1. Basic Statistics Concepts Marketing Logistics

2. Basic Statistics Concepts Including: histograms, means, normal distributions, standard deviations.

3. Basic Statistics Concepts Developing a histogram.

4. Developing a Histogram Let’s say we are looking at the test scores of 42 students. For the sake of our discussion, we will call each test score an “observation.” Therefore, we have 42 observations. The next slide shows the 42 test scores or observations.

5. Observations

6. Plotting Scores on a Histogram We decide to start figuring out how many times students made a specific score. In other words, how many students got a score of 85? How many got a 95? And so on… We list all the scores, then begin recording how many times a student got that score. The next slide shows a list of all the scores.

7. 55 60 65 70 75 80 85 90 95

8. Scores made by students

9. Back to Our Observations.

10. Observations

11. Back to Our Observations. How many times did someone get a 70? Look on the next slide and count the number of scores of 70.

12. Observations

13. There are six scores of 70 Observations

14. Back to Our List of Scores

15. 55 60 65 70 75 80 85 90 95 Back to Our List of Scores

16. 55 60 65 70 75 80 85 90 95 1 Back to Our List of Scores For each of the scores of 70 we make one mark.

17. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 We continue to count the number of specific observations having a specific score.

18. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 We continue to count the number of specific observations having a specific score. We are making what is called a “histogram.”

19. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Histogram Many times our histogram will end up looking much like this:

20. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Histogram This is what is called a “normal distribution.”

21. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” When things occur at what we would call random, they frequently fall into a normal distribution. Histogram

22. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” In a normal distribution the highest number of observations occurs at the mean. Histogram

23. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” In a normal distribution the highest number of observations occurs at the mean. There were seven scores of 75. Histogram

24. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” Then they tend to taper off as you go to the higher scores… Histogram

25. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” Then they tend to taper off as you go to the higher scores. Only six scores of 80… Histogram

26. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” Then they tend to taper off as you go to the higher scores. Only four scores of 85… Histogram

27. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” Then they tend to taper off as you go to the higher scores. Three scores of 90. Histogram

28. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” Then they tend to taper off as you go to the higher scores. Two scores of 95. Histogram

29. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” Below the mean, scores tend to taper off, usually at about an identical rate as the scores we just looked at that were above the mean. Histogram

30. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” This phenomenon often occurs in events that we consider to be at random… Histogram

31. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” …the scores tend to be distributed in a predictable way… Histogram

32. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” …so we say it’s a… Histogram

33. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” …so we say it’s a… Histogram

34. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 This is what is called a “normal distribution.” Histogram It is usually graphed somewhat like this:

35. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Histogram It is usually graphed somewhat like this: This is what is called a “normal distribution.”

36. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Histogram It is usually graphed somewhat like this: While this is a rather crude graphing, the next slide shows several examples of normal distributions.

37. Total Order Cycle with Variability 2. Order entry and processing Frequency: 1 2 3 1. Order preparation and transmittal Frequency: 1 2 3 3. Order picking or production Frequency: 1 9 Frequency: TOTAL 3.5 days 8 20 days 5. Transportation Frequency: 1 3 5 6. Customer receiving Frequency:.5 1 1.5 From instructional material from “Strategic Logistics Management” by Stock and Lambert (2001).

38. Total Order Cycle with Variability 2. Order entry and processing Frequency: 1 2 3 1. Order preparation and transmittal Frequency: 1 2 3 3. Order picking or production Frequency: 1 9 Frequency: TOTAL 3.5 days 8 20 days 5. Transportation Frequency: 1 3 5 6. Customer receiving Frequency:.5 1 1.5 The line in the middle of each normal distribution indicates the average or, more correctly, the “mean.” It’s the place where we have the highest number of observations. From instructional material from “Strategic Logistics Management” by Stock and Lambert (2001).

