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Alliance Stat Class Understanding Measures of Center and Spread

Alliance Stat Class Understanding Measures of Center and Spread. What is the median? How do you define and how do you find the value? What happens if you have an odd number of data values? What happens if you have an even number of data values?

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Alliance Stat Class Understanding Measures of Center and Spread

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  1. Alliance Stat Class Understanding Measures of Center and Spread

  2. What is the median? How do you define and how do you find the value? What happens if you have an odd number of data values? What happens if you have an even number of data values? Example: find the median for the following sets of values: 14 15 15 16 18 21 25 27 30 30 30 30 30 26 12 15 9 27 9 13 13 13 18

  3. Practice finding the Median Using the back to back stemplot of the battery data from your assignment Find the median for each battery type.

  4. What is the mean? Can you define or tell me what it means without saying the word average?

  5. How do you determine the mean? VersusHow do you interpret the mean?

  6. Level A ActivityLength of First NameA Conceptual Activity for:• Developing an Understanding of the Mean as the “Fair Share” value• Developing a Measure of Variation from “Fair Share”

  7. A Statistical QuestionHow long are the first names of students in class?• Nine students were asked what was the length of their first name.• Each student represented her/his name length with a collection snap cubes.

  8. Snap Cube Representation for Nine Name Lengths

  9. How might we examine the data on the name length for these nine children?

  10. 2 3 3 4 4 5 6 7 9 Ordered Snap Cube & Numerical Representations of Nine Name Lengths

  11. Notice that the name lengths vary. What if we used all our name lengths and tried to make all names the same length, in which case there is no variability.How many people would be in each name length?

  12. How can we go about creating these new groups?We might start by separating all the names into one large group.

  13. All 43 letters in the students’ names

  14. Create Nine “New” Groups

  15. Create Nine “New” Groups

  16. Original Question:What if we used all our name lengths and tried to make all names the same length, in which case there is no variability.How many people would be in each name length?

  17. This is the value of the mean number of letters in the 9 students names. What is the formula to find the mean? ∑x = ____ n How does this formula compare with how we found the mean through the idea of fair share?

  18. A New ProblemWhat if the fair share value or the mean number of letters for nine children is 6? What are some different snap cube representations that might produce a fair share value of 6? How many total number of cubes do you need?

  19. Snap Cube Representation of Nine Families, Each of Size 6

  20. Two Examples with Fair Share Value of 6. Which group is “closer” to being “fair?”

  21. How might we quantity “how close” a group of name lengths is to being fair?

  22. Steps to Fair• One step occurs when a snap cube is removed from a stack higher than the fair share value and placed on a stack lower than the fair share value .• A measure of the degree of fairness in a snap cube distribution is the “number of steps” required to make it fair.Note -- Fewer steps indicates closer to fair

  23. Number of Steps to Make Fair: 8 Number of Steps to Make Fair: 9

  24. Students completing Level A understand:• the notion of “fair share” for a set of numeric data• the fair share value is also called the mean value• the algorithm for finding the mean• the notion of “number of steps” to make fair as a measure of variability about the mean• the fair share/mean value provides a basis for comparison between two groups of numerical data with different sizes (thus can’t use total)

  25. Data About Us Investigation 3: What do we mean by Mean? Data Distributions Investigation 2: Making Sense of Measures of Center 2.1 The mean as an Equal Share 2.2 The Mean as a Balance Point in a Distribution

  26. Level B ActivityThe Name Length Problem • How long are the first names of students in class? A Conceptual Activity for:• Developing an Understanding of the Mean as the “Balance Point” of a Distribution• Developing Measures of Variation about the Mean

  27. Level B ActivityHow long is your first name?Nine children were asked this question. The following dot plot is one possible result for the nine children:

  28. -+--+--+--+--+--+--+--+--+- 2 3 4 5 6 7 8 9 10

  29. Do These Distributions Balance? Why? -+--+--+--+--+--+--+--+--+- 2 3 4 5 6 7 8 9 10 -+--+--+--+--+--+--+--+--+- 2 3 4 5 6 7 8 9 10

  30. In which group do the data (name length) vary (differ) more from the mean value of 6?

  31. 1 2 4 2 1 0 1 2 3 -+--+--+--+--+--+--+--+--+- 2 3 4 5 6 7 8 9 10 0 0 4 3 2 0 2 3 4 -+--+--+--+--+--+--+--+--+- 2 3 4 5 6 7 8 9 10

  32. In Distribution 1, the Total Distance from the Mean is 16. In Distribution 2, the Total Distance from the Mean is 18.Consequently, the data in Distribution 2 differ more from the mean than the data in Distribution 1.

  33. The SAD is defined to be:The Sum of the Absolute DeviationsNote the relationship between SAD and Number of Steps to Fair from Level A: SAD = 2xNumber of Steps

  34. 1 2 4 2 1 0 1 2 3 -+--+--+--+--+--+--+--+--+- 2 3 4 5 6 7 8 9 10 Note the total distance for the values below the mean of 6 is 8, the same as the total distance for the values above the mean. Hence the distribution will “balance” at 6 (the mean)‏

  35. An Illustration where the SAD doesn’t work!

  36. 4 4 -+--+--+--+--+--+--+--+--+- 2 3 4 5 6 7 8 9 10 1 1 1 1 1 1 1 1 -+--+--+--+--+--+--+--+--+- 2 3 4 5 6 7 8 9 10

  37. Finding SAD Since both points are 4 from the mean SAD = 8 Since all 8 are 1 from the mean SAD = 8 Find the “average” distance from mean 2 data points: SAD/ 2 = 4 8 data points: SAD / 8 = 1 We now have found MAD = Mean Absolute Deviation

  38. Adjusting the SAD for group sizes yields the:MAD = Mean Absolute Deviation

  39. Summary of Level B • Mean as the balance point of a distribution• Mean as a “central” point• Various measures of variation about the mean.

  40. Other measures of variability Range = Highest – Lowest MAD – Mean Absolute Deviation Standard Deviation

  41. -+--+--+--+--+--+--+--+--+- 2 3 4 5 6 7 8 9 10

  42. Sum of Absolute Deviations = 18 Mean Absolute Deviation (MAD) = 18/9 = 2

  43. Interpret MAD On “average” how far is the data from the mean. Example: Find MAD 35 60 63 Find the mean Complete table 3. Find the mean of the absolute deviations

  44. What is the sum of the x – xbar column? Find the sum of the (x – x bar) column. Divide by 4 MAD = _______ Interpret this value

  45. Another Measure of Spread: Standard Deviation This is the most common measure of Variability

  46. What is the sum of the x – xbar column? Find the sum of the (x – x bar)^2 column. Divide by 4 Take the square root Standard Deviation = _______ Interpret this value

  47. Steps to find standard deviation Find the mean Complete the table. Divide by n Take the square root This is the standard deviation What does standard deviation mean? “Average” distance a data point is from the mean.

  48. Practice Measures of Variability Set of Data Length of first and middle names in my family 14, 9, 10, 12 Find and interpret Standard Deviation

  49. Practice Standard Deviation 14, 9, 10, 12 The mean = 11.25 Find the sum of the squared deviations = 14.75 Divide by 4 = 3.6875 Square root = 1.92

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