1. 3 - 1 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

2. 3 - 2 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 1. Calculate the arithmetic mean, the weighted mean, the median, the mode, and the geometric mean of a given data set. 2. Identify the relative positions of the arithmetic mean, median and mode for both symmetric and skewed distributions. When you have completed this chapter, you will be able to: Explain your choice of the measure of central tendency of data. 3. Point out the proper uses and common misuses of each measure. 4. 5. Explain the result of your analysis.

3. 3 - 3 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Five Measures of Central Tendency median arithmetic mean weighted mean mode geometric mean Average price of a house in Ottawa (2000) was $126 000 The average income of two parent families with children in Canada was $65,847 in 1995 and $72,910 in 1999. (StatCan) The average price of a house in Toronto in 1996 was $238,511 (StatCan) My grade point average for last semester was 4.0

4. 3 - 4 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Arithmetic Mean All values are used It is unique The sum of the deviations from the mean is 0 It is calculated by summing the values and dividing by the number of values It requires the interval scale …is the most widely used measure of location.

5. 3 - 5 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Population Mean Formula … is the population mean ( pronounced mu) … is the total number of observations … is a particular value … indicates the operation of adding (sigma)  N x 

6. 3 - 6 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. T erminology Parameter … is a measurable characteristic of a P opulation … is a measurable characteristic of a P opulation Statistic … is a measurable characteristic of a Sample … is a measurable characteristic of a Sample

7. 3 - 7 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. = 48 500 The Kiers family owns four cars. The following is the current mileage on each of the four cars: Find the mean mileage for the cars. Population Mean 56,000 23,000 42,000 73,000 56000 + 23000 + 42000 + 73000 4 56000 + 23000 + 42000 + 73000 4 = = Formula

8. 3 - 8 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Sample Mean … is the sample mean ( read “x bar”) … is the number of sample observations … is a particular value … indicates the operation of adding (sigma) n x  Formula

9. 3 - 9 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. A sample of five executives received the following bonuses last year ($000): 14.0 15.0 17.0 16.0 15.0 Determine the average bonus given last year: 14 + 15 + 17 + 16 + 15 5 14 + 15 + 17 + 16 + 15 5 = = 15.4 The average bonus given last year was $15 400 = 77 / 5 Formula

10. 3 - 10 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Properties of an Arithmetic Mean …Every set of interval-level and ratio- level data has a mean … All the values are included in computing the mean …A set of data has a unique mean …The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero! … The mean is affected by unusually large or small data values

11. 3 - 11 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. = 5 = 5 Arithmetic Mean as a Balance Point Illustrate the mean of the values 3, 8 and 4. = 15 / 3 = 15 / 3

12. 3 - 12 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 5 11 3 9 Determining the Mean in Excel

13. 3 - 13 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. See Using Click on Tools Click on DATA ANALYSIS See…

14. 3 - 14 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Highlight DESCRIPTIVE STATISTICS …Click OK Using See… See

15. 3 - 15 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using See INPUT NEEDS A3:A42 Click on OK See…

16. 3 - 16 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using See Solution Alternate solution…

17. 3 - 17 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using CLICK ON PASTE FUCTION See… CLICK ON See

18. 3 - 18 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using See… SCROLL DOWN TO STATISTICAL

19. 3 - 19 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using Click on OK HIGHLIGHT AVERAGE IN RIGHT MENU See See…

20. 3 - 20 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using See The mean (average) is placed in the cell on the worksheet where your cursor was when you began.

21. 3 - 21 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Weighted Mean The weighted mean of a set of numbers x 1, x 2,... x n, with corresponding weights w 1, w 2,...,w n, is computed from the following formula:

22. 3 - 22 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 89.0$  50.44$  )50.0($5 15 5  ).1($15  )90.0($15  )75.0($15   w μ During a one hour period on a hot Saturday afternoon cabana boy Chris served fifty drinks. He sold: …five drinks for $0.50 …fifteen for $0.75 …fifteen for $0.90 …fifteen for $1.10 Compute: - the weighted mean of the price of the drinks -

23. 3 - 23 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The Median is the midpoint of the values after they have been ordered from the smallest to the largest There are as many values above the median as below it in the data array The Median For an even set of values, the median will be the arithmetic average of the two middle numbers

24. 3 - 24 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The ages for a sample of five college students are: 21, 25, 19, 20, 22 The heights of four basketball players, in inches, are: 76, 73, 80, 75 Thus the median is 21 Thus the median is 75.5 Arranging the data in ascending order gives: 19, 20, 21, 22, 25 Arranging the data in ascending order gives: 73, 75, 76, 80

25. 3 - 25 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Properties of the Median There is a unique median for each data set It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur It can be computed for ratio-level, interval-level, and ordinal-level data It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class

26. 3 - 26 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The Mode The Mode is the value of the observation that appears most frequently used The exam scores for ten students are: 81, 93, 84, 75, 68, 87, 81, 75, 81, 87 The score of 81 occurs the most often …it is the Mode!

27. 3 - 27 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The Geometric Mean (GM) of a set of n numbers is defined as the nth root of the product of the n numbers. The geometric mean is used to average percents, indexes, and relatives. The formula is: Geometric Mean GMxxxx n n  ()( )...() 123

28. 3 - 28 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The interest rate on three bonds was 5, 21, and 4 percent The Geometric Mean is: Geometric Mean 49.7)4)(21)(5( 3  GM The arithmetic mean is (5+21+4)/3 =10.0 The GM gives a more conservative profit figure because it is not heavily weighted by the rate of 21percent

29. 3 - 29 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another. Geometric Mean continued… The formula is:  n GM = (Value at end of period) (Value at beginning of period)

30. 3 - 30 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in 2000. Geometric Mean continued… i.e. the Geometric Mean rate of increase is 1.27%.