39. Total Order Cycle with Variability 2. Order entry and processing Frequency: 1 2 3 1. Order preparation and transmittal Frequency: 1 2 3 3. Order picking or production Frequency: 1 9 Frequency: TOTAL 3.5 days 8 20 days 5. Transportation Frequency: 1 3 5 6. Customer receiving Frequency:.5 1 1.5 Mean or average of about 2. Mean of just under 1 Mean of about 10. Mean of 1 Mean of about 3 The line in the middle of each normal distribution indicates the average or, more correctly, the “mean.” It’s the place where we have the highest number of observations. From instructional material from “Strategic Logistics Management” by Stock and Lambert (2001).

40. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Histogram If the highest part of our histrogram is on 75, it stands to reason that 75 is our mean of our test scores.

41. Observations Sure enough, if you were to average out all of our observations…

42. Mean = 75 Observations Sure enough, if you were to average out all of our observations… You get a mean of 75.

43. A mean can be a good predictor… A mean or average can often help me predict what will happen in the future. For instance, if students usually get a mean of 75 on tests, by giving basically the same kinds of tests, an instructor can usually predict that in the future students will usually score an average of 75 on the test.

44. A mean can be a good predictor, but… Sometimes a mean is not enough for a prediction or determination.

45. A mean can be a good predictor, but… Sometimes a mean is not enough for a prediction or determination. For instance, if I tell you that the average, or mean, depth of the Mississippi River is about 18 feet, I’m not giving you a clear picture of the nature of the river.

46. A mean can be a good predictor, but… Sometimes a mean is not enough for a prediction or determination. For instance, if I tell you that the average, or mean, depth of the Mississippi River is about 18 feet, I’m not giving you a clear picture of the nature of the river. That’s because at its headwaters, the river averages around 3 feet deep. But in certain places around New Orleans, it is 200 feet deep.

47. Mississippi River Surface Mississippi River Bottom 3 feet 200 feet HeadwatersNew Orleans Mean depth = 18 feet

48. Mississippi River Surface Mississippi River Bottom 3 feet 200 feet Mean depth = 18 feet There is a big difference between 3 feet and 200 feet. HeadwatersNew Orleans

49. Mississippi River Surface Mississippi River Bottom 3 feet 200 feet Mean depth = 18 feet There is a big difference between 3 feet and 200 feet. HeadwatersNew Orleans That means my description of the Mississippi River as having an 18-foot mean depth really doesn’t tell me much.

50. Mississippi River Surface Mississippi River Bottom 3 feet 200 feet Mean depth = 18 feet There is a big difference between 3 feet and 200 feet. HeadwatersNew Orleans To get an accurate picture or prediction of Mississippi River depth, I need to find out how much variation there is around the mean. In other words, at different places along the river, how much different is the depth at that place, compared to the mean? That means my description of the Mississippi River as having an 18-foot mean depth really doesn’t tell me much.

51. Mississippi River Surface Mississippi River Bottom 3 feet 200 feet Mean depth = 18 feet HeadwatersNew Orleans To get an accurate picture or prediction of Mississippi River depth, I need to find out how much variation there is around the mean. In other words, at different places along the river, how much different is the depth at that place, compared to the mean. Depth 5 feet, 13 feet less than mean of 18 feet

52. Mississippi River Surface Mississippi River Bottom 3 feet 200 feet Mean depth = 18 feet HeadwatersNew Orleans To get an accurate picture or prediction of Mississippi River depth, I need to find out how much variation there is around the mean. In other words, at different places along the river, how much different is the depth at that place, compared to the mean. Depth 100 feet, 82 feet more than mean of 18 feet

53. Mississippi River Surface Mississippi River Bottom 3 feet 200 feet Mean depth = 18 feet HeadwatersNew Orleans To get an accurate picture or prediction of Mississippi River depth, I need to find out how much variation there is around the mean. In other words, at different places along the river, how much different is the depth at that place, compared to the mean. Depth 190 feet, 172 feet more than mean of 18 feet

54. Mississippi River Surface Mississippi River Bottom 3 feet 200 feet Mean depth = 18 feet HeadwatersNew Orleans To get an accurate picture or prediction of Mississippi River depth, I need to find out how much variation there is around the mean. In other words, at different places along the river, how much different is the depth at that place, compared to the mean. And so on…

55. Mississippi River Surface Mississippi River Bottom 3 feet 200 feet Mean depth = 18 feet HeadwatersNew Orleans To get an accurate picture or prediction of Mississippi River depth, I need to find out how much variation there is around the mean. In other words, at different places along the river, how much different is the depth at that place, compared to the mean. If I can now average all of these measurements -- how far away each depth is from the mean of 18 feet – I can get a clearer picture of how deep the Mississippi River actually is.

