1. 1-1 What is Statistics? GOALS When you have completed this Part, you will be able to: ONE Understand why we study statistics. TWO Explain what is meant by descriptive statistics and inferential statistics. THREE Distinguish between a qualitative variable and a quantitative variable. FOUR Distinguish between a discrete variable and a continuous variable. FIVE Distinguish among the nominal, ordinal, interval, and ratio levels of measurement. SIX Define the terms mutually exclusive and exhaustive. Goals

2. 1-2 What is Meant by Statistics? Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting numerical data to assist in making more effective decisions.

3. 1-3 Who Uses Statistics? Statistical techniques are used extensively by marketing, accounting, quality control, consumers, professional sports people, hospital administrators, educators, politicians, physicians, and many others.

4. 1-4 Types of Statistics EXAMPLE 2: According to Consumer Reports, General Electric washing machine owners reported 9 problems per 100 machines during 2001. The statistic 9 describes the number of problems out of every 100 machines. Descriptive Statistics Descriptive Statistics : Methods of organizing, summarizing, and presenting data in an informative way. EXAMPLE 1: A Gallup poll found that 49% of the people in a survey knew the name of the first book of the Bible. The statistic 49 describes the number out of every 100 persons who knew the answer.

5. 1-5 Types of Statistics Population Collection A Population is a Collection of all possible individuals, objects, or measurements of interest. Sample A Sample is a portion, or part, of the population of interest Inferential Statistics: Inferential Statistics: A decision, estimate, prediction, or generalization about a population, based on a sample.

6. 1-6 Types of Statistics (examples of inferential statistics) Example 2: Wine tasters sip a few drops of wine to make a decision with respect to all the wine waiting to be released for sale. Example 1: TV networks constantly monitor the popularity of their programs by hiring Nielsen and other organizations to sample the preferences of TV viewers. Example 3: The accounting department of a large firm will select a sample of the invoices to check for accuracy for all the invoices of the company.

7. 1-7 Types of Variables Qualitative Attribute Variable For a Qualitative or Attribute Variable the characteristic being studied is nonnumeric.

8. 1-8 Types of Variables Number of children in a family Quantitative Variable In a Quantitative Variable information is reported numerically. Balance in your checking account Minutes remaining in class

9. 1-9 Types of Variables Discrete Variables: Discrete Variables: can only assume certain values and there are usually “gaps” between values. Example: the number of bedrooms in a house, or the number of hammers sold at the local Home Depot (1,2,3,…,etc). DiscreteContinuous Quantitative variables can be classified as either Discrete or Continuous.

10. 1-10 Types of Variables The height of students in a class. Continuous Variable A Continuous Variable can assume any value within a specified range. The pressure in a tire The weight of a pork chop

11. 1-11 Summary of Types of Variables

12. 1-12 Levels of Measurement There are four levels of dataNominalOrdinalIntervalRatio

13. 1-13 Nominal data Nominal level Nominal level Data that is classified into categories and cannot be arranged in any particular order.

14. 1-14 Levels of Measurement Mutually exclusive An individual, object, or measurement is included in only one category. Nominal level variables must be: Exhaustive Each individual, object, or measurement must appear in one of the categories.

15. 1-15 Levels of Measurement During a taste test of 4 soft drinks, Coca Cola was ranked number 1, Dr. Pepper number 2, Pepsi number 3, and Root Beer number 4. Ordinal level Ordinal level : involves data arranged in some order, but the differences between data values cannot be determined or are meaningless.

16. 1-16 Levels of Measurement Temperature on the Fahrenheit scale. Interval level Similar to the ordinal level, with the additional property that meaningful amounts of differences between data values can be determined. There is no natural zero point.

17. 1-17 Levels of Measurement Ratio level: Ratio level: the interval level with an inherent zero starting point. Differences and ratios are meaningful for this level of measurement.

18. 1-18 Describing Data: Frequency Distributions and Graphic Presentation GOALS When you have completed this Part, you will be able to: ONE Organize data into a frequency distribution. TWO Portray a frequency distribution in a histogram, frequency polygon, and cumulative frequency polygon. THREE Present data using such graphic techniques as line charts, bar charts, and pie charts. Goals

19. 1-19 Frequency Distribution Frequency Distribution A Frequency Distribution is a grouping of data into mutually exclusive categories showing the number of observations in each class.

