84 179.56 204.49 75.6 958.94 Bluman, Chapter 3 19

  20. Measures of Variation: Coefficient of Variation The coefficient of variation is the standard deviation divided by the mean, expressed as a percentage. Use CVAR to compare standard deviations when the units are different. Bluman, Chapter 3 20

  21. Chapter 3Data Description Section 3-2 Example 3-25 Page #132 Bluman, Chapter 3 21

  22. Example 3-25: Sales of Automobiles The mean of the number of sales of cars over a 3-month period is 87, and the standard deviation is 5. The mean of the commissions is $5225, and the standard deviation is $773. Compare the variations of the two. Commissions are more variable than sales. Bluman, Chapter 3 22

  23. Measures of Variation: Range Rule of Thumb The Range Rule of Thumb approximates the standard deviation as when the distribution is unimodal and approximately symmetric. Bluman, Chapter 3 24

  24. Measures of Variation: Range Rule of Thumb Use to approximate the lowest value and to approximate the highest value in a data set. Bluman, Chapter 3 25

  25. Measures of Variation: Chebyshev’s Theorem The proportion of values from any data set that fall within k standard deviations of the mean will be at least 1-1/k2, where k is a number greater than 1 (k is not necessarily an integer). Bluman, Chapter 3 26

  26. Measures of Variation: Chebyshev’s Theorem The proportion of values from any data set that fall within k standard deviations of the mean will be at least 1-1/k2, where k is a number greater than 1 (k is not necessarily an integer). # of Minimum Proportion Minimum Percentage standard within k standard within k standard deviations, k deviations deviations 2 75% 1-1/4=3/4 88.89% 1-1/9=8/9 3 93.75% 1-1/16=15/16 4 Bluman, Chapter 3 27

  27. Measures of Variation: Chebyshev’s Theorem Bluman, Chapter 3 28

  28. Chapter 3Data Description Section 3-2 Example 3-27 Page #135 Bluman, Chapter 3 29

  29. Example 3-27: Prices of Homes The mean price of houses in a certain neighborhood is $50,000, and the standard deviation is $10,000. Find the price range for which at least 75% of the houses will sell. Chebyshev’s Theorem states that at least 75% of a data set will fall within 2 standard deviations of the mean. 50,000 – 2(10,000) = 30,000 50,000 + 2(10,000) = 70,000 At least 75% of all homes sold in the area will have a price range from $30,000 and $75,000. Bluman, Chapter 3 30

  30. Chapter 3Data Description Section 3-2 Example 3-28 Page #135 Bluman, Chapter 3 31

  31. Example 3-28: Travel Allowances A survey of local companies found that the mean amount of travel allowance for executives was $0.25 per mile. The standard deviation was 0.02. Using Chebyshev’s theorem, find the minimum percentage of the data values that will fall between $0.20 and $0.30. At least 84% of the data values will fall between $0.20 and $0.30. Bluman, Chapter 3 32

  32. Measures of Variation: Empirical Rule (Normal) The percentage of values from a data set that fall within k standard deviations of the mean in a normal (bell-shaped) distribution is listed below. Bluman, Chapter 3 33

  33. Measures of Variation: Empirical Rule (Normal) Bluman, Chapter 3 34

  34. Assignment • Pg 137: 1-4, 8, 26 • Pg 137: 7, 10, 12, 18, 21, 23, 29, 30, 33, 34, 37

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