56. Mississippi River Surface Mississippi River Bottom 3 feet 200 feet Mean depth = 18 feet HeadwatersNew Orleans To get an accurate picture or prediction of Mississippi River depth, I need to find out how much variation there is around the mean. In other words, at different places along the river, how much different is the depth at that place, compared to the mean. In other words, what I want to know is the “standard deviation” – what is the average all of the depth measurements are away from the mean of 18 feet.

57. In other words, what I want to know is the “standard deviation” – what is the average all of the depth measurements are away from the mean of 18 feet. Let’s go back to our test scores.

58. Mean = 75 Observations

59. Standard Deviation How far away from the mean do the observations generally fall?

60. Standard Deviation How far away from the mean do the observations generally fall? There is a formula to show us…

61. Standard Deviation How far away from the mean do the observations generally fall? SD = (Observation – mean) 2 N-1

62. Standard Deviation How far away from the mean do the observations generally fall? SD = (Observation – mean) 2 N-1 To find standard deviation…

63. Standard Deviation How far away from the mean do the observations generally fall? We take the square root of… SD = (Observation – mean) 2 N-1

64. Standard Deviation How far away from the mean do the observations generally fall? SD = (Observation – mean) 2 N-1 The sum of…

65. Standard Deviation How far away from the mean do the observations generally fall? SD = (Observation – mean) 2 N-1 The difference between the observation minus the mean…

66. Standard Deviation How far away from the mean do the observations generally fall? SD = (Observation – mean) 2 N-1 The difference between the observation minus the mean… …Squared …

67. Standard Deviation How far away from the mean do the observations generally fall? SD = (Observation – mean) 2 N-1 …divided by one less than the number of observations

68. Standard Deviation Note for the Statistics Police, but something you don’t have to worry about… SD = (Observation – mean) 2 N-1 …divided by one less than the number of observations N-1 may not always be technically correct. In some cases it should be just N, the number of observations. However, in this class we will always use N-1.* *For population use N; for sample use N-1.

69. Standard Deviation Let’s find the standard deviation for our test score observations… SD = (Observation – mean) 2 N-1

70. Standard Deviation Let’s begin with just this part of the formula… SD = (Observation – mean) 2 N-1

71. Standard Deviation Let’s begin with just this part of the formula… (Observation – mean) 2

72. Standard Deviation Let’s begin with just this part of the formula and look at the test score of 70, one of our observations. (Observation – mean) 2

73. Standard Deviation Let’s begin with just this part of the formula and look at the test score of 70, one of our observations. (Observation – mean) 2 70

74. Standard Deviation And let’s include our mean, which we determined to be 75. (Observation – mean) 2 70 75

75. Standard Deviation (Observation – mean) 2 70 75 Subtract the mean from the observation, or take 75 away from 70. -

76. Standard Deviation (Observation – mean) 2 70 75 Subtract the mean from the observation, or take 75 away from 70. It equals minus 5. -= -5

77. Standard Deviation (Observation – mean) 2 70 75 -= -5 We cannot have negative numbers, so to make -5 positive, we square it, since a negative times a negative equals a positive.

78. Standard Deviation (Observation – mean) 2 70 75 -= -5 X - 5 = 25 We cannot have negative numbers, so to make -5 positive, we square it, since a negative times a negative equals a positive.

79. Standard Deviation (Observation – mean) 2 70 75 -= -5 X - 5 = 25 We cannot have negative numbers, so to make -5 positive, we square it, since a negative times a negative equals a positive. This is called a “square.”

