Sec4-2-3 - 4.2 Measures of Central Tendency We often...

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1 4.2 Measures of Central Tendency We often represent the data set by numerical summary measures, usually called the typical values . A measure of central tendency gives the average of a data set or center of a histogram or a frequency distribution curve. 3 different measures of central tendency: The mean The median The mode 4.2.1 The Mean ( arithmetic mean ) Calculating Mean for Ungrouped Data: The sample mean of a numerical sample, , ..., , 3 , 2 , 1 n x x x x denoted by x , is n x x = = points data of number the points data the of sum the where n is the sample size. The population mean of a numerical data set, denoted by μ , is the average of all x values (data points) in the entire population, which is calculated by N x = μ where N is the population size. Example 1 : The Table gives the 2002 payrolls of five Major League Baseball (MLB) teams. Find the mean of the 2002 payrolls of these five MLB teams. MLB Team 2002 Total Payroll (millions of dollars) Anaheim Angles 62 Atlanta Braves 93 New York Yankees 126 St. Louis Cardinals 75 Tampa Bay Devil Rays 34
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2 Example 2 : The following are the ages of eight randomly selected employees of a company: 53 32 61 27 39 44 49 57 Find the mean age of these employees. If the data are given in the form of a frequency table, we can not obtain the sum of individual values directly. . Calculating Mean for Grouped Data 1. single valued groups The population mean: N xf = μ The sample mean: n xf x = where x is the single value in each group 2. interval groups The population mean: N mf = The sample mean: n mf x = where m is the midpoint of an interval and f is the frequency of the interval. Example 3 : The following table gives the frequency distribution of the vehicles owned by households in a large city. A sample of 40 households is randomly selected. Find the mean of the vehicles owned by households. Vehicles Owned Number of Households ( f ) 0 2 1 18 2 11 3 4 4 3 5 2
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3 Example 4 : The following table gives the frequency distribution of the daily commuting times (in minutes) from home to work for 25 randomly selected employees of a company. Find the mean of the daily commuting times. Solution: Daily Commuting Time (minutes) f m mf 10 0 < x 4 20 10 < x 9 30 20 < x 6 40 30 < x 4 50 40 < x 2 n = = mf Example 5 : During a two week period 10 houses were sold in Fancytown. Example 6 : During a two week period 10 houses were sold in Lowtown. Daily Commuting Time (minutes) f 10 0 < x 4 20 10 < x 9 30 20 < x 6 40 30 < x 4 50 40 < x 2 H o u s e P r ic e in L o w t o w n x 9 7 ,0 0 0 9 3 ,0 0 0 1 1 0 , 0 0 0 1 2 1 , 0 0 0 1 1 3 , 0 0 0 9 5 ,0 0 0 1 0 0 , 0 0 0 1 2 2 , 0 0 0 9 9 ,0 0 0 2 , 0 0 0 ,0 0 0 2 , 9 5 0 ,0 0 0 x = H o u s e P r ic e in F a n c y t o w n x 2 3 1 , 0 0 0 3 1 3 , 0 0 0 2 9 9 , 0 0 0 3 1 2 , 0 0 0 2 8 5 , 0 0 0 3 1 7 , 0 0 0 2 9 4 , 0 0 0 2 9 7 , 0 0 0 3 1 5 , 0 0 0 2 8 7 , 0 0 0 2 ,9 5 0 ,0 0 0 x = 500000 1000000 1500000 2000000 Dotplots for Fancytown and Lowtown Fancytown Lowtown 295000
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Sec4-2-3 - 4.2 Measures of Central Tendency We often...

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