1. Some definitions In Statistics

2. A sample: Is a subset of the population

3. In statistics: One draws conclusions about the population based on data collected from a sample

4. Reasons: Cost It is less costly to collect data from a sample then the entire population Accuracy

5. Data from a sample sometimes leads to more accurate conclusions then data from the entire population Costs saved from using a sample can be directed to obtaining more accurate observations on each case in the population

6. Types of Samples different types of samples are determined by how the sample is selected.

7. Convenience Samples In a convenience sample the subjects that are most convenient to the researcher are selected as objects in the sample. This is not a very good procedure for inferential Statistical Analysis but is useful for exploratory preliminary work.

8. Quota samples In quota samples subjects are chosen conveniently until quotas are met for different subgroups of the population. This also is useful for exploratory preliminary work.

9. Random Samples Random samples of a given size are selected in such that all possible samples of that size have the same probability of being selected.

10. Convenience Samples and Quota samples are useful for preliminary studies. It is however difficult to assess the accuracy of estimates based on this type of sampling scheme. Sometimes however one has to be satisfied with a convenience sample and assume that it is equivalent to a random sampling procedure

11. Population Sample Case  Variables X Y Z

12. Some other definitions

13. A population statistic (parameter): Any quantity computed from the values of variables for the entire population.

14. A sample statistic: Any quantity computed from the values of variables for the cases in the sample.

15. Since only cases from the sample are observed –only sample statistics are computed –These are used to make inferences about population statistics –It is important to be able to assess the accuracy of these inferences

16. To download lectures 1.Go to the stats 244 web site a)Through PAWS or b)by going to the website of the department of Mathematics and Statistics -> people -> faculty -> W.H. Laverty -> Stats 244-. Lectures. 2.Then a)select the lecture b)Right click and choose Save as

17. To print lectures 1.Open the lecture using MS Powerpoint 2.Select the menu item File -> Print

18. The following dialogue box appear

19. In the Print what box, select handouts

20. Set Slides per page to 6 or 3.

21. 6 slides per page will result in the least amount of paper being printed 12 34 56

22. 3 slides per page leaves room for notes. 1 2 3

23. Organizing and describing Data

24. Techniques for continuous variables

25. The Grouped frequency table: The Histogram

26. To Construct A Grouped frequency table A Histogram

27. 1.Find the maximum and minimum of the observations. 2.Choose non-overlapping intervals of equal width (The Class Intervals) that cover the range between the maximum and the minimum. 3.The endpoints of the intervals are called the class boundaries. 4.Count the number of observations in each interval (The cell frequency - f). 5.Calculate relative frequency relative frequency = f/N

28. Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial Reading Acheivement Score, and Final Reading Acheivement Score for 23 students who have recently completed a reading improvement program InitialFinal VerbalMathReadingReading StudentIQIQAcheivementAcheivement 186941.11.7 21041031.51.7 386921.51.9 41051002.02.0 51181151.93.5 6961021.42.4 790871.51.8 8951001.42.0 9105961.71.7 1084801.61.7 1194871.61.7 121191161.73.1 1382911.21.8 1480931.01.7 151091241.82.5 161111191.43.0 1789941.61.8 18991171.62.6 1994931.41.4 20991101.42.0 2195971.51.3 221021041.73.1 23102931.61.9

29. In this example the upper endpoint is included in the interval. The lower endpoint is not.

30. Histogram – Verbal IQ

31. Histogram – Math IQ

32. Example In this example we are comparing (for two drugs A and B) the time to metabolize the drug. 120 cases were given drug A. 120 cases were given drug B. Data on time to metabolize each drug is given on the next two slides

33. Drug A

34. Drug B

35. Grouped frequency tables

36. Histogram – drug A (time to metabolize)

37. Histogram – drug B (time to metabolize)

38. Some comments about histograms The width of the class intervals should be chosen so that the number of intervals with a frequency less than 5 is small. This means that the width of the class intervals can decrease as the sample size increases

39. If the width of the class intervals is too small. The frequency in each interval will be either 0 or 1 The histogram will look like this

40. If the width of the class intervals is too large. One class interval will contain all of the observations. The histogram will look like this

41. Ideally one wants the histogram to appear as seen below. This will be achieved by making the width of the class intervals as small as possible and only allowing a few intervals to have a frequency less than 5.

42. As the sample size increases the histogram will approach a smooth curve. This is the histogram of the population

43. N = 25

44. N = 100

45. N = 500

46. N = 2000

47. N = ∞

48. Comment: the proportion of area under a histogram between two points estimates the proportion of cases in the sample (and the population) between those two values.