20. 1-20 Determining the question to be addressed Constructing a frequency distribution involves: Constructing a frequency distribution

21. 1-21 Determining the question to be addressed Constructing a frequency distribution involves: Collecting raw data Constructing a frequency distribution

22. 1-22 Determining the question to be addressed Constructing a frequency distribution involves: Collecting raw data Organizing data (frequency distribution) Constructing a frequency distribution

23. 1-23 Determining the question to be addressed Constructing a frequency distribution involves: Collecting raw data Organizing data (frequency distribution) Presenting data (graph) Constructing a frequency distribution

24. 1-24 Determining the question to be addressed Constructing a frequency distribution involves: Collecting raw data Organizing data (frequency distribution) Presenting data (graph) Drawing conclusions Constructing a frequency distribution

25. 1-25 Collecting raw data Organizing data (frequency distribution) Presenting data (graph) Drawing conclusions 1.53.55.5 7.59.5 11.513.5 5 10 15 20 Constructing a frequency distribution

26. 1-26 Class Midpoint: Class Midpoint: A point that divides a class into two equal parts. This is the average of the upper and lower class limits. Class Frequency Class Frequency : The number of observations in each class. Class interval Class interval : The class interval is obtained by subtracting the lower limit of a class from the lower limit of the next class. The class intervals should be equal. Definitions

27. 1-27 EXAMPLE 1 15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6. Organize the data into a frequency distribution. Dr. Tillman is Dean of the School of Business Socastee University. He wishes prepare to a report showing the number of hours per week students spend studying. He selects a random sample of 30 students and determines the number of hours each student studied last week.

28. 1-28 Example 1 continued Step One: Step One: Decide on the number of classes using the formula 2 k > n where k=number of classes n=number of observations oThere are 30 observations so n=30. oTwo raised to the fifth power is 32. oTherefore, we should have at least 5 classes, i.e., k=5.

29. 1-29 where H=highest value, L=lowest value 33.8 – 10.3 5 = 4.7 Step Two Step Two: Determine the class interval or width using the formula H – L k i > = Round up for an interval of 5 hours. Set the lower limit of the first class at 7.5 hours, giving a total of 6 classes. Example 1 continued

30. 1-30 EXAMPLE 1 continued Step Three Step Three: Set the individual class limits and Steps Four and Five Steps Four and Five: Tally and count the number of items in each class.

31. 1-31 Class Midpoint Class Midpoint : find the midpoint of each interval, use the following formula: Upper limit + lower limit 2 Example 1 continued

32. 1-32 Example 1 continued Relative Frequency Distribution A Relative Frequency Distribution shows the percent of observations in each class.

33. 1-33 Graphic Presentation of a Frequency Distribution A Histogram is a graph in which the class midpoints or limits are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars and the bars are drawn adjacent to each other. Histograms, Frequency Polygons Cumulative Frequency The three commonly used graphic forms are Histograms, Frequency Polygons, and a Cumulative Frequency distribution.

34. 1-34 Histogram for Hours Spent Studying midpoint

35. 1-35 Graphic Presentation of a Frequency Distribution Frequency Polygon A Frequency Polygon consists of line segments connecting the points formed by the class midpoint and the class frequency.

36. 1-36 Frequency Polygon for Hours Spent Studying

37. 1-37 Cumulative Frequency Distribution A Cumulative Frequency Distribution is used to determine how many or what proportion of the data values are below or above a certain value. Cumulative Frequency Distribution To create a cumulative frequency polygon, scale the upper limit of each class along the X-axis and the corresponding cumulative frequencies along the Y-axis. Cumulative Frequency distribution

38. 1-38 Cumulative Frequency Table for Hours Spent Studying Cumulative frequency table

39. 1-39 Cumulative Frequency Distribution For Hours Studying Cumulative frequency distribution

40. 1-40 Line graphs are typically used to show the change or trend in a variable over time. Line Graphs

41. 1-41 Example 3 continued

42. 1-42 Construct a bar chart for the number of unemployed per 100,000 population for selected cities during 2001 ar Chart A Bar Chart can be used to depict any of the levels of measurement (nominal, ordinal, interval, or ratio). Bar Chart

43. 1-43 Bar Chart for the Unemployment Data

44. 1-44 Pie Chart A sample of 200 runners were asked to indicate their favorite type of running shoe. Draw a pie chart based on the following information. Pie Chart A Pie Chart is useful for displaying a relative frequency distribution. A circle is divided proportionally to the relative frequency and portions of the circle are allocated for the different groups.