80. ObservationMean Observation Minus mean Observation minus mean squared known as a“square.” Expressed another way: S

81. Observation Mean Observation minus mean Observation minus mean squared We go through all our observations, subtracting the mean from them and squaring the results. Squares First six observations…

82. Observation Mean Observation minus mean Observation minus mean squared Squares …next seven observations. Rather than have slides showing all the observations…

83. Observation Mean Observation minus mean Observation minus mean squared …I will skip to the final four observations. Squares

84. Standard Deviation And come to the next part of our formula… SD = (Observation – mean) 2 N-1

85. Standard Deviation And come to the next part of our formula… …adding up all of the squares. SD = (Observation – mean) 2 N-1

86. Standard Deviation And come to the next part of our formula… …adding up all of the squares. SD = (Observation – mean) 2

87. Observation Mean Observation minus mean Observation minus mean squared Add up all squares First six observations… ADDADD

88. Observation Mean Observation minus mean Observation minus mean squared Add up all squares ADDADD …next seven observations. Rather than have slides showing all the observations…

89. Observation Mean Observation minus mean Observation minus mean squared ADDADD ADDADD …I will skip to the final four observations.

90. Observation Mean Observation minus mean Observation minus mean squared Sum of squares …total of all observations.

91. Standard Deviation We now have some data for our formula… SD = (Observation – mean) 2 N-1

92. Standard Deviation We now have some data for our formula… SD = (Observation – mean) 2 N-1 Sum of squares

93. Standard Deviation We now have some data for our formula… SD = 4300 N-1 Sum of squares

94. Standard Deviation Now for the next part of our formula… SD = 4300 N-1

95. Standard Deviation Now to the next part of our formula… …divide the sum of squares by the number of observations minus 1. SD = N-1 4300

96. Count the number of observations…

97. Observation Mean Observation minus mean Observation minus mean squared First six observations… 123456123456 Count the number of observations…

98. Observation Mean Observation minus mean Observation minus mean squared …next seven observations. Rather than have slides showing all the observations… 7 8 9 10 11 12 13 Count the number of observations…

99. Observation Mean Observation minus mean Observation minus mean squared Sum of squares 39 40 41 42 Count the number of observations… …I will skip to the final four observations.

100. Observation Mean Observation minus mean Observation minus mean squared 39 40 41 42 Count the number of observations… There are 42 observations Sum of squares

101. Standard Deviation SD = N-1 4300 More data for the formula…

102. Standard Deviation SD = N-1 4300 More data for the formula… We have 42 observations so n is 42.

103. Standard Deviation SD = 42-1 4300 More data for the formula… We have 42 observations so n is 42.

104. Standard Deviation SD = 42-1 4300 More data for the formula… 42 minus 1 is 41.

105. Standard Deviation SD = 41 4300 More data for the formula… 42 minus 1 is 41.

106. Standard Deviation SD = 41 4300 Process part of the formula… 4300 divided by 41…

107. Standard Deviation SD = 41 4300 Process part of the formula… 4300 divided by 41… = 104.87 …equals 104.87

108. Standard Deviation SD = Finish the formula… Find the square root of 104.87 104.87

109. Standard Deviation SD = Finish the formula… Find the square root of 104.87 = 104.87 = 10.24 Which is 10.24

110. Standard Deviation SD = Finish the formula… Therefore, our standard deviation is 10.24 10.24

111. Standard Deviation SD = This means that our observations average 10.24 away from the mean. Therefore, our standard deviation is 10.24 10.24

112. To Review…

113. We developed a histogram…

114. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Histogram

115. To Review… We examined a normal distribution...

116. 55 60 65 70 75 80 85 90 95 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Normal Distribution

117. Total Order Cycle with Variability 2. Order entry and processing Frequency: 1 2 3 1. Order preparation and transmittal Frequency: 1 2 3 3. Order picking or production Frequency: 1 9 Frequency: TOTAL 3.5 days 8 20 days 5. Transportation Frequency: 1 3 5 6. Customer receiving Frequency:.5 1 1.5 Normal Distribution From instructional material from “Strategic Logistics Management” by Stock and Lambert (2001).

118. To Review… We learned how to determine standard deviation, or the average of how far observations are different from the mean.

119. Standard Deviation How far away from the mean do the observations generally fall? SD = (Observation – mean) 2 N-1

120. End of Program.