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Section 2.4

Section 2.4. Measures of Variation. Larson/Farber 4th ed. Section 2.4 Objectives. Determine the range of a data set Determine the variance and standard deviation of a population and of a sample Use the Empirical Rule and Chebychev’s Theorem to interpret standard deviation

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Section 2.4

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  1. Section 2.4 Measures of Variation Larson/Farber 4th ed.

  2. Section 2.4 Objectives • Determine the range of a data set • Determine the variance and standard deviation of a population and of a sample • Use the Empirical Rule and Chebychev’s Theorem to interpret standard deviation • Approximate the sample standard deviation for grouped data Larson/Farber 4th ed.

  3. Range Range • The difference between the maximum and minimum data entries in the set. • The data must be quantitative. • Range = (Max. data entry) – (Min. data entry) Larson/Farber 4th ed.

  4. Example: Finding the Range A sample of annual salaries (in thousands of dollars) for private school teachers. Find the range of the salaries. 21.8 18.4 20.3 17.6 19.7 18.3 19.4 20.8 Larson/Farber 4th ed.

  5. minimum maximum Solution: Finding the Range • Ordering the data helps to find the least and greatest salaries. 17.6 18.3 18.4 19.4 19.7 20.3 20.8 21.8 • Range = (Max. salary) – (Min. salary) = 21.8 – 17.6 = 4.2 The range of starting salaries is 4.2 or $4,200. Larson/Farber 4th ed.

  6. Deviation, Variance, and Standard Deviation Deviation • The difference between the data entry, x, and the mean of the data set. • Population data set: • Deviation of x = x – μ • Sample data set: • Deviation of x = x – x Larson/Farber 4th ed.

  7. Example: Finding the Deviation A sample of annual salaries (in thousands of dollars) for private school teachers. Find the range of the salaries. 21.8 18.4 20.3 17.6 19.7 18.3 19.4 20.8 • Solution: • First determine the mean annual salary. Larson/Farber 4th ed.

  8. Solution: Finding the Deviation • Determine the deviation for each data entry. Σ(x – μ) = 0 Larson/Farber 4th ed.

  9. Finding the Sample Variance & Standard Deviation In Words In Symbols • Find the mean of the sample data set. • Find deviation of each entry. • Square each deviation. • Add to get the sum of squares. Larson/Farber 4th ed.

  10. In Words In Symbols Finding the Sample Variance & Standard Deviation • Divide by n – 1 to get the sample variance. • Find the square root to get the sample standard deviation. Larson/Farber 4th ed.

  11. Finding the Population Variance & Standard Deviation In Words In Symbols • Find the mean of the population data set. • Find deviation of each entry. • Square each deviation. • Add to get the sum of squares. x – μ (x – μ)2 SSx = Σ(x – μ)2 Larson/Farber 4th ed.

  12. In Words In Symbols Finding the Population Variance & Standard Deviation • Divide by N to get the population variance. • Find the square root to get the population standard deviation. Larson/Farber 4th ed.

  13. Compare Variance

  14. Example: Finding the Standard Deviation A sample of annual salaries (in thousands of dollars) for private school teachers. Find the range of the salaries. 21.8 18.4 20.3 17.6 19.7 18.3 19.4 20.8 Larson/Farber 4th ed.

  15. Solution: Finding the Standard Deviation • Determine SSx • n = 8 Larson/Farber 4th ed.

  16. Solution: Finding the Sample Variance Sample Variance Population Variance The sample variance is 1.99 or roughly 2 or 1,990. Larson/Farber 4th ed.

  17. Solution: Finding the Sample Standard Deviation Sample Standard Deviation The sample standard deviation is about 1.41 or 1410. Larson/Farber 4th ed.

  18. Interpreting Standard Deviation • Do Problem #26 Larson/Farber 4th ed.

  19. Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics: • About 68% of the data lie within one standard deviation of the mean. • About 95% of the data lie within two standard deviations of the mean. • About 99.7% of the data lie within three standard deviations of the mean. Larson/Farber 4th ed.

  20. 68% within 1 standard deviation 95% within 2 standard deviations 99.7% within 3 standard deviations 34% 34% 13.5% 13.5% 2.35% 2.35% Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) Larson/Farber 4th ed.

  21. Example: Using the Empirical Rule The mean value of land and buildings per acre from a sample of farms is $2400, with a standard deviation of $450. Between what values do about 95% of the data lie? What percent of the values are between $2400 and $3300? 2400 + 2(450) = 3300 2400 - 2(450) = 1500 Larson/Farber 4th ed.

  22. Solution: Using the Empirical Rule • Because the distribution is bell-shaped, you can use the Empirical Rule. 34% 13.5% $1050 $1500 $1950 $2400 $2850 $3300 $3750 34% + 13.5% = 47.5% of land values are between $2400 and $3300. Larson/Farber 4th ed.

  23. Chebychev’s Theorem • The portion of any data set lying within k standard deviations (k > 1) of the mean is at least: • k = 2: In any data set, at least of the data lie within 2 standard deviations of the mean. • k = 3: In any data set, at least of the data lie within 3 standard deviations of the mean. Larson/Farber 4th ed.

  24. Example: Using Chebychev’s Theorem The mean time in a women’s 400-meter dash is 57.07 seconds, with a standard deviation of 1.05. Using Chebychev’s Theorem for k = 2, 4, 6. 57.07 - 2(1.05) = 54.97 57.07 + 2(1.05) = 59.17 75% of the women came in between 54.97 and 59.17 seconds. Larson/Farber 4th ed.

  25. Standard Deviation for Grouped Data Sample standard deviation for a frequency distribution • When a frequency distribution has classes, estimate the sample mean and standard deviation by using the midpoint of each class. where n= Σf (the number of entries in the data set) Larson/Farber 4th ed.

  26. Example: Finding the Standard Deviation for Grouped Data Do #40 on page 97 Larson/Farber 4th ed.

  27. Section 2.4 Summary • Determined the range of a data set • Determined the variance and standard deviation of a population and of a sample • Used the Empirical Rule and Chebychev’s Theorem to interpret standard deviation • Approximated the sample standard deviation for grouped data • Homework 2.4 EOO Larson/Farber 4th ed.

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