Frequency Distribution Mean Variance Standard Deviation F Given

Frequency Distribution: Mean, Variance, Standard Deviation F Given: Number of credit hours a sample of 25 fulltime students are taking this semester was collected and is summarized here as a frequency distribution: x f 12 13 14 15 16 5 7 6 4 3 F Find: a) The values for n, x and x 2 using the summations: f, xf and x 2 f b) The mean, variance and standard deviation 1

Understanding a Frequency Distribution F A sample of 25 data is summarized here as a frequency distribution: x 12 13 14 15 16 f 5 7 6 4 3 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} F 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 2

Finding the Extensions & Summations F Use a table format to find the extensions for each x value and the 3 summations f, xf and x 2 f : x 12 13 14 15 16 f 5 7 6 4 3 f = 25 xf 12 12 x 5=60 5 13 13 x 7=91 7 14 14 x 6=84 6 15 15 x 4=60 4 16 16 x 3=48 3 xf = 343 x 2 f 122 = 144 144 x 5=720 5 132 = 169 169 x 7=1183 7 142 = 196 196 x 6=1176 6 152 = 225 225 x 4=900 4 162 = 256 256 x 3=768 3 x 2 f = 4747 1. Find n ; The sample size n is f, the sum of the frequencies The sample size n = f = 25 2. Find the sum of all data by finding xf ; Find xf for each x The sum of all data = xf = 343 3. Find the sum of all squared data by finding x 2 f ; First, find x 2 for each x ; Second, find x 2 f for each x The sum of all squared data = 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 3

Finding the Sample Mean xf F Formula 2. 11 will be used: x = f F Previously determined values: f = 25, xf = 343 xf 343 x= = = 13. 72 f 25 The sample mean is 13. 7 credit hours 4

Finding the Sample Variance 2 ( xf) x 2 f f 2 F Formula 2. 16 will be used: s = f - 1 F Previously determined values: x 2 f = 4747 xf = 343 f = 25 2 2 ( ) ( xf) 343 x 2 f (4747) f 41. 04 25 2 s = = 1. 71 f - 1 25 - 1 24 The sample variance is 1. 71 5

Finding the Sample Standard Deviation F The standard deviation is the square root of variance: s= s 2 F Therefore, the standard deviation is: s = 1. 71 = 1. 3077 = 1. 3 2 The standard deviation is 1. 3 credit hours 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 6

Using a Grouped Frequency Distribution F 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 Interval f 2. 50 - 7. 50 - 12. 50 - 17. 50 - 22. 50 - 27. 50 6 9 5 3 2 F 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 7

Finding the Class Midpoint F 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: Class Interval Midpoint f f 2. 50 - 7. 50 - 12. 50 - 17. 50 - 22. 50 - 27. 50 5. 0 6 10. 0 9 15. 0 5 20. 0 3 25. 0 2 6 9 5 3 2 F Find: The class midpoints, one class at a time (lower boundary + upper boundary) / 2 2. 50 + 7. 50 10. 00 = = 5. 0 2 2 7. 50 + 12. 50 20. 00 = = 10. 0 2 2 The midpoint for each class will the be the class’s representative value and be used for finding the extensions 8

Finding the Extensions & Summations F Use a table format to find the extensions for each x value and the 3 summations f, xf and x 2 f : x 5 10 15 20 25 x 2 52 = 25 102 = 100 152 = 225 202 = 400 252 = 625 meaningless totals f 6 9 5 3 2 f = 25 xf 55 x 6=30 6 10 x 9=90 10 9 15 15 x 5=75 5 20 20 x 3=60 3 25 25 x 2=50 2 xf = 305 x 2 f 25 25 x 6=150 6 100 x 9=900 100 9 225 5 225 x 5=1125 400 x 3=1200 400 3 625 x 2=1250 2 x 2 f = 4625 1. For x 2, multiply each x by itself 2. For xf, multiply each x by its frequency f 3. For x 2 f, multiply each x 2 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 9

Finding the Sample Mean xf F Formula 2. 11 will be used: x = f F Previously determined values: f = 25, xf = 305 xf 305 x= = = 12. 2 f 25 The sample mean is $12. 20 10

Finding the Sample Variance 2 ( xf) x 2 f f 2 F Formula 2. 16 will be used: s = f - 1 F Previously determined values: x 2 f = 4625 xf = 305 2 2 ( ) ( xf) 305 x 2 f (4625) f 25 2 s = = = f - 1 25 - 1 f = 25 904 = 37. 6666 = 37. 7 24 The sample variance is 37. 7 11

Finding the Sample Standard Deviation F The standard deviation is the square root of variance: s= s 2 F Therefore, the standard deviation is: s = 37. 6666 = 6. 1367 = 6. 14 2 The standard deviation is $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 12