Đang tải... (xem toàn văn)
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; 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 your choice of the measure of central tendency of data.
3 1 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 3 2 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 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 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Five Measures of Measures of Five Central Central Tendency Tendency arithmetic mean mode median weighted mean Average price of a house in Average price of a house in Ottawa (2000) was $126 000 Ottawa (2000) was $126 000 The average income of two The average income of two parent families with children in parent families with children in Canada was $65,847 in 1995 and Canada was $65,847 in 1995 and $72,910 in 1999. (StatCan) (StatCan) $72,910 in 1999. Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 3 3 geometric mean The average price of a The average price of a house in Toronto in 1996 house in Toronto in 1996 was $238,511 (StatCan) (StatCan) was $238,511 My grade point average My grade point average for last semester was 4.0 for last semester was 4.0 Arithmetic Mean Arithmetic Mean 3 4 …is the most widely used measure of location …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 It is unique The sum of the deviations from the mean is 0 Copyrightâ2004byTheMcGrawưHillCompanies,Inc.Allrightsreserved. PopulationMean PopulationMean Formula Formula x = N isthepopulationmean (pronounced mu) N … is the total number of observations x … is a particular value … indicates the operation of adding (sigma) Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 3 5 Terminology Parameter …is a measurable characteristic of a is a measurable characteristic of a … opulation PPopulation Statistic …is a measurable characteristic of a is a measurable characteristic of a … Sample Sample Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 3 6 Population Mean Population Mean Formula Formula The Kiers family The Kiers family 3 7 x µ = N owns four cars. Find the mean owns four cars. Find the mean mileage for the cars The following is mileage for the cars The following is the current mileage the current mileage on each of the four on each of the four 56000 + 23000 + 42000 + 73000 cars: 56000 + 23000 + 42000 + 73000 cars: = 4 56,000 23,000 4 42,000 73,000 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 48 500 == 48 500 Sample Mean Sample Mean Formula Formula x x = 3 8 x n …is the sample mean (read “x bar”) n … is the number of sample observations x … is a particular value … indicates the operation of adding (sigma) Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 3 9 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: Formula Formula x = x n 14 + 15 + 17 + 16 + 15 14 + 15 + 17 + 16 + 15 = 5 5 77 / 5 == 15.4 15.4 == 77 / 5 The average bonus given last year was $15 400 The average bonus given last year was $15 400 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Properties of an of an Properties 3 10 Arithmetic Mean Arithmetic Mean …Every set of intervallevel 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 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. The Mean The Mean of of Grouped Data Grouped Data The mean of a sample of data organized The mean of a sample of data organized in a frequency distribution is in a frequency distribution is computed by the following formula: computed by the following formula: x Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. fx N 3 44 The Mean The Mean of of Grouped Data Grouped Data 3 45 A sample of ten movie theatres in a metropolitan A sample of ten movie theatres in a metropolitan area tallied the total number of movies area tallied the total number of movies showing last week. showing last week. Compute the mean number of movies showing Compute the mean number of movies showing per theatre. per theatre. Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. The Mean The Mean 3 46 fx x N of Grouped Data of Grouped Data Continued… Class (f)(x) Midpoint Movies Showing Frequency 1 to under 3 2 3 to under 5 5 to under 7 18 7 to under 9 8 9 to under 11 10 30 Total 10 f Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 66 The Mean The Mean 3 47 fx x N of Grouped Data of Grouped Data Movies Showing Frequency Total 10 f Formula Formula Continued… Class (f)(x) Midpoint 66 X Xf n 66 10 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. = 6.6 = 6.6 The Mean The Mean 3 48 fx x N of Grouped Data of Grouped Data Determine the average student study time Determine the average student study time Frequency Class (f)(x) Midpoint f Hours Studying 10 to under 15 12.5 62.5 15 to under 20 12 17.5 210 20 to under 25 25 to under 30 Formula Formula x 5fx 30 to under 35 N Total 30 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 22.5 61027.5 = 20.33 30 32.5 135 137.5 65 610 Finding the Median of Finding the Median of Grouped Data Grouped Data 3 49 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 determine which class will contain the 25th value Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Finding the Median of Finding the Median of Grouped Data Grouped Data 3 50 Estimate the median value within chosen class… N CF Median = L + (i) f L … is the lower limit of the median class CF … is the cumulative frequency as you enter the median class f … is the frequency of the median class i … is the class interval or size Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Finding the Median of Finding the Median of Grouped Data Grouped Data Movies Showing Frequency Cumulative f 1 to under 3 f 3 to under 5 Median class L 5 to under 7 7 to under 9 CF 9 to under 11 Total i = 2 3 10 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. f 10 3 51 N CF L + (i ) f 10 3 = 5 + 2 = 6.33 = 6.33 The Mode of Grouped Data The Mode of Grouped Data 3 52 The mode for grouped data is approximated by the midpoint of the class with the largest class frequency Movies Showing Frequency f Class Midpoint 1 to under 3 3 to under 5 5 to under 7 7 to under 9 9 to under 11 10 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. This is This is considered considered to be to be BiModal BiModal The Mode of Grouped Data The Mode of Grouped Data Approximate the Mode of this distribution Approximate the Mode of this distribution Hours Studying Frequency f Class Midpoint 10 to under 15 12.5 15 to under 20 12 17.5 20 The modal class is 15 to under 20, to under 25 22.5 The modal class is 15 to under 20, 25 to underapproximately 17.5 30 27.5 approximately 17.5 30 to under 35 Total 30 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 32.5 3 53 Symmetric Distribution zero skewness mode = median = mean Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 3 54 Right Skewed Distribution e Sk Mean and Median are to the right of the Mode Positively skewed d we ig R ht Mode< Median< Mean Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 3 55 Left Skewed Distribution Sk ew ed le f t Mean and Median are to the left of the Mode Negatively skewed