49. Example: The following histogram displays the birth weight (in Kg’s) of n = 100 births

50. Find the proportion of births that have a birthweight less than 0.34 kg.

51. Proportion = (1+1+3+10+11+19+17)/100 = 0.62

52. The Characteristics of a Histogram Central Location (average) Spread (Variability, Dispersion) Shape

53. Central Location

54. Spread, Dispersion, Variability

55. Shape – Bell Shaped (Normal)

56. Shape – Positively skewed

57. Shape – Negatively skewed

58. Shape – Platykurtic

59. Shape – Leptokurtic

60. Shape – Bimodal

61. The Stem-Leaf Plot An alternative to the histogram

62. Each number in a data set can be broken into two parts – A stem – A Leaf

63. Example Verbal IQ = 84 84 –Stem = 10 digit = 8 – Leaf = Unit digit = 4 Leaf Stem

64. Example Verbal IQ = 104 104 –Stem = 10 digit = 10 – Leaf = Unit digit = 4 Leaf Stem

65. To Construct a Stem- Leaf diagram Make a vertical list of “all” stems Then behind each stem make a horizontal list of each leaf

66. Example The data on N = 23 students Variables Verbal IQ Math IQ Initial Reading Achievement Score Final Reading Achievement Score

67. Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial Reading Acheivement Score, and Final Reading Acheivement Score for 23 students who have recently completed a reading improvement program InitialFinal VerbalMathReadingReading StudentIQIQAcheivementAcheivement 186941.11.7 21041031.51.7 386921.51.9 41051002.02.0 51181151.93.5 6961021.42.4 790871.51.8 8951001.42.0 9105961.71.7 1084801.61.7 1194871.61.7 121191161.73.1 1382911.21.8 1480931.01.7 151091241.82.5 161111191.43.0 1789941.61.8 18991171.62.6 1994931.41.4 20991101.42.0 2195971.51.3 221021041.73.1 23102931.61.9

68. We now construct: a stem-Leaf diagram of Verbal IQ

69. A vertical list of the stems 8 9 10 11 12 We now list the leafs behind stem

70. 8 9 10 11 12 8610486 511896909510584 94119828010911189999499 95102 2

71. 8 9 10 11 12 8610486 511896909510584 94119828010911189999499 95102 2

72. 86 6 4 2 0 9 96 0 5 4 9 4 9 5 104 5 5 9 2 2 118 9 1 12

73. 8 0 2 4 6 6 9 9 0 4 4 5 5 6 9 9 10 2 2 4 5 5 9 11 1 8 9 12 The leafs may be arranged in order

74. 8 0 2 4 6 6 9 9 0 4 4 5 5 6 9 9 10 2 2 4 5 5 9 11 1 8 9 12 The stem-leaf diagram is equivalent to a histogram

75. 8 0 2 4 6 6 9 9 0 4 4 5 5 6 9 9 10 2 2 4 5 5 9 11 1 8 9 12 The stem-leaf diagram is equivalent to a histogram

76. Rotating the stem-leaf diagram we have 8090100110120

77. The two part stem leaf diagram Sometimes you want to break the stems into two parts for leafs 0,1,2,3,4 * for leafs 5,6,7,8,9

78. Stem-leaf diagram for Initial Reading Acheivement 1.01234444455556666677789 2.0 This diagram as it stands does not give an accurate picture of the distribution

79. We try breaking the stems into two parts 1.*012344444 1. 55556666677789 2.* 0 2.

80. The five-part stem-leaf diagram If the two part stem-leaf diagram is not adequate you can break the stems into five parts for leafs 0,1 tfor leafs 2,3 ffor leafs 4, 5 s for leafs 6,7 *for leafs 8,9

81. We try breaking the stems into five parts 1.*01 1.t23 1.f444445555 1.s66666777 1. 89 2.* 0

82. Stem leaf Diagrams Verbal IQ, Math IQ, Initial RA, Final RA

83. Some Conclusions Math IQ, Verbal IQ seem to have approximately the same distribution “bell shaped” centered about 100 Final RA seems to be larger than initial RA and more spread out Improvement in RA Amount of improvement quite variable

84. Numerical Measures Measures of Central Tendency (Location) Measures of Non Central Location Measure of Variability (Dispersion, Spread) Measures of Shape

