1. Chapter 3.4 Measures of Central Tendency Measures of Central Tendency

2. Measures of Central Tendency include mean, median and mode Measures of Central Tendency include mean, median and mode

3. The mean is the sum of values in a set divided by the number of values in the set of data. The mean is the sum of values in a set divided by the number of values in the set of data.

4. The median is the middle value when the data is ordered from least to greatest. The median is the middle value when the data is ordered from least to greatest.

5. The mode is the number that occurs most often in the set of data. The mode is the number that occurs most often in the set of data.

6. Example 1 Example: Suppose a class received the following test scores out of 100: Example: Suppose a class received the following test scores out of 100: 61, 76, 89, 72, 65, 71, 61, 83, 45, 68, 62, 59, 71, 68, 69, 86 61, 76, 89, 72, 65, 71, 61, 83, 45, 68, 62, 59, 71, 68, 69, 86

7. a) What is the mean? What is the mean?

8. Mean = Total of scores Mean = Total of scores Number of scores Number of scores ___________________________

9. Mean = Total of scores Mean = Total of scores Number of scores Number of scores = 1106 = 1106 16 16 ___________________________ _______

10. Mean = Total of scores Mean = Total of scores Number of scores Number of scores = 1106 = 1106 16 16 = 69.125 = 69.125 ___________________________ _______

11. b) What is the median? What is the median?

12. b) First, order the data from least to greatest First, order the data from least to greatest

13. b) What is the median? What is the median? First, order the data from least to greatest First, order the data from least to greatest 45, 59, 61, 61, 62, 65, 68, 68, 69, 71, 71, 72, 76, 83, 86, 89 45, 59, 61, 61, 62, 65, 68, 68, 69, 71, 71, 72, 76, 83, 86, 89

14. Then, figure out what the middle value is or what the middle values are. Then, figure out what the middle value is or what the middle values are. If there is only one middle value, then that number is the median. If there is only one middle value, then that number is the median.

15. If there are two middle values, then you determine the median by adding those values and dividing by 2. If there are two middle values, then you determine the median by adding those values and dividing by 2.

16. Going back to our numbers, placed in order: Going back to our numbers, placed in order: 45, 59, 61, 61, 62, 65, 68, 68, 69, 71, 71, 72, 76, 83, 86, 89 45, 59, 61, 61, 62, 65, 68, 68, 69, 71, 71, 72, 76, 83, 86, 89

17. The middle numbers are 68 and 69, so our median is The middle numbers are 68 and 69, so our median is

18. Median = 68 + 69 Median = 68 + 69 2 _____________

19. The middle numbers are 68 and 69, so our median is The middle numbers are 68 and 69, so our median is Median = 68 + 69 Median = 68 + 69 2 = 68.5 = 68.5 _____________

20. c) What is the mode? What is the mode?

21. c) The mode is the number which occurs most often. Since 61, 68 and 71 each occur twice, the three modes are 61, 68 and 71. The mode is the number which occurs most often. Since 61, 68 and 71 each occur twice, the three modes are 61, 68 and 71.

22. Example 2 Suppose a charity fundraiser gave out the following prizes:

23. Example 2 Suppose a charity fundraiser gave out the following prizes: One $5000 prize, four $1000 prizes, eight $500 prizes, and eighty $10 prizes.

24. a) What is the mean?

25. A total of 93 prizes were given out.

26. The total amount of money given out is calculated as follows …

27. $5000 x 1 = $5000

28. $1000 x 4 = $4000

29. $5000 x 1 = $5000 $1000 x 4 = $4000 $500 x 8 = $4000 $500 x 8 = $4000

30. $5000 x 1 = $5000 $1000 x 4 = $4000 $500 x 8 = $4000 $500 x 8 = $4000 $10 x 80 = $800 $10 x 80 = $800

31. $5000 x 1 = $5000 $1000 x 4 = $4000 $500 x 8 = $4000 $500 x 8 = $4000 $10 x 80 = $800 $10 x 80 = $800 Total = $13800 Total = $13800

32. Mean = Total Money Given Out Number of Prizes Given Number of Prizes Given ____________________________________

33. Mean = Total Money Given Out Number of Prizes Given Number of Prizes Given = $13800 93 ___________________________________ __________

34. Mean = Total Money Given Out Number of Prizes Given Number of Prizes Given = $13800 93 = $148.39 ____________________________________ __________

35. b) What is the median?

36. There are 93 prizes, 80 of which have a value of $10. Therefore, the median prize value is $10.

37. c) What is the mode?

38. The number which occurs most often is 10, so the mode is 10.

39. Chapter 3.5 Measures of Spread

40.  Measures of Spread include range and standard deviation.

41.  The range is the difference between the greatest and least values in a set of data.

42.  The standard deviation is the typical distance of a particular value from the mean. The greater the standard deviation, the greater the spread of the data.

43.  The formula for standard deviation is quite complicated.

44.  Standard deviation =

45. (x 1 -mean) 2 + (x 2 -mean) 2 + … + (x n -mean) 2 (x 1 -mean) 2 + (x 2 -mean) 2 + … + (x n -mean) 2 n √ _________________________________________________ ________________________________________________

46. Example  The Pittsburgh Penguins have recorded the following point totals in the past five years

47. Example  58, 105, 102, 99, 101

48.  What is the range of points, and what is the standard deviation?

49.  The highest point total is 105 and the lowest point total is 58. The range is 47

50.  To determine the standard deviation, we first need to determine the mean.

51.  Mean = 58 + 105 + 102 + 99 + 101 5 _______________________________________

52.  Mean = 58 + 105 + 102 + 99 + 101 5 = 465 5 _______________________________________ _______

53.  Mean = 58 + 105 + 102 + 99 + 101 5 = 465 5 = 93 = 93 _______________________________________ _______

54.  Refer to table in your notes to calculate standard deviation.