1. Understanding Basic Statistics Fourth Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Three Averages and Variation

2. Copyright © Houghton Mifflin Company. All rights reserved.3 | 2 Measures of Central Tendency Mode Median Mean

3. Copyright © Houghton Mifflin Company. All rights reserved.3 | 3 The Mode the value that occurs most frequently in a data set

4. Copyright © Houghton Mifflin Company. All rights reserved.3 | 4 Find the mode: The mode is 6. 6, 7, 2, 3, 4, 6, 2, 6

5. Copyright © Houghton Mifflin Company. All rights reserved.3 | 5 Find the mode: There is no mode for this data. 6, 7, 2, 3, 4, 5, 9, 8

6. Copyright © Houghton Mifflin Company. All rights reserved.3 | 6 The Median the central value of an ordered distribution

7. Copyright © Houghton Mifflin Company. All rights reserved.3 | 7 To find the median of raw data: Order the data from smallest to largest. For an odd number of values pick the middle value. or For an even number of values compute the average of the middle two values

8. Copyright © Houghton Mifflin Company. All rights reserved.3 | 8 Find the median: Rearrange:1, 2, 2, 3, 4, 5, 7 The median is 3. Data:5, 2, 7, 1, 4, 3, 2

9. Copyright © Houghton Mifflin Company. All rights reserved.3 | 9 Find the Median: Rearrange:12, 22, 31, 43, 50, 57 The median is the average of the middle two values = Data:31, 57, 12, 22, 43, 50

10. Copyright © Houghton Mifflin Company. All rights reserved.3 | 10 Finding the median for a large data set For an ordered data set of n values: Position of the middle value =

11. Copyright © Houghton Mifflin Company. All rights reserved.3 | 11 The Mean An average that uses the exact value of each entry Sometimes called the arithmetic mean

12. Copyright © Houghton Mifflin Company. All rights reserved.3 | 12 The Mean The mean of a collection of data is found by: summing all the entries dividing by the number of entries

13. Copyright © Houghton Mifflin Company. All rights reserved.3 | 13 6, 7, 2, 3, 4, 5, 2, 8 Find the Mean:

14. Copyright © Houghton Mifflin Company. All rights reserved.3 | 14 Sigma Notation The symbol  means “sum the following.”  is the Greek letter (capital) sigma.

15. Copyright © Houghton Mifflin Company. All rights reserved.3 | 15 Notations for mean Sample meanPopulation mean Greek letter (mu)

16. Copyright © Houghton Mifflin Company. All rights reserved.3 | 16 Number of entries in a set of data If the data represents a sample, the number of entries = n. If the data represents an entire population, the number of entries = N.

17. Copyright © Houghton Mifflin Company. All rights reserved.3 | 17 Sample mean

18. Copyright © Houghton Mifflin Company. All rights reserved.3 | 18 Population mean

19. Copyright © Houghton Mifflin Company. All rights reserved.3 | 19 Resistant Measure a measure that is not influenced by extremely high or low data values

20. Copyright © Houghton Mifflin Company. All rights reserved.3 | 20 Which is less resistant? Mean Median The mean is less resistant. It can be made arbitrarily large by increasing the size of one value.

21. Copyright © Houghton Mifflin Company. All rights reserved.3 | 21 Trimmed Mean a measure of center that is more resistant than the mean but is still sensitive to specific data values

22. Copyright © Houghton Mifflin Company. All rights reserved.3 | 22 To calculate a (5 or 10%) trimmed mean Order the data from smallest to largest. Delete the bottom 5 or 10% of the data. Delete the same percent from the top of the data. Compute the mean of the remaining 80 or 90% of the data.

23. Copyright © Houghton Mifflin Company. All rights reserved.3 | 23 Compute a 10% trimmed mean: 15, 17, 18, 20, 20, 25, 30, 32, 36, 60 Delete the top and bottom 10% New data list: 17, 18, 20, 20, 25, 30, 32, 36 10% trimmed mean =

24. Copyright © Houghton Mifflin Company. All rights reserved.3 | 24 Weighted Average An average where more importance or weight is assigned to some of the numbers

25. Copyright © Houghton Mifflin Company. All rights reserved.3 | 25 Weighted Average If x is a data value and w is the weight assigned to that value Weighted average =

26. Copyright © Houghton Mifflin Company. All rights reserved.3 | 26 Calculating a Weighted Average In a pageant, the interview is worth 30% and appearance is worth 70%. Find the weighted average for a contestant with an interview score of 90 and an appearance score of 80.

