1. 1 M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 2-4 Measures of Center

2. 2 Objectives Day 1 Given a data set, determine the mean, median, and mode.

3. 3 a value at the center or middle of a data set Measures of Center

4. 4 Mean (Arithmetic Mean) AVERAGE the number obtained by adding the values and dividing the total by the number of values Definitions

5. 5 Notation  denotes the addition 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

6. 6 Notation is pronounced ‘x-bar’ and denotes the mean of a set of sample values x = n  x x x

7. 7 Notation µ is pronounced ‘mu’ and denotes the mean of all values in a population is pronounced ‘x-bar’ and denotes the mean of a set of sample values Calculators can calculate the mean of data x = n  x x x N µ =  x x

8. 8 Definitions  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude

9. 9 Definitions  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude  often denoted by x (pronounced ‘x-tilde’) ~

10. 10 Definitions  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude  often denoted by x (pronounced ‘ x -tilde’)  is not affected by an extreme value ~

11. 11 6.72 3.46 3.606.44 3.46 3.60 6.446.72 no exact middle -- shared by two numbers 3.60 + 6.44 2 (even number of values) MEDIAN is 5.02

12. 12 6.72 3.46 3.606.44 26.70 3.46 3.60 6.446.72 26.70 (in order - odd number of values) exact middle MEDIAN is 6.44 6.72 3.46 3.606.44 3.46 3.60 6.446.72 no exact middle -- shared by two numbers 3.60 + 6.44 2 (even number of values) MEDIAN is 5.02

13. 13 Definitions  Mode the score that occurs most frequently Bimodal Multimodal No Mode denoted by M the only measure of central tendency that can be used with nominal data

14. 14 a. 5 5 5 3 1 5 1 4 3 5 b. 1 2 2 2 3 4 5 6 6 6 7 9 c. 1 2 3 6 7 8 9 10 Examples  Mode is 5  Bimodal - 2 and 6  No Mode

15. 15  Midrange the value midway between the highest and lowest values in the original data set Definitions

16. 16  Midrange the value midway between the highest and lowest values in the original data set Definitions Midrange = highest score + lowest score 2

17. 17 Carry one more decimal place than is present in the original set of values Round-off Rule for Measures of Center

18. 18 use class midpoint of classes for variable x Mean from a Frequency Table

19. 19 use class midpoint of classes for variable x Mean from a Frequency Table x = Formula 2-2 f  (f x) 

20. 20 use class midpoint of classes for variable x Mean from a Frequency Table x = class midpoint f = frequency  f = n f = n x = Formula 2-2 f  (f x) 

21. 21 Example Qwerty Keyboard Word Ratings Word Ratings Interval Midpoints Frequency 0-2120 3-541456 6-8715105 9-1110220 12-14131 x fxf

22. 22 Pages 65-66 3,5,9,11

23. 23 Homework Solutions pg 65 #3 Mean Median 35 46 55 65 74 83 88 93 99 107 108 119 Occurs between the 6 th and 7 th data values Mode- none

24. 24 Homework Solutions pg 65 #5 Mean Median Mode Jefferson Valley Providence 7.70

25. 25 Homework Solutions pg 65 #9 40-49 50-59 60-69 70-79 80-89 90-99 100-109 freqx*f midpts

26. 26 Homework Solutions pg 65 #11 Mean Midpts x Freq f x*f

27. 27 Objectives Day 2 Given a data set where the scores vary in importance, compute a weighted mean. Determine how extreme values affect measures of center. Understand the relationship between the shape of a distribution and the relative location of the mean and median.

28. 28 Weighted Mean – used when scores vary in importance Formula

29. 29 Example Weighted Mean The final grade computation for a freshman statistics course is based on weighted components. Tests 20% each Final 40% If the Test scores are 83%, 73%, 82%, and a final exam score of 91% … What is the final grade?

30. 30 Weighted Mean common error The weights do not need to sum to 100

31. 31 Example Weighted Mean Two algebra classes had the following average test scores. Period 2 had a mean score of 40 with 24 students and period 8 had a mean score of 34 with 16 students. What is the mean of the two classes combined?

32. 32 Example Weighted Mean Your first semester grades at college are as follows: SubjectGradeCredits BiologyA3 CalculusB4 College Writing B3 ArcheryC1 ChemistryA3 What is your GPA?

33. 33 Advantages - Disadvantages Table 2-13 Best Measure of Center

34. 34 Extreme Values and Measures of Center Consider the following data set of salaries at a small shipping company. 30,000 30,000 30,000 30,000 30,000 40,000 45,000 125,000 What is the mean? The median? The mode? Now change the $125,000 to $250,000

35. 35  Symmetric Data is symmetric if the left half of its histogram is roughly a mirror of its right half.  Skewed Data is skewed if it is not symmetric and if it extends more to one side than the other. Definitions

36. 36 Skewness Mode = Mean = Median SYMMETRIC Figure 2-13 (b)

37. 37 Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median Figure 2-13 (b) Figure 2-13 (a)

38. 38 Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median SKEWED RIGHT (positively) Mean Mode Median Figure 2-13 (b) Figure 2-13 (a) Figure 2-13 (c)

39. 39 Review of Concepts The mean of a data set is the balance point of the values. Think of the mean as a give and take The median is the middle value of an ordered data set. The mode is the data value that occurs most frequently. There may be no mode, one mode, or more than one mode. If a distribution has few values or it is skewed, then the measures of center may not actually be near the center of the distribution. You must make appropriate decisions as to use which measure of center is most appropriate.

40. 40 Page 68 #20, 21, 24 Page 107 # 2, 3 Handout data analysis of mean