1. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Measures of Central Tendency Section 2-4 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statisitics Addison Wesley Longman

2. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2 Measure of Central Tendency

3. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3 a value at the center or middle of a data set Measure of Central Tendency

4. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4 Mean Arithmetic Mean AVERAGE Definitions

5. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 5 FIGURE 2-7 Mean as a Balance Point

6. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6 Mean FIGURE 2-7 Mean as a Balance Point

7. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7 Notation

8. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8 Notation  denotes the summation of a set of values

9. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 9 Notation  denotes the summation of a set of values x is the variable usually used to represent the individual data values

10. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 10 Notation  denotes the summation of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample

11. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 11 Notation  denotes the summation of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample N represents the number of data values in a population

12. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 12 Notation x is pronounced ‘x-bar’ and denotes the mean of a set of sample values  denotes the summation of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample N represents the number of data values in a population

13. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 13 µ is pronounced ‘mu’ and denotes the mean of all values Notation x is pronounced ‘x-bar’ and denotes the mean of a set of sample values  denotes the summation of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample N represents the number of data values in a population in a population

14. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14 Definitions  Mean

15. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15 Definitions  Mean the value obtained by adding the scores and dividing the total by the number of scores

16. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16 x = Definitions  Mean the value obtained by adding the scores and dividing the total by the number of scores n  x Sample

17. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 17 x = Definitions  Mean the value obtained by adding the scores and dividing the total by the number of scores n  x x Sample N µ =  x x Population

18. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 18 Calculators can calculate the mean of data Definitions  Mean the value obtained by adding the scores and dividing the total by the number of scores n x =  x x Sample N µ =  x x Population

19. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 19 Examples

20. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 20 use class mark of classes for variable x Mean from a Frequency Table

21. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 21 use class mark of classes for variable x Mean from a Frequency Table x = Formula 2-2 f  (f x) 

22. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 22 use class mark of classes for variable x Mean from a Frequency Table x = Formula 2-2 f  (f x)  x = class mark

23. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 23 use class mark of classes for variable x Mean from a Frequency Table x = Formula 2-2 f  (f x)  x = class mark f = frequency

24. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 24 use class mark of classes for variable x Mean from a Frequency Table x = Formula 2-2 f  (f x)  x = class mark f = frequency  f = n f = n

25. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 25 Weighted Mean

26. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 26 Weighted Mean x = w  (w x) 

27. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 27 Examples

28. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 28 Definitions  Median the middle value when scores are arranged in (ascending or descending) order

29. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 29 Definitions  Median the middle value when scores are arranged in (ascending or descending) order often denoted by x (pronounced ‘x-tilde’) ~

30. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 30 Definitions  Median the middle value when scores are arranged in (ascending or descending) order often denoted by x (pronounced ‘x-tilde’) is not affected by an extreme value ~

31. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 31 5 5 5 3 1 5 1 4 3 5 2 1 1 2 3 3 4 5 5 5 5 5 (in order) exact middle Examples MEDIAN is 4

32. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 32 1 1 3 3 4 5 5 5 5 5 no exact middle -- shared by two numbers 4 + 5 2 = 4.5 5 5 5 3 1 5 1 4 3 5 2 1 1 2 3 3 4 5 5 5 5 5 (in order) exact middle MEDIAN is 4 Examples MEDIAN is 4.5

33. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 33 Definitions  Mode the score that occurs most frequently Bimodal Multimodal No Mode

34. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 34 Examples Mode is 5 Bimodal (2 & 6) No Mode a. 5 5 5 3 1 5 1 4 3 5 b. 2 2 2 3 4 5 6 6 6 7 9 c. 2 3 6 7 8 9 10

35. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 35 Examples Mode is 5 Bimodal (2 & 6) No Mode a. 5 5 5 3 1 5 1 4 3 5 b. 2 2 2 3 4 5 6 6 6 7 9 c. 2 3 6 7 8 9 10 d. 2 2 3 3 3 4 e. 2 2 3 3 4 4 5 5

36. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 36 Examples Mode is 5 Bimodal (2 & 6) No Mode a. 5 5 5 3 1 5 1 4 3 5 b. 2 2 2 3 4 5 6 6 6 7 9 c. 2 3 6 7 8 9 10 d. 2 2 3 3 3 4 e. 2 2 3 3 4 4 5 5 Mode is 3 No Mode

37. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 37 Mode is “O” blood type Blood types: O 35 A 14 B 16 AB 10 Remark: Mode is the only measure of central tendency that can be used with nominal data

38. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 38  Midrange the value halfway between the highest and lowest scores Definitions

39. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 39  Midrange the value halfway between the highest and lowest scores Definitions Midrange = highest score + lowest score 2

40. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 40 5 5 5 3 1 5 1 4 3 5 2 1 1 2 3 3 4 5 5 5 5 5 (in order) Midrange is (5 + 1)/2 = 3 Examples

41. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 41 Carry one more decimal place than is present in the original set of data Round-off rule for measures of central tendency

42. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 42 Advantages - Disadvantages Best Measure of Central Tendency Table 2-6

43. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 43 Table 2-6

44. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 44 Skewness Mode = Mean = Median SYMMETRIC Figure 2-8 (b)

45. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 45 Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median Figure 2-8 (b) Figure 2-8 (a)

46. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 46 Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median SKEWED RIGHT (positively) Mean Mode Median Figure 2-8 (b) Figure 2-8 (a) Figure 2-8 (c)