1. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics Seventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Three Averages and Variation

2. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 Measures of Central Tendency Mode Median Mean

3. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3 The Mode the value or property that occurs most frequently in the data

4. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 4 Find the mode: 6, 7, 2, 3, 4, 6, 2, 6 The mode is 6.

5. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 5 Find the mode: 6, 7, 2, 3, 4, 5, 9, 8 There is no mode for this data.

6. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 6 The Median the central value of an ordered distribution

7. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 7 To find the median of raw data: Order the data from smallest to largest. For an odd number of data values, the median is the middle value. For an even number of data values, the median is found by dividing the sum of the two middle values by two.

8. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 8 Find the median: Data:5, 2, 7, 1, 4, 3, 2 Rearrange:1, 2, 2, 3, 4, 5, 7 The median is 3.

9. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9 Find the median: Data:31, 57, 12, 22, 43, 50 Rearrange:12, 22, 31, 43, 50, 57 The median is the average of the middle two values =

10. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 10 The Mean The mean of a collection of data is found by: summing all the entries dividing by the number of entries

11. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 Find the mean: 6, 7, 2, 3, 4, 5, 2, 8

12. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 Sigma Notation The symbol  means “sum the following.”  is the Greek letter (capital) sigma.

13. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 13 Notations for mean Sample mean “x bar” Population mean Greek letter (mu)

14. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 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.

15. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 15 Sample mean

16. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 Population mean

17. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 17 Resistant Measure a measure that is not influenced by extremely high or low data values

18. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 Which is less resistant? Mean Median The mean is less resistant. It can be made arbitrarily large by increasing the size of one value.

19. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 Trimmed Mean a measure of center that is more resistant than the mean but is still sensitive to specific data values

20. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 20 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.

21. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 21 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 =

22. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 22 Measures of Variation Range Standard Deviation Variance

23. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 23 The Range the difference between the largest and smallest values of a distribution

24. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 24 Find the range: 10, 13, 17, 17, 18 The range = largest minus smallest = 18 minus 10 = 8

25. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 25 The standard deviation a measure of the average variation of the data entries from the mean

26. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 26 Standard deviation of a sample n = sample size mean of the sample

27. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 27 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.

28. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 28 The Variance the square of the standard deviation

29. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 29 Variance of a Sample

30. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 30 Find the standard deviation and variance x 30 26 22 4 0  4 16 0 16 ___ 32 78 mean= 26 Sum = 0

31. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 31 = 32  2 =16 The variance

32. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 32 The standard deviation s =

33. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 33 Find the mean, the standard deviation and variance x45574x45574 25  1 0 2  1 Find the mean, the standard deviation and variance 1 0 4 1 6 mean = 5

34. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 34 The mean, the standard deviation and variance Mean = 5

35. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 35 Computation formula for sample standard deviation:

36. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 36 To find Square the x values, then add.

37. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 37 To find Sum the x values, then square.

38. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 38 Use the computing formulas to find s and s 2 x45574x45574 x 2 16 25 49 16 25 131 n = 5 (S x ) 2 = 25 2 = 625 S x 2 = 131 SS x = 131 – 625/5 = 6 s 2 = 6/(5 –1) = 1.5 s = 1.22

39. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 39 Population Mean and Standard Deviation

40. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 40 COEFFICIENT OF VARIATION: a measurement of the relative variability (or consistency) of data

41. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 41 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 pounds. Which (newborns or 3-month-olds) are more variable in weight?

42. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 42 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

43. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 43 Use Coefficient of Variation To compare two groups of data, to answer: Which is more consistent? Which is more variable?

44. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 44 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:

45. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 45 CHEBYSHEV'S THEOREM for k = 2 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 2) standard deviations of the mean? At least of the data falls within 2 standard deviations of the mean.

46. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 46 CHEBYSHEV'S THEOREM for k = 3 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 3) standard deviations of the mean? At least of the data falls within 3 standard deviations of the mean.

47. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 47 CHEBYSHEV'S THEOREM for k =4 According to Chebyshev’s Theorem, at least what fraction of the data falls within “k” (k = 4) standard deviations of the mean? At least of the data falls within 4 standard deviations of the mean.

48. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 48 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?

49. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 49 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

50. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 50 Mean and Standard Deviation of Grouped Data Make a frequency table Compute the midpoint (x) for each class. Count the number of entries in each class (f). Sum the f values to find n, the total number of entries in the distribution. Treat each entry of a class as if it falls at the class midpoint.

51. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 51 Sample Mean for a Frequency Distribution x = class midpoint

52. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 52 Sample Standard Deviation for a Frequency Distribution

53. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 53 Computation Formula for Standard Deviation for a Frequency Distribution

54. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 54 Calculation of the mean of grouped data Ages: f 30 - 34 4 35 - 39 5 40 - 44 2 45 - 49 9 x 32 37 42 47 xf 128 185 84 423  xf = 820  f = 20

55. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 55 Mean of Grouped Data

56. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 56 Calculation of the standard deviation of grouped data Ages: f 30 - 34 4 35 - 39 5 40 - 44 2 45 - 49 9  x 32 37 42 47 x – mean – 9 – 4 1 6 Mean  (x – mean) 2 81 16 1 36 (x – mean) 2 f 324 80 2 324  f = 20  (x – mean) 2 f = 730

57. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 57 Calculation of the standard deviation of grouped data  f = n = 20

58. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 58 Computation Formula for Standard Deviation for a Frequency Distribution

59. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 59 Computation Formula for Standard Deviation f 4 5 2 9 x 32 37 42 47 xf 128 185 84 423  xf = 820  f = 20 x 2 f 4096 6845 3528 19881  x 2 f = 34350

60. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 60 Computation Formula for Standard Deviation for a Frequency Distribution

61. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 61 Weighted Average Average calculated where some of the numbers are assigned more importance or weight

62. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 62 Weighted Average

63. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 63 Compute the Weighted Average: Midterm grade = 92 Term Paper grade = 80 Final exam grade = 88 Midterm weight = 25% Term paper weight = 25% Final exam weight = 50%

64. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 64 Compute the Weighted Average: xwxw Midterm 92.2523 Term Paper 80.2520 Final exam 88.5044 1.0087

65. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 65 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 above the Pth percentile will be (100 – P)%.

66. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 66 Percentiles 40% of data Lowest value Highest value P 40 60% of data

67. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 67 Quartiles Percentiles that divide the data into fourths Q 1 = 25th percentile Q 2 = the median Q 3 = 75th percentile

68. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 68 Quartiles Q1Q1 Median = Q 2 Q3Q3 Inter-quartile range = IQR = Q 3 — Q 1 Lowest value Highest value

69. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 69 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.

70. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 70 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.

71. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 71 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.

72. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 72 Find the quartiles: 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.

73. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 73 Find the quartiles: 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.

74. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 74 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

75. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 75 Five-Number Summary of Data Lowest value First quartile Median Third quartile Highest value

76. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 76 Box-and-Whisker Plot a graphical presentation of the five- number summary of data

77. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 77 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.

78. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 78 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 = 24Q3 = 33 Highest = 51

79. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 79 Box-and-Whisker Plot Lowest = 12 Q 1 = 17 median = 24 Q3 = 33 Highest = 51 60 - 55 - 50 - 45 - 40 - 35 - 30 - 25 - 20 - 15 - 10 -