27. Copyright © Houghton Mifflin Company. All rights reserved.3 | 27 Measures of Variation Range Standard Deviation Variance

28. Copyright © Houghton Mifflin Company. All rights reserved.3 | 28 The Range the difference between the largest and smallest values of a distribution

29. Copyright © Houghton Mifflin Company. All rights reserved.3 | 29 Find the range: 10, 13, 17, 17, 18 The range = largest minus smallest = 18 minus 10 = 8

30. Copyright © Houghton Mifflin Company. All rights reserved.3 | 30 The Standard Deviation a measure of the average variation of the data entries from the mean

31. Copyright © Houghton Mifflin Company. All rights reserved.3 | 31 Standard deviation of a sample n = sample size mean of the sample

32. Copyright © Houghton Mifflin Company. All rights reserved.3 | 32 To calculate standard deviation of a sample Calculate the mean of the sample. Find the difference between each entry (x) and the mean. These differences will add up to zero. Square the deviations from the mean. Sum the squares of the deviations from the mean. Divide the sum by (n  1) to get the variance. Take the square root of the variance to get the standard deviation.

33. Copyright © Houghton Mifflin Company. All rights reserved.3 | 33 The Variance the square of the standard deviation

34. Copyright © Houghton Mifflin Company. All rights reserved.3 | 34 Variance of a Sample

35. Copyright © Houghton Mifflin Company. All rights reserved.3 | 35 Find the standard deviation and variance x 30 26 22 4 0 -4 16 0 16 ___ 32 78 Sum = 0 Mean = 26

36. Copyright © Houghton Mifflin Company. All rights reserved.3 | 36 = 32  2 =16 The variance

37. Copyright © Houghton Mifflin Company. All rights reserved.3 | 37 The standard deviation s =

38. Copyright © Houghton Mifflin Company. All rights reserved.3 | 38 x45574x45574 25  1 0 2  1 1 0 4 1 6 mean = 5 Find the mean, the standard deviation and variance

39. Copyright © Houghton Mifflin Company. All rights reserved.3 | 39 The mean, the standard deviation and variance Mean = 5

40. Copyright © Houghton Mifflin Company. All rights reserved.3 | 40 Computation Formulas for Sample Variance and Standard Deviation:

41. Copyright © Houghton Mifflin Company. All rights reserved.3 | 41 To find  x 2 Square the x values, then add.

42. Copyright © Houghton Mifflin Company. All rights reserved.3 | 42 To find (  x ) 2 Sum the x values, then square.

43. Copyright © Houghton Mifflin Company. All rights reserved.3 | 43 Use the computing formulas to find s and s 2 x45574x45574 x 2 16 25 49 16 25 131

44. Copyright © Houghton Mifflin Company. All rights reserved.3 | 44 Population Mean

45. Copyright © Houghton Mifflin Company. All rights reserved.3 | 45 Population Standard Deviation

46. Copyright © Houghton Mifflin Company. All rights reserved.3 | 46 Coefficient Of Variation: A measurement of the relative variability (or consistency) of data.

47. Copyright © Houghton Mifflin Company. All rights reserved.3 | 47 CV is used to compare variability or consistency A sample of newborn infants had a mean weight of 6.2 pounds with a standard deviation of 1 pound. A sample of three-month-old children had a mean weight of 10.5 pounds with a standard deviation of 1.5 pound. Which (newborns or 3-month-olds) are more variable in weight?

48. Copyright © Houghton Mifflin Company. All rights reserved.3 | 48 To compare variability, compare Coefficient of Variation For newborns: For 3-month-olds: CV = 16% CV = 14% Higher CV: more variable Lower CV: more consistent

49. Copyright © Houghton Mifflin Company. All rights reserved.3 | 49 Use Coefficient of Variation To compare two groups of data, to answer: Which is more consistent? Which is more variable?