85. Measures of Central Tendency (Location) Mean Median Mode Central Location

86. Measures of Non-central Location Quartiles, Mid-Hinges Percentiles Non - Central Location

87. Measure of Variability (Dispersion, Spread) Variance, standard deviation Range Inter-Quartile Range Variability

88. Measures of Shape Skewness Kurtosis

89. Measures of Central Location (Mean) Summation Notation Let x 1, x 2, x 3, … x n denote a set of n numbers. Then the symbol denotes the sum of these n numbers x 1 + x 2 + x 3 + …+ x n

90. Example Let x 1, x 2, x 3, x 4, x 5 denote a set of 5 denote the set of numbers in the following table. i12345 xixi 101521713

91. Then the symbol denotes the sum of these 5 numbers x 1 + x 2 + x 3 + x 4 + x 5 = 10 + 15 + 21 + 7 + 13 = 66

92. Meaning of parts of summation notation Quantity changing in each term of the sum Starting value for i Final value for i each term of the sum

93. Example Again let x 1, x 2, x 3, x 4, x 5 denote a set of 5 denote the set of numbers in the following table. i12345 xixi 101521713

94. Then the symbol denotes the sum of these 3 numbers = 15 3 + 21 3 + 7 3 = 3375 + 9261 + 343 = 12979

95. Mean Let x 1, x 2, x 3, … x n denote a set of n numbers. Then the mean of the n numbers is defined as:

96. Example Again let x 1, x 2, x 3, x 4, x 5 denote a set of 5 denote the set of numbers in the following table. i12345 xixi 101521713

97. Then the mean of the 5 numbers is:

98. Interpretation of the Mean Let x 1, x 2, x 3, … x n denote a set of n numbers. Then the mean,, is the centre of gravity of those the n numbers. That is if we drew a horizontal line and placed a weight of one at each value of x i, then the balancing point of that system of mass is at the point.

99. x1x1 x2x2 x3x3 x4x4 xnxn

100. 10715 21 13 In the Example 100 20

101. The mean,, is also approximately the center of gravity of a histogram

102. The Median Let x 1, x 2, x 3, … x n denote a set of n numbers. Then the median of the n numbers is defined as the number that splits the numbers into two equal parts. To evaluate the median we arrange the numbers in increasing order.

103. If the number of observations is odd there will be one observation in the middle. This number is the median. If the number of observations is even there will be two middle observations. The median is the average of these two observations

104. Example Again let x 1, x 2, x 3, x 3, x 4, x 5 denote a set of 5 denote the set of numbers in the following table. i12345 xixi 101521713

105. The numbers arranged in order are: 710131521 Unique “Middle” observation – the median

106. Example 2 Let x 1, x 2, x 3, x 4, x 5, x 6 denote the 6 denote numbers: 23411219648 Arranged in increasing order these observations would be: 81219234164 Two “Middle” observations

107. Median = average of two “middle” observations =

108. Example The data on N = 23 students Variables Verbal IQ Math IQ Initial Reading Achievement Score Final Reading Achievement Score

109. Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial Reading Acheivement Score, and Final Reading Acheivement Score for 23 students who have recently completed a reading improvement program InitialFinal VerbalMathReadingReading StudentIQIQAcheivementAcheivement 186941.11.7 21041031.51.7 386921.51.9 41051002.02.0 51181151.93.5 6961021.42.4 790871.51.8 8951001.42.0 9105961.71.7 1084801.61.7 1194871.61.7 121191161.73.1 1382911.21.8 1480931.01.7 151091241.82.5 161111191.43.0 1789941.61.8 18991171.62.6 1994931.41.4 20991101.42.0 2195971.51.3 221021041.73.1 23102931.61.9

110. Computing the Median Stem leaf Diagrams Median = middle observation =12 th observation

111. Summary

112. Some Comments The mean is the centre of gravity of a set of observations. The balancing point. The median splits the obsevations equally in two parts of approximately 50%

113. The median splits the area under a histogram in two parts of 50% The mean is the balancing point of a histogram 50% median

114. For symmetric distributions the mean and the median will be approximately the same value 50% Median &

115. 50% median For Positively skewed distributions the mean exceeds the median For Negatively skewed distributions the median exceeds the mean 50%

116. An outlier is a “wild” observation in the data Outliers occur because –of errors (typographical and computational) –Extreme cases in the population

117. The mean is altered to a significant degree by the presence of outliers Outliers have little effect on the value of the median This is a reason for using the median in place of the mean as a measure of central location Alternatively the mean is the best measure of central location when the data is Normally distributed (Bell-shaped)