45. 1-45 Pie Chart for Running Shoes

46. 1-46 Describing Data: Numerical Measures GOALS When you have completed this Part, you will be able to: ONE Calculate the arithmetic mean, median, mode, weighted mean, and the geometric mean. TWO Explain the characteristics, uses, advantages, and disadvantages of each measure of location. THREE Identify the position of the arithmetic mean, median, and mode for both a symmetrical and a skewed distribution. Goals 3- 46

47. 1-47 FOUR Compute and interpret the range, the mean deviation, the variance, and the standard deviation of ungrouped data. Describing Data: Numerical Measures FIVE Explain the characteristics, uses, advantages, and disadvantages of each measure of dispersion. SIX Understand Chebyshev’s theorem and the Empirical Rule as they relate to a set of observations. Goals 3- 47

48. 1-48 Characteristics of the Mean It is calculated by summing the values and dividing by the number of values.  It requires the interval scale.  All values are used.  It is unique.  The sum of the deviations from the mean is 0. Arithmetic Mean The Arithmetic Mean is the most widely used measure of location and shows the central value of the data. The major characteristics of the mean are: 3- 48

49. 1-49 Population Mean where  µ is the population mean  N is the total number of observations.  X is a particular value.   indicates the operation of adding. Population Mean For ungrouped data, the Population Mean is the sum of all the population values divided by the total number of population values: 3- 49

50. 1-50 Example 1 Find the mean mileage for the cars. Parameter A Parameter is a measurable characteristic of a population. The Kiers family owns four cars. The following is the current mileage on each of the four cars. 56,000 23,000 42,000 73,000 3- 50

51. 1-51 Sample Mean where n is the total number of values in the sample. For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values: 3- 51

52. 1-52 Example 2 statistic A statistic is a measurable characteristic of a sample. A sample of five executives received the following bonus last year ($000): 14.0, 15.0, 17.0, 16.0, 15.0 3- 52

53. 1-53 Properties of the Arithmetic Mean  Every set of interval-level and ratio-level data has a mean.  All the values are included in computing the mean.  A set of data has a unique mean.  The mean is affected by unusually large or small data values.  The arithmetic mean is the only measure of location where the sum of the deviations of each value from the mean is zero. Properties Properties of the Arithmetic Mean 3- 53

54. 1-54 Example 3 mean Consider the set of values: 3, 8, and 4. The mean is 5. Illustrating the fifth property 3- 54

55. 1-55 Weighted Mean Weighted Mean The Weighted Mean of a set of numbers X 1, X 2,..., X n, with corresponding weights w 1, w 2,...,w n, is computed from the following formula: 3- 55

56. 1-56 Example 4 During a one hour period on a hot Saturday afternoon cabana boy Chris served fifty drinks. He sold five drinks for $0.50, fifteen for $0.75, fifteen for $0.90, and fifteen for $1.10. Compute the weighted mean of the price of the drinks. 3- 56

57. 1-57 The Median There are as many values above the median as below it in the data array. For an even set of values, the median will be the arithmetic average of the two middle numbers and is found at the (n+1)/2 ranked observation. Median The Median is the midpoint of the values after they have been ordered from the smallest to the largest. 3- 57

58. 1-58 The ages for a sample of five college students are: 21, 25, 19, 20, 22. Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21. The median (continued) 3- 58

59. 1-59 Example 5 Arranging the data in ascending order gives: 73, 75, 76, 80 Thus the median is 75.5. The heights of four basketball players, in inches, are: 76, 73, 80, 75. The median is found at the (n+1)/2 = (4+1)/2 =2.5 th data point. 3- 59

60. 1-60 Properties of the Median  There is a unique median for each data set.  It is not affected by extremely large or small values and is therefore a valuable measure of location when such values occur.  It can be computed for ratio-level, interval- level, and ordinal-level data.  It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class. Properties of the Median 3- 60