50. Copyright © Houghton Mifflin Company. All rights reserved.3 | 50 CHEBYSHEV'S THEOREM For any set of data and for any number k, greater than one, the proportion of the data that lies within k standard deviations of the mean is at least:

51. Copyright © Houghton Mifflin Company. All rights reserved.3 | 51 Results of Chebyshev’s Theorem

52. Copyright © Houghton Mifflin Company. All rights reserved.3 | 52 Using Chebyshev’s Theorem A mathematics class completes an examination and it is found that the class mean is 77 and the standard deviation is 6. According to Chebyshev's Theorem, between what two values would at least 75% of the grades be?

53. Copyright © Houghton Mifflin Company. All rights reserved.3 | 53 Mean = 77 Standard deviation = 6 At least 75% of the grades would be in the interval: 77 – 2(6) to 77 + 2(6) 77 – 12 to 77 + 12 65 to 89

54. Copyright © Houghton Mifflin Company. All rights reserved.3 | 54 Percentiles For any whole number P (between 1 and 99), the Pth percentile of a distribution is a value such that P% of the data fall at or below it. The percent falling at or above the Pth percentile will be (100 – P)%.

55. Copyright © Houghton Mifflin Company. All rights reserved.3 | 55 A histogram showing the 60 th Percentile

56. Copyright © Houghton Mifflin Company. All rights reserved.3 | 56 Percentiles

57. Copyright © Houghton Mifflin Company. All rights reserved.3 | 57 Quartiles Percentiles that divide the data into fourths Q 1 = 25th percentile Q 2 = the median Q 3 = 75th percentile

58. Copyright © Houghton Mifflin Company. All rights reserved.3 | 58 Quartiles

59. Copyright © Houghton Mifflin Company. All rights reserved.3 | 59 Computing Quartiles Order the data from smallest to largest. Find the median, the second quartile. Find the median of the data falling below Q 2. This is the first quartile. Find the median of the data falling above Q 2. This is the third quartile.

60. Copyright © Houghton Mifflin Company. All rights reserved.3 | 60 Inter-quartile range

61. Copyright © Houghton Mifflin Company. All rights reserved.3 | 61 Find the quartiles: 12 15 16 16 17 18 22 22 23 24 25 30 32 33 33 34 41 45 51 The data has been ordered. The median is 24.

62. Copyright © Houghton Mifflin Company. All rights reserved.3 | 62 12 15 16 16 17 18 22 22 23 24 25 30 32 33 33 34 41 45 51 The data has been ordered. The median is 24. Find the quartiles:

63. Copyright © Houghton Mifflin Company. All rights reserved.3 | 63 12 15 16 16 17 18 22 22 23 24 25 30 32 33 33 34 41 45 51 For the data below the median, the median is 17. 17 is the first quartile. Find the quartiles:

64. Copyright © Houghton Mifflin Company. All rights reserved.3 | 64 12 15 16 16 17 18 22 22 23 24 25 30 32 33 33 34 41 45 51 For the data above the median, the median is 33. 33 is the third quartile. Find the quartiles:

65. Copyright © Houghton Mifflin Company. All rights reserved.3 | 65 Find the interquartile range: 12 15 16 16 17 18 22 22 23 24 25 30 32 33 33 34 41 45 51 IQR = Q 3 – Q 1 = 33 – 17 = 16

66. Copyright © Houghton Mifflin Company. All rights reserved.3 | 66 Five-Number Summary of Data Lowest value First quartile Median Third quartile Highest value

67. Copyright © Houghton Mifflin Company. All rights reserved.3 | 67 Box-and-Whisker Plot a graphical presentation of the five- number summary of data

68. Copyright © Houghton Mifflin Company. All rights reserved.3 | 68 Box-and-Whisker Plot

69. Copyright © Houghton Mifflin Company. All rights reserved.3 | 69 Making a Box-and-Whisker Plot Draw a vertical scale including the lowest and highest values. To the right of the scale, draw a box from Q 1 to Q 3. Draw a solid line through the box at the median. Draw lines (whiskers) from Q 1 to the lowest and from Q 3 to the highest values.

70. Copyright © Houghton Mifflin Company. All rights reserved.3 | 70 Construct a Box-and-Whisker Plot: 12 15 16 16 17 18 22 22 23 24 25 30 32 33 33 34 41 45 51 Lowest = 12Q 1 = 17 Median = 24Q 3 = 33 Highest = 51

71. Copyright © Houghton Mifflin Company. All rights reserved.3 | 71 Box-and-Whisker Plot Highest = 51 Q 3 = 33 Median = 24 Q 1 = 17 Lowest = 12