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Understanding Measures of Central Tendency in Statistics

This chapter explores various measures of central tendency in statistics, including arithmetic mean, median, mode, weighted mean, and geometric mean. Topics covered include calculating means, properties of means, and practical examples to illustrate these concepts.

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Understanding Measures of Central Tendency in Statistics

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  1. Describing Data: Measures of Central Tendency Chapter 3

  2. FiveMeasures ofCentral Tendency • median arithmetic mean mode weighted mean geometric mean Example My grade point average for last semester was 4.0

  3. 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

  4. 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)

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

  6. 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

  7. 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)

  8. 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

  9. 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!

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

  11. 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

  12. + + + 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 -

  13. 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

  14. 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

  15. 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

  16. 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!

  17. = 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:

  18. = = 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

  19. 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

  20. 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?

  21. 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

  22. 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:

  23. 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.

  24. 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

  25. 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

  26. 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

  27. Symmetric Distribution zero skewnessmode =median= mean

  28. Skewed Right Right Skewed Distribution Mean and Median are to the right of the Mode Positively skewed Mode< Median< Mean

  29. Skewed left < Mode < Median Mean Left Skewed Distribution Mean and Median are to the left of the Mode Negatively skewed

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