1. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 1 of 19 Chapter 3 Section 3 Measures of Central Tendency and Dispersion from Grouped Data

2. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 2 of 19 Chapter 3 – Section 3 ●Learning objectives  The mean from grouped data  The weighted mean  The variance and standard deviation for grouped data 1 2 3

3. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 3 of 19 Chapter 3 – Section 3 ●Data may come in groups rather than individually ●The values may have been summarized in frequency distributions  Ranges of ages (20 – 29, 30 – 39,...)  Ranges of incomes ($10,000 – $19,999, $20,000 – $39,999, $40,000 – $79,999,...) ●The exact values for the mean, variance, and standard deviation cannot be calculated

4. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 4 of 19 Chapter 3 – Section 3 ●Learning objectives  The mean from grouped data  The weighted mean  The variance and standard deviation for grouped data 1 2 3

5. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 5 of 19 Chapter 3 – Section 3 ●To compute the mean for grouped data  Assume that, within each class, the mean of the data is equal to the class midpoint  Use the class midpoint in the formula for the mean  The number of times the class midpoint value is used is equal to the frequency of the class ●To compute the mean for grouped data  Assume that, within each class, the mean of the data is equal to the class midpoint  Use the class midpoint in the formula for the mean  The number of times the class midpoint value is used is equal to the frequency of the class ●If 6 values are in the interval [ 8, 10 ], then we proceed as if all 6 values are equal to 9 (the midpoint of [ 8, 10 ]

6. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 6 of 19 Chapter 3 – Section 3 ●As an example, for the following frequency table, we calculate the mean as if  The value 1 occurred 3 times  The value 3 occurred 7 times  The value 5 occurred 6 times  The value 7 occurred 1 time Class0 – 1.92 – 3.94 – 5.96 – 7.9 Midpoint1357 Frequency3761

7. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 7 of 19 Chapter 3 – Section 3 ●The calculation for the mean would be or Class0 – 1.92 – 3.94 – 5.96 – 7.9 Midpoint1357 Frequency3761

8. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 8 of 19 Chapter 3 – Section 3 ●Evaluating this formula ●The mean is about 3.6 ●In mathematical notation ●This would be μ for the population mean and for the sample mean

9. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 9 of 19 Chapter 3 – Section 3 ●Learning objectives  The mean from grouped data  The weighted mean  The variance and standard deviation for grouped data 1 2 3

10. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 10 of 19 Chapter 3 – Section 3 ●Sometimes not all data values are equally important ●To compute a grade point average (GPA), a grade in a 4 credit class is worth more than a grade in a 1 credit class ●Sometimes not all data values are equally important ●To compute a grade point average (GPA), a grade in a 4 credit class is worth more than a grade in a 1 credit class ●The weights w i quantify the relative importance of the different values ●Higher weights correspond to more important values

11. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 11 of 19 Chapter 3 – Section 3 ●As an example, the following grades would yield a GPA (on a 4 point scale) of CourseCreditsGrade Statistics3A French Literature3B Biochemistry5B Badminton1D

12. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 12 of 19 Chapter 3 – Section 3 ●In mathematical notation, if w i is the weight corresponding to the data value x i, then the weighted mean is ●This formula looks similar to one for the mean for grouped data, and the concepts are similar

13. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 13 of 19 Chapter 3 – Section 3 ●Learning objectives  The mean from grouped data  The weighted mean  The variance and standard deviation for grouped data 1 2 3

14. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 14 of 19 Chapter 3 – Section 3 ●To compute the variance for grouped data  Assume again that, within each class, the mean of the data is equal to the class midpoint  Use the class midpoint in the formula for the variance  The number of times the class midpoint value is used is equal to the frequency of the class ●To compute the variance for grouped data  Assume again that, within each class, the mean of the data is equal to the class midpoint  Use the class midpoint in the formula for the variance  The number of times the class midpoint value is used is equal to the frequency of the class ●If 6 values are in the interval [ 8, 10 ], then we assume that all 6 values are equal to 9 (the midpoint of [ 8, 10 ] ●The same approach as for the mean

15. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 15 of 19 Chapter 3 – Section 3 ●As an example, for the following frequency table, we calculate the variance as if  The value 1 occurred 3 times  The value 3 occurred 7 times  The value 5 occurred 6 times  The value 7 occurred 1 time Class0 – 1.92 – 3.94 – 5.96 – 7.9 Midpoint1357 Frequency3761

16. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 16 of 19 Chapter 3 – Section 3 ●From our previous example, the mean is 3.6 ●Just as for the mean, the calculation for the variance would then be Class0 – 1.92 – 3.94 – 5.96 – 7.9 Midpoint1357 Frequency3761

17. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 17 of 19 Chapter 3 – Section 3 ●Evaluating this formula ●The variance is about 2.7 ●The standard deviation would be about

18. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 18 of 19 Chapter 3 – Section 3 ●In mathematical notation ●The population variance would be ●The sample variance would be ●The standard deviations would be the corresponding square roots

19. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 3 – Slide 19 of 19 Summary: Chapter 3 – Section 3 ●The mean for grouped data  Use the class midpoints  Obtain an approximation for the mean ●The variance and standard deviation for grouped data  Use the class midpoints  Obtain an approximation for the variance and standard deviation