Standard Deviation. Lecture 18 Section Robb T. Koether. Hampden-Sydney College. Mon, Sep 26, 2011

Transcription

1 Standard Deviation Lecture 18 Section Robb T. Koether Hampden-Sydney College Mon, Sep 26, 2011 Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

2 Outline 1 Variability 2 The Standard Deviation Examples Alternate Formula 3 TI-83 Standard Deviations 4 Interpreting the Standard Deviation 5 Assignment 6 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

3 Quintiles RT-D Editorial Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

4 Outline 1 Variability 2 The Standard Deviation Examples Alternate Formula 3 TI-83 Standard Deviations 4 Interpreting the Standard Deviation 5 Assignment 6 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

5 Variability Our ability to estimate a parameter accurately depends on the variability of the estimator. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

6 Variability Our ability to estimate a parameter accurately depends on the variability of the estimator. That, in turn, depends on the variability that is inherent in the population. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

7 Variability Our ability to estimate a parameter accurately depends on the variability of the estimator. That, in turn, depends on the variability that is inherent in the population. The more variability in the population, the more variable the estimator. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

8 Variability Our ability to estimate a parameter accurately depends on the variability of the estimator. That, in turn, depends on the variability that is inherent in the population. The more variability in the population, the more variable the estimator. What do we mean by variability in the population? Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

9 Variability Our ability to estimate a parameter accurately depends on the variability of the estimator. That, in turn, depends on the variability that is inherent in the population. The more variability in the population, the more variable the estimator. What do we mean by variability in the population? How do we measure it? Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

10 An Example A person offers you $100 if you can predict the high temperature on March 15, 2012 or on July 15, 2012 to within 5. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

11 An Example A person offers you $100 if you can predict the high temperature on March 15, 2012 or on July 15, 2012 to within 5. Your choice of dates. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

12 An Example A person offers you $100 if you can predict the high temperature on March 15, 2012 or on July 15, 2012 to within 5. Your choice of dates. For which date should you choose to predict the high temperature? Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

13 An Example A person offers you $100 if you can predict the high temperature on March 15, 2012 or on July 15, 2012 to within 5. Your choice of dates. For which date should you choose to predict the high temperature? On which date is the high temperature less variable? Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

14 An Example A person offers you $100 if you can predict the high temperature on March 15, 2012 or on July 15, 2012 to within 5. Your choice of dates. For which date should you choose to predict the high temperature? On which date is the high temperature less variable? Naturally, you should choose the date with less variability. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

15 An Example Here are boxplots for the daily highs on March 15 and July 15 for the past 30 years. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

16 An Example Here are boxplots for the daily highs on March 15 and July 15 for the past 30 years. July 15 March Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

17 An Example Here are boxplots for the daily highs on March 15 and July 15 for the past 30 years. July 15 March Which is more variable? Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

18 An Example Here are boxplots for the daily highs on March 15 and July 15 for the past 30 years. July 15 March Which is more variable? How much more variable is it? Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

19 Measures of Variability The IQR is one measure of variability. IQR for March 15 data = = 17. IQR for July 15 data = = 7. By that measure, the July temperatures are about variable as the March temperatures. We will also measure variability by using the variance. the standard deviation. times as Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

20 Deviations from the Mean Definition (Deviation) The deviation of an observation x is the difference between x and the sample mean x. deviation of x = x x. For a member of the population, the deviation is measured from the population mean: deviation of x = x µ. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

21 Deviations from the Mean mean Deviations from the mean Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

22 Deviations from the Mean deviation = Deviations from the mean Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

23 Deviations from the Mean deviation = Deviations from the mean Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

24 Deviations from the Mean dev = Deviations from the mean Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

25 Deviations from the Mean deviation = Deviations from the mean Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

26 Deviations from the Mean deviation = Deviations from the mean Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

27 Outline 1 Variability 2 The Standard Deviation Examples Alternate Formula 3 TI-83 Standard Deviations 4 Interpreting the Standard Deviation 5 Assignment 6 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

28 Deviations from the Mean How do we obtain one number that is representative of the whole set of individual deviations? Normally we use an average to summarize a set of numbers. Why will the average not work in this case? Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

29 Deviations from the Mean How do we obtain one number that is representative of the whole set of individual deviations? Normally we use an average to summarize a set of numbers. Why will the average not work in this case? It will not work because (x x) = 0. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

30 Sum of Squared Deviations Rather than average the deviations, we will average their squares. That way, there will be no canceling. So we compute first the sum of the squared deviations. Definition (Sum of squared deviations) The sum of squared deviations, denoted SSX, of a set of numbers is the sum of the squares of their deviations from the mean of the set. SSX = (x x) 2. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

31 Sum of Squared Deviations To find SSX 1 Find the average: x = x n. 2 Find the deviations from the average: x x. 3 Square the deviations: (x x) 2. 4 Add them up: SSX = (x x) 2. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

32 Sum of Squared Deviations Example (Calculating SSX) Let the sample be {1, 4, 7, 8, 10}. Then SSX = (1 6) 2 + (4 6) 2 + (7 6) 2 + (8 6) 2 + (10 6) 2 = ( 5) 2 + ( 2) = = 50. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

33 Sum of Squared Deviations Practice Let the sample be {1, 5, 6, 9, 14}. Calculate The sample mean. The deviations. The squared deviations. The sum of the squared deviations. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

34 The Variance Definition (Variance of a population) The variance of a population, denoted σ 2, is the average of the squared deviations of the members of the population. σ 2 = (x µ) 2. N Definition (Variance of a sample) The variance of a sample, denoted s 2, is the sum of the squared deviations of the members of the sample, divided by 1 less than the sample size. (x x) s 2 2 =. n 1 Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