31. 3 - 31 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Determining the Median, Mode or Geometric Mean in Excel

32. 3 - 32 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using Click on Tools See… Click DATA ANALYSIS

33. 3 - 33 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using Highlight DESCRIPTIVE STATISTICS SUMMARY STATISTICS INPUT NEEDS See SOLUTION

34. 3 - 34 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using Solution Alternate solution… The geometric mean doesn’t show up in summary statistics!

35. 3 - 35 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using CLICK ON PASTE FUCTION See… CLICK ON See

36. 3 - 36 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using SCROLL DOWN to STATISTICAL See…

37. 3 - 37 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using OR… HIGHLIGHT MEDIAN IN RIGHT MENU See

38. 3 - 38 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using See OR… HIGHLIGHT GEOMETRIC MEAN IN RIGHT MENU

39. 3 - 39 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using See See… HIGHLIGHT MODE IN RIGHT MENU

40. 3 - 40 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The calculated values are placed in the cell on the worksheet where your cursor was when you began Using New

41. 3 - 41 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Countries VisitedExpenditures ($Cdn millions) Australia 227 Cuba 265 Dominican Rep. 122 France 506 Germany 183 Hong Kong 138 Ireland 114 Italy 283 Japan 150 Mexico 557 Netherlands 107 Spain 105 Switzerland 91 United Kingdom 1009 United States 8401 The following table shows the expenditures of Canadians in 15 countries they visited in 1999 Source: Statistics Canada, Tourism and the Centre for Education Statistics

42. 3 - 42 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Is the mean or median expenditure a more accurate reflection of the “average” Canadian out-of-country expenditure? What happens to the values of the mean and median when you remove the United States expenditures from the sample? …if you remove both the UK and US from the sample? What happens to the values of the mean and median when you remove the United States expenditures from the sample? …if you remove both the UK and US from the sample?

43. 3 - 43 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using The mean is strongly affected by the inclusion of these two OUTLIERS … therefore, the median is a more appropriate measure of “average” in this case

44. 3 - 44 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The Mean of Grouped Data The mean of a sample of data organized in a frequency distribution is computed by the following formula: N fx  

45. 3 - 45 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. A sample of ten movie theatres in a metropolitan area tallied the total number of movies showing last week. Compute the mean number of movies showing per theatre. The Mean of Grouped Data

46. 3 - 46 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Continued… 6610Total 301039 to under 11 8817 to under 9 18635 to under 7 8423 to under 5 2211 to under 3 (f)(x) Class Midpoint Frequency f Movies Showing The Mean of Grouped Data The Mean of Grouped Data N fx  

47. 3 - 47 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. = 6.6 10 66  Continued… (f)(x) Class Midpoint Frequency f Movies Showing 6610Total Formula n Xf   The Mean of Grouped Data The Mean of Grouped Data N fx  

48. 3 - 48 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 610 6532.5 137.527.5 13522.5 21017.5 62.512.5 30Total 230 to under 35 525 to under 30 620 to under 25 12 15 to under 20 510 to under 15 (f)(x) Class Midpoint Frequency f Hours Studying Determine the average student study time The Mean of Grouped Data The Mean of Grouped Data N fx   = 20.33 30 610  Formula N fx  

49. 3 - 49 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Finding the Median of Grouped Data 1. Construct a cumulative f requency distribution To determine the median class for Grouped Data: 2. Divide the total number of data values by 2 3. Determine which class will contain this value E.g. If n = 50, 50/2 = 25, then determine which class will contain the 25 th value

50. 3 - 50 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. )( 2  i f CF N L +Median = … is the lower limit of the median class … is the cumulative frequency as you enter the median class … is the frequency of the median class … is the class interval or size L CF f i Finding the Median of Grouped Data Estimate the median value within chosen class…

51. 3 - 51 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Cumulative f 10Total 1039 to under 11 717 to under 9 635 to under 7 323 to under 5 111 to under 3 Frequency f Movies Showing 2 - 3 10 = 5 + = 6.33 3 2 )( 2  i f CF N L + CF Median class L f i = 2 Finding the Median of Grouped Data

52. 3 - 52 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The Mode of Grouped Data The mode for grouped data is approximated by the midpoint of the class with the largest class frequency 1039 to under 11 817 to under 9 635 to under 7 423 to under 5 211 to under 3 Class Midpoint Frequency f Movies Showing This is considered to be BiModal

53. 3 - 53 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Approximate the Mode of this distribution The Mode of Grouped Data 32.5 27.5 22.5 17.5 12.5 30Total 230 to under 35 525 to under 30 620 to under 25 12 15 to under 20 510 to under 15 Class Midpoint Frequency f Hours Studying The modal class is 15 to under 20, approximately 17.5

54. 3 - 54 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. zero skewness mode = median = mean Symmetric Distribution

55. 3 - 55 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Right Skewed Distribution Mean and Median are to the right of the Mode Skewed Right Positively skewed Mode< Median< Mean

56. 3 - 56 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Left Skewed Distribution Mean and Median are to the left of the Mode Negatively skewed Skewed left < Mode < Median Mean

57. 3 - 57 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Test your learning … www.mcgrawhill.ca/college/lind Click on… Online Learning Centre for quizzes extra content data sets searchable glossary access to Statistics Canada’s E-Stat data …and much more!

58. 3 - 58 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. This completes Chapter 3