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Chapter 3

Describing Data: Measures of Central Tendency. Chapter 3. 1. 2. 3. 4. 5. Chapter Goals. When you have completed this chapter, you will be able to:. Calculate the arithmetic mean , the weighted mean , the median , the mode , and the geometric mean of a given data set.

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Chapter 3

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  1. Describing Data: Measures of Central Tendency Chapter 3

  2. 1. 2. 3. 4. 5. Chapter Goals When you have completed this chapter, you will be able to: Calculate the arithmeticmean, the weightedmean, the median, the mode, and the geometricmean of a given data set. Identify the relative positions of the arithmetic mean, median and mode for both symmetric and skewed distributions. Point out the proper uses and common misuses of each measure. Explain your choice of the measure of central tendency of data. Explain the result of your analysis.

  3. FiveMeasures ofCentral Tendency • median arithmetic mean mode weighted mean geometric mean Examples Average price of a house in Ottawa (2000) was $126 000 The average price of a house in Toronto in 1996 was $238,511 (StatCan) The average income of two parent families with children in Canada was $65,847 in 1995 and $72,910 in 1999. (StatCan) My grade point average for last semester was 4.0

  4. Arithmetic Mean …is the most widely used measure of location. It is calculated by summing the values and dividing by the number of values It requires the interval scale All values are used Characteristics It is unique The sum of the deviations from the mean is 0

  5. Formula m N x  Population Mean where … is the population mean (pronouncedmu) … is the total number of observations … is a particular value … indicates the operation of adding (sigma)

  6. Terminology Parameter …is a measurable characteristic of a Population Statistic …is a measurable characteristic of a Sample

  7. Formula Q The Kiers family owns four cars. The following is the current mileage on each of the four cars: 56000 + 23000 + 42000 + 73000 4 = Population Mean Find the mean mileage for the cars. 56,000 23,000 42,000 73,000 = 48 500

  8. Formula n x  Sample Mean where …is the sample mean (read “x bar”) … is the number of sample observations … is a particular value … indicates the operation of adding (sigma)

  9. Q uestion A 14 + 15 + 17 + 16 + 15 5 = Formula 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: = 77 / 5 = 15.4 The average bonus given last year was $15 400

  10. 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 mean is affected by unusually large or small data values …The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero!

  11. Arithmetic Mean as a Balance Point Illustrate the mean of the values 3, 8 and 4. = 15 / 3 = 5

  12. Determining the Mean in Excel 2 9 5 11 3

  13. Using Click on Tools See Click on DATA ANALYSIS See…

  14. Using See HighlightDESCRIPTIVE STATISTICS …Click OK See…

  15. Using Click on OK INPUT NEEDS See A3:A42 See…

  16. Using See Solution Alternate solution…

  17. Using CLICK ON CLICK ONPASTE FUCTION See See…

  18. Using SCROLL DOWN TO STATISTICAL See…

  19. Using HIGHLIGHT AVERAGE IN RIGHT MENU Click on OK See See…

  20. Using See The mean (average) is placed in the cell on the worksheet where your cursor was when you began.

  21. Weighted Mean The weighted meanof a set of numbers x1, x2, ... xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula: Example

  22. + + + 5 ($ 0 . 50 ) 15 ($ 0 . 75 ) 15 ($ 0 . 90 ) 15 ($ 1 . 15 ) + + + 5 15 15 15 $ 44 . 50 = = $ 0 . 89 50 = μ w Example 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. The Median The Medianis themidpoint of the values after they have been ordered from the smallest to the largest Note There are as many values above the median as below it in the data array For an even set of values, the median will be the arithmetic average of the two middle numbers

  24. Examples The ages for a sample of five college students are: 21, 25, 19, 20, 22 A. Arranging the data in ascending order gives: 19, 20, 21, 22, 25 Thus the median is 21 The heights of four basketball players, in inches, are: 76, 73, 80, 75 B. Arranging the data in ascending order gives: 73, 75, 76, 80 Thus the median is 75.5

  25. 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 mediandoes not lie in an open-ended class

  26. TheMode TheModeis the value of the observation that appears most frequently used Example 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 theMode!

  27. = GM ( x )( x )( x ). . . ( x ) n 1 2 3 n Geometric Mean 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:

  28. = = 3 GM ( 5 )( 21 )( 4 ) 7 . 49 Geometric Mean Example The interest rate on three bonds was 5, 21, and 4 percent The Geometric Mean is: 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. Geometric Mean Examples continued… (Value at end of period) - 1 n GM = (Value at beginning of period) 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. The formula is:

  30. Geometric Mean Examples continued… The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in 2000. i.e. the Geometric Mean rate of increase is 1.27%.

  31. Determining the Median, Mode or Geometric Mean in Excel

  32. Using Click on Tools ClickDATA ANALYSIS See…

  33. Using HighlightDESCRIPTIVE STATISTICS INPUT NEEDS SUMMARY STATISTICS See SOLUTION

  34. Using Note Solution The geometric meandoesn’t show up in summary statistics! Alternate solution…

  35. Using CLICK ON CLICK ONPASTE FUCTION See See…

  36. Using SCROLL DOWN to STATISTICAL See…

  37. Using HIGHLIGHT MEDIAN IN RIGHT MENU See OR…

  38. Using HIGHLIGHT GEOMETRICMEAN IN RIGHT MENU See OR…

  39. Using HIGHLIGHT MODE IN RIGHT MENU See See…

  40. Using New The calculated values are placed in the cell on the worksheet where your cursor was when you began

  41. Source: Statistics Canada, Tourism and the Centre for Education Statistics The following table shows the expenditures of Canadians in 15 countries they visited in 1999

  42. Is the meanor 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?

  43. Using A The meanis strongly affected by the inclusion of these two OUTLIERS … therefore, the median is a more appropriate measure of “average” in this case

  44. fx S = N The Mean of Grouped Data The mean of a sample of data organized in a frequency distribution is computed by the following formula:

  45. The Mean of Grouped Data Example 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.

  46. fx S The Mean of Grouped Data = N Example Continued… Movies Showing Frequency f Class Midpoint (f)(x) 1 to under 3 1 2 2 3 to under 5 2 4 8 5 to under 7 3 6 18 7 to under 9 1 8 8 9 to under 11 3 10 30 Total 10 66

  47. fx S The Mean of Grouped Data = Xf S = N n Example Continued… Movies Showing Frequency f Class Midpoint (f)(x) A Total 10 66 Formula 66 = 10 = 6.6

  48. fx fx S S The Mean of Grouped Data = = N N Hours Studying Frequency f Class Midpoint (f)(x) 10 to under 15 5 12 15 to under 20 20 to under 25 6 610 Formula 25 to under 30 5 = 20.33 = 30 30 to under 35 2 Total 30 Determine the average student study time Example 2 12.5 62.5 17.5 210 22.5 135 27.5 137.5 32.5 65 610

  49. Finding the Median of Grouped Data To determine the median class for Grouped Data: 1. Construct a cumulative frequency distribution 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 determinewhich class will contain the 25th value

  50. N - CF 2 Median = L + ( i ) f L CF f i Finding the Median of Grouped Data Estimate the median value within chosen class… where … 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

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