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 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. 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. minimum maximum

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 Larson/Farber 4th ed. Solution: First determine the mean annual salary.

8. Solution: Finding the Deviation Larson/Farber 4th ed. Determine the deviation for each data entry. Salary, x Deviation: x – μ 19.54 17.617.6 - 19.54 =-1.94 18.318.3 - 19.54 =-1.24 18.418.4 - 19.54 =-1.14 19.419.4 - 19.54 =-0.14 19.719.7 - 19.54 =0.16 20.320.3 - 19.54 =0.76 20.820.8 - 19.54 =1.26 21.821.8 - 19.54 =2.26 Σx =156.30.00 Σ(x – μ) = 0

9. Finding the Sample Variance & Standard Deviation In Words In Symbols Larson/Farber 4th ed. 1.Find the mean of the sample data set. 2.Find deviation of each entry. 3.Square each deviation. 4.Add to get the sum of squares.

10. Finding the Sample Variance & Standard Deviation Larson/Farber 4th ed. 5.Divide by n – 1 to get the sample variance. 6.Find the square root to get the sample standard deviation. In Words In Symbols

11. Finding the Population Variance & Standard Deviation In Words In Symbols Larson/Farber 4th ed. 1.Find the mean of the population data set. 2.Find deviation of each entry. 3.Square each deviation. 4.Add to get the sum of squares. x – μ (x – μ) 2 SS x = Σ(x – μ) 2

12. Finding the Population Variance & Standard Deviation Larson/Farber 4th ed. 5.Divide by N to get the population variance. 6.Find the square root to get the population standard deviation. In Words In Symbols

13. Compare Variance PopulationSample

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 Larson/Farber 4th ed. Determine SS x n = 8 Salary, xDeviation: x – μ 19.54 117.617.6 - 19.54 =-1.943.75 218.318.3 - 19.54 =-1.241.53 318.418.4 - 19.54 =-1.141.29 419.419.4 - 19.54 =-0.140.02 519.719.7 - 19.54 =0.160.03 620.320.3 - 19.54 =0.760.58 720.820.8 - 19.54 =1.261.59 821.821.8 - 19.54 =2.265.12 Σx =156.313.92

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

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

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: Larson/Farber 4th ed. 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.

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

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? Larson/Farber 4th ed. 2400 + 2(450) = 3300 2400 - 2(450) = 1500

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

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

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. Larson/Farber 4th ed. 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.

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. Larson/Farber 4th ed. where n= Σf (the number of entries in the data set)

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

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.