1. Sect.2.4 Measures of variation Objective: SWBAT find the range of a data set Find the variance and standard deviation of a population and of a sample How to approximate the sample standard deviation for grouped data

2. Definition The range of a data set is the difference between the maximum and minimum data entries in the set. Range=(Maximum entry)-(Minimum entry)

3. Starting salaries for corporations A(100’s of dollars) Salary41383945474144413742 Salary40234150493241295258 Starting salaries for corporations B (100’s of dollars)

4. Solution: Ordering the data helps to find the least and greatest salaries. 3738 39 41 41 41 42 44 45 47 Range = 47 – 37 = 10 So the range of starting salaries for Corporation A is 10 or $10,000

5. Definition The deviation of an entry x in a population data set is the difference between the entry and the mean µ of the data set. Deviation of x = x - µ.

6. Finding the deviations of a data set Solution: The mean starting salary is µ= 415/10 = 41.5 To find out how much each salary deviates from the mean, subtract 41.5 from each salary. For instance, the deviation of 41 (or $41,000) is 41 - 41.5 = -0.5 (or -$500)

7. Salary (1000’s of Dollars) Deviation (1000 of dollars) 41 38 39 45 47 41 44 41 37 42 -0.5 -3.5 -2.5 3.5 5.5 -0.5 2.5 -0.5 -4.5 0.5 ∑ x = 415∑ (x- µ)= 0 Definition The population variance of a population data set of N entries is Population variance is

8. Population Standard Deviation Guidelines Finding the Population variance and Standard Deviation 1.Find the mean of the population data set 2.Find the deviation of each entry 3.Square each deviation 4.Add to get the sum of the squares 5.Divide by N to get the variance. 6. Find the square root of the variance to get The population standard deviation

9. Sum of Squares of starting salaries SalaryDeviationSquares 41 38 39 45 47 41 44 41 37 42 -0.5 -3.5 -2.5 3.5 5.5 -0.5 2.5 -0.5 -4.5 0.5 0.25 12.25 6.25 12.25 30.25 0.25 6.25 0.25 20.25 0.25

10. Homework – Copy the following 1.) 11 10 8 4 6 7 11 6 11 7 2.) 13 23 15 13 18 13 15 14 20 20 18 17 20 13 3.) 15 8 12 5 19 14 8 6 13 4.) 24 26 27 23 9 14 8 8 26 15 15 27 11

11. Measures of Variation Section 2.4

12. Closing prices for two stocks were recorded on ten successive Fridays. Calculate the mean, median and mode for each. Mean = 61.5 Median = 62 Mode = 67 Mean = 61.5 Median = 62 Mode = 67 56 33 56 42 57 48 58 52 61 57 63 67 67 77 67 82 67 90 Stock AStock B Two Data Sets

13. Range for A = 67 – 56 = $11 Range = Maximum value – Minimum value Range for B = 90 – 33 = $57 The range is easy to compute but only uses two numbers from a data set. Measures of Variation

14. The deviation for each value x is the difference between the value of x and the mean of the data set. In a population, the deviation for each value x is: Measures of Variation To learn to calculate measures of variation that use each and every value in the data set, you first want to know about deviations. In a sample, the deviation for each value x is:

15. – 5.5 – 4.5 – 3.5 – 0.5 1.5 5.5 56 57 58 61 63 67 Deviations 56 – 61.5 57 – 61.5 58 – 61.5 Stock ADeviation The sum of the deviations is always zero.

16. Population Variance Sum of squares – 5.5 – 4.5 – 3.5 – 0.5 1.5 5.5 x 56 57 58 61 63 67 30.25 20.25 12.25 0.25 2.25 30.25 188.50 Population Variance: The sum of the squares of the deviations, divided by N.

17. Population Standard Deviation Population Standard Deviation: The square root of the population variance. The population standard deviation is $4.34.

18. Sample Variance and Standard Deviation To calculate a sample variance divide the sum of squares by n – 1. The sample standard deviation, s, is found by taking the square root of the sample variance.

19. Summary Range = Maximum value – Minimum value Sample Standard Deviation Sample Variance Population Standard Deviation Population Variance

20. Data with symmetric bell-shaped distribution have the following characteristics. About 68% of the data lies within 1 standard deviation of the mean About 99.7% of the data lies within 3 standard deviations of the mean About 95% of the data lies within 2 standard deviations of the mean –4–3–2–101234 Empirical Rule (68-95-99.7%) 13.5% 2.35%

21. The mean value of homes on a street is $125 thousand with a standard deviation of $5 thousand. The data set has a bell shaped distribution. Estimate the percent of homes between $120 and $135 thousand. Using the Empirical Rule $120 thousand is 1 standard deviation below the mean and $135 thousand is 2 standard deviations above the mean. 68% + 13.5% = 81.5% So, 81.5% have a value between $120 and $135 thousand. 125130135120140145115110105

22. Chebychev’s Theorem For k = 3, at least 1 – 1/9 = 8/9 = 88.9% of the data lie within 3 standard deviation of the mean. For any distribution regardless of shape the portion of data lying within k standard deviations (k > 1) of the mean is at least 1 – 1/k 2. For k = 2, at least 1 – 1/4 = 3/4 or 75% of the data lie within 2 standard deviation of the mean.

23. Chebychev’s Theorem The mean time in a women’s 400-meter dash is 52.4 seconds with a standard deviation of 2.2 sec. Apply Chebychev’s theorem for k = 2. 52.454.656.85950.24845.8 2 standard deviations At least 75% of the women’s 400-meter dash times will fall between 48 and 56.8 seconds. Mark a number line in standard deviation units. A

24. Homework 1-12 and 13-23 all, 29-34 pgs. 84-88