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1 Frequency Distribution: Mean, Variance, Standard Deviation Given: Number of credit hours a sample of 25 full- time students are taking this semester was collected and is summarized here as a frequency distribution: Find:a)The values for n, x and x 2 using the summations: f, xf and x 2 f x f b)The mean, variance and standard deviation

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2 Understanding a Frequency Distribution A sample of 25 data is summarized here as a frequency distribution: x f For the above frequency distribution, a) What do the entries x = 12 and f = 5 mean? {The x-value 12 occurred 5 times in the sample} b) If you total the values listed in the x-column, what would this total represent? {It would be the sum of the 5 distinct x-values occurring in the sample, not the sum of all 25 values} Remember, the x represents the sum of all data values for the sample - this sample has 25 data, not 5 as listed in the x-column

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3 1. Find n ; Finding the Extensions & Summations Use a table format to find the extensions for each x value and the 3 summations f, xf and x 2 f : x f 3 The sample size n is f, the sum of the frequencies The sample size n = f = 25 f = Find the sum of all data by finding xf ; The sum of all data = xf = x5=60 13x7=91 14x6=84 15x4=60 16x3= xf = Find the sum of all squared data by finding x 2 f ; x2x2 x2fx2f 144x5= x7= x6= x4= x3= = = = = = x 2 f = 4747 Notes:Save these 3 summations for future formula work DO NOT find the summations of the x and x 2 columns xf Find xf for each x First, find x 2 for each x ; Second, find x 2 f for each x The sum of all squared data = x 2 f =

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Finding the Sample Mean Formula 2.11 will be used: x = xf f Previously determined values: f = 25, xf = 343 x = xf f = = The sample mean is 13.7 credit hours

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f - 1 s 2 = x 2 f - f ( xf) 2 = - 1 ( ) - ( ) 2 25 Finding the Sample Variance Formula 2.16 will be used: Previously determined values: The sample variance is 1.71 f - 1 s 2 = x 2 f - f ( xf) 2 x 2 f = 4747 xf = 343 f = 25 = = 1.71

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6 Finding the Sample Standard Deviation The standard deviation is the square root of variance: s = s 2 Therefore, the standard deviation is: s = s 2 = = The standard deviation is 1.3 credit hours = 1.3 Notes:1)The unit of measure for the standard deviation is the unit of the data 2)Use a non-rounded value of variance when calculating the standard deviation 1.71

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7 Using a Grouped Frequency Distribution Find:a)The class midpoint for each class b)Estimate the values for n, x and x 2 using the summations: f, xf and x 2 f c)The mean, variance and standard deviation Given: Twenty-five men were asked, How much did you spend at the barber shop during your last visit? The data is summarized using intervals and is listed here as a grouped frequency: Class Intervalf

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8 Class Intervalf MidpointClass Intervalf Finding the Class Midpoint Find:The class midpoints, one class at a time (lower boundary + upper boundary) / 2 Each class interval contains several different data values. In order to use the frequency distribution, a class midpoint must be determined for each class. This center value for the class will be used to approximate the value of each data that belongs to that class. The class midpoints are found by averaging the extreme values for each class: The midpoint for each class will the be the classs representative value and be used for finding the extensions 2.50 = = 5.0 = =

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9 5 2 = = = = = x6= x9= x5= x3= x2=1250 5x6=30 10x9=90 15x5=75 20x3=60 25x2=50 xf 1. For x 2, multiply each x by itself 2. For xf, multiply each x by its frequency f 4. Find the summations by totaling the columns Notes:Save these 3 summations for future formula work DO NOT find the summations for the x and x 2 columns Finding the Extensions & Summations Use a table format to find the extensions for each x value and the 3 summations f, xf and x 2 f : x2fx2f x2x2 x f For x 2 f, multiply each x 2 by its frequency f f = 25 xf = 305 x 2 f = 4625 meaningless totals

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Finding the Sample Mean Formula 2.11 will be used: x = xf f Previously determined values: f = 25, xf = 305 x = xf f = = 12.2 The sample mean is $12.20

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f - 1 s 2 = x 2 f - f ( xf) 2 = - 1 ( ) - ( ) Finding the Sample Variance Formula 2.16 will be used: Previously determined values: The sample variance is 37.7 f - 1 s 2 = x 2 f - f ( xf) 2 x 2 f = 4625 xf = 305 f = 25 = = = 37.7

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12 Finding the Sample Standard Deviation The standard deviation is the square root of variance: s = s 2 Therefore, the standard deviation is: s = s 2 = = The standard deviation is $6.14 = 6.14 Notes:1)The unit of measure for the standard deviation is the unit of the data 2)Use a non-rounded value of variance when calculating the standard deviation

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