35 The Sample Variance Theory shows that if we divide (x x) 2 by n 1 instead of n, then s 2 will be a better estimator of σ 2. If we divide by n, then s 2 will systematically underestimate σ 2. Therefore, we do divide by n 1. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

36 The Sample Variance Definition (Standard deviation of a population) The standard deviation of a population, denoted σ, is the square root of the population variance. (x µ) 2 σ = N. Definition (Standard deviation of a sample) The standard deviation of a sample, denoted s, is the square root of the sample variance. (x x) 2 s = n 1. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

37 Outline 1 Variability 2 The Standard Deviation Examples Alternate Formula 3 TI-83 Standard Deviations 4 Interpreting the Standard Deviation 5 Assignment 6 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

38 Example Example (Calculating s) For the sample {1, 4, 7, 8, 10}, we found that SSX = 50. Therefore, and so s 2 = 50 4 = 12.5 s = 12.5 = Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

39 Sum of Squared Deviations Practice Let the sample be {1, 5, 6, 9, 14}. Calculate s 2 and s. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

40 Example How does s compare to the individual deviations? We will interpret s as being representative of the deviations in the sample. Does that seem reasonable for the previous examples? Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

41 Example It turns out that the standard deviation of the March 15 high temperatures is The standard deviation of the July 15 high temperatures is 4.5. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

42 Outline 1 Variability 2 The Standard Deviation Examples Alternate Formula 3 TI-83 Standard Deviations 4 Interpreting the Standard Deviation 5 Assignment 6 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

43 Alternate Formula for SSX An alternate formula for SSX is Then, as before and SSX = x 2 ( x) 2. n s 2 = SSX n 1 s = SSX n 1. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

44 Example Example (Alternate formula for SSX) Let the sample be {1, 4, 7, 8, 10}. Then x = 30 and x 2 = = 230. So SSX = = = 50. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

45 Sum of Squared Deviations Practice Let the sample be {1, 5, 6, 9, 14}. Find x. Find x 2. Use the alternate formula to find SSX, s 2, and s. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

46 Outline 1 Variability 2 The Standard Deviation Examples Alternate Formula 3 TI-83 Standard Deviations 4 Interpreting the Standard Deviation 5 Assignment 6 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

47 TI-83 - Standard Deviations TI-83 Standard Deviations Follow the procedure for computing the mean. The display shows Sx and σx. Sx is the sample standard deviation. σx is the population standard deviation. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

48 Example Example TI-83 Standard Deviations Let the sample be {1, 4, 7, 8, 10}. We get Sx = σx = Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

49 Sum of Squared Deviations Practice Let the sample be {1, 5, 6, 9, 14}. Use the TI-83 to find s and s 2. What are the values of x and x 2? Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

50 Outline 1 Variability 2 The Standard Deviation Examples Alternate Formula 3 TI-83 Standard Deviations 4 Interpreting the Standard Deviation 5 Assignment 6 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

51 Interpreting the Standard Deviation Observations that deviate from x by much more than s are unusually far from the mean. Observations that deviate from x by much less than s are unusually close to the mean. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

52 Interpreting the Standard Deviation x Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

53 Interpreting the Standard Deviation s s x - s x x + s Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

54 Interpreting the Standard Deviation Close, but not unusually close to x x - s x x + s Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

55 Interpreting the Standard Deviation Unusually close to x x - s x x + s Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

56 Interpreting the Standard Deviation s s x - 2s x - s x x + s x + 2s Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

57 Interpreting the Standard Deviation Far, but not unusually far from x s s x - 2s x - s x x + s x + 2s Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

58 Interpreting the Standard Deviation Unusually far from x x - 2s x - s x x + s x + 2s Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

59 Outline 1 Variability 2 The Standard Deviation Examples Alternate Formula 3 TI-83 Standard Deviations 4 Interpreting the Standard Deviation 5 Assignment 6 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

60 Assignment Homework Read Section 5.3.4, pages Let s Do It! 5.13, 5.14, Page 333, exercises 10, 11, 14, 16-18, 20, 21. Chapter 5 review, p. 345, exercises 29-32, 36-40, 42-44, 47, 52, 53, 55. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

61 Outline 1 Variability 2 The Standard Deviation Examples Alternate Formula 3 TI-83 Standard Deviations 4 Interpreting the Standard Deviation 5 Assignment 6 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

62 Answers to Even-numbered Exercises Page 333, Exercises 10, 14, 16, 18, , 5, 5, (a) false. (b) false. (c) true. (d) false (a) Shelf 1: IQR= 8, s = Shelf 2: IQR= 6, s = Shelf 3: IQR= 6.5, s = (b) Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

63 Answers to Even-numbered Exercises Page 345, Exercises 30, 32, 36, 40, 42, 44, (a) Data Set I: mean = 4.1, median = 6, mode = 6. Data Set II: mean = 11.86, median = 8, mode = 6 and 8. (b) The mean is a poor measure for Data Set II because its value is greater than all but one of the data values (a) (i) Median = 7. (ii) Mean = 10. (iii) Range = 6. (iv) Standard deviation = 2. (v) 80 th percentile = 11. (b) The range and the standard deviation. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

64 Answers to Even-numbered Exercises Page 345, Exercises 30, 32, 36, 40, 42, 44, (a) True. (b) True. (c) False. (d) False. (It almost always changes.) (e) False. (It is almost always greater.) 5.40 (a) Min = 10, Q1 = 10.5, Med = 12, Q3 = 14, Max = (b) Launcher C. (c) Launcher B. (d) No. The mean cannot be determined from a five-number summary. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42

65 Answers to Even-numbered Exercises Page 345, Exercises 30, 32, 36, 40, 42, 44, (a) Min = 440, Q1 = 585, Med = 705, Q3 = 795, Max = (b) C. (c) A. Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, / 42