61. 1-61 The Mode: Example 6 Example 6 : Example 6 : The exam scores for ten students are: 81, 93, 84, 75, 68, 87, 81, 75, 81, 87. Because the score of 81 occurs the most often, it is the mode. Data can have more than one mode. If it has two modes, it is referred to as bimodal, three modes, trimodal, and the like. Mode The Mode is another measure of location and represents the value of the observation that appears most frequently. 3- 61

62. 1-62 Symmetric distribution Symmetric distribution : A distribution having the same shape on either side of the center Skewed distribution Skewed distribution : One whose shapes on either side of the center differ; a nonsymmetrical distribution. Can be positively or negatively skewed, or bimodal The Relative Positions of the Mean, Median, and Mode 3- 62

63. 4-63 Skewness is the measurement of the lack of symmetry of the distribution. The coefficient of skewness can range from -3.00 up to 3.00 when using the following formula: A value of 0 indicates a symmetric distribution. Some software packages use a different formula which results in a wider range for the coefficient.  s MedianX sk   3

64. 1-64 The Relative Positions of the Mean, Median, and Mode: Symmetric Distribution Zero skewness Mean =Median =Mode 3- 64

65. 1-65 The Relative Positions of the Mean, Median, and Mode: Right Skewed Distribution  Positively skewed: Mean and median are to the right of the mode. Mean>Median>Mode 3- 65

66. 1-66 Negatively Skewed: Mean and Median are to the left of the Mode. Mean<Median<Mode The Relative Positions of the Mean, Median, and Mode: Left Skewed Distribution 3- 66

67. 1-67 Geometric Mean The geometric mean is used to average percents, indexes, and relatives. Geometric Mean The Geometric Mean (GM) of a set of n numbers is defined as the nth root of the product of the n numbers. The formula is: 3- 67

68. 1-68 Example 7 The interest rate on three bonds were 5, 21, and 4 percent. The arithmetic mean is (5+21+4)/3 =10.0. The geometric mean is The GM gives a more conservative profit figure because it is not heavily weighted by the rate of 21percent. 3- 68

69. 1-69 Geometric Mean continued Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another. 3- 69

70. 1-70 Example 8 The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in 2000. That is, the geometric mean rate of increase is 1.27%. 3- 70

71. 1-71 Variance: Variance: the arithmetic mean of the squared deviations from the mean. Standard deviation Standard deviation: The square root of the variance. Variance and standard Deviation 3- 71

72. 1-72 Not influenced by extreme values.  The units are awkward, the square of the original units.  All values are used in the calculation. The major characteristics of the Population Variance Population Variance are: Population Variance 3- 72

73. 1-73 Population Variance Population Variance formula: X is the value of an observation in the population m is the arithmetic mean of the population N is the number of observations in the population Population Standard Deviation Population Standard Deviation formula: Variance and standard deviation 3- 73

74. 1-74 In Example 9, the variance and standard deviation are: Example 9 continued 3- 74

75. 1-75 Sample variance (s 2 ) Sample standard deviation (s) Sample variance and standard deviation 3- 75

76. 1-76 Example 11 The hourly wages earned by a sample of five students are: $7, $5, $11, $8, $6. Find the sample variance and standard deviation. 3- 76

77. 4-77 Using the twelve stock prices, we find the mean to be 84.42, standard deviation, 7.18, median, 84.5. Coefficient of variation = 8.5% Coefficient of skewness = -.035 Example 2 revisited  s MedianX sk   3

78. 1-78 Chebyshev’s theorem: Chebyshev’s theorem: For any set of observations, the minimum proportion of the values that lie within k standard deviations of the mean is at least:  where k is any constant greater than 1. Chebyshev’s theorem 3- 78

79. 1-79 Empirical Rule Empirical Rule : For any symmetrical, bell-shaped distribution:  About 68% of the observations will lie within 1s the mean  About 95% of the observations will lie within 2s of the mean  Virtually all the observations will be within 3s of the mean Interpretation and Uses of the Standard Deviation 3- 79

80. 1-80      68% 95% 99.7% Interpretation and Uses of the Standard Deviation 3- 80

81. 1-81 The Mean of Grouped Data Mean  The Mean of a sample of data organized in a frequency distribution is computed by the following formula: 3- 81

82. 1-82 Example 12 A sample of ten movie theaters in a large metropolitan area tallied the total number of movies showing last week. Compute the mean number of movies showing. 3- 82

83. 1-83 The Median of Grouped Data where L is the lower limit of the median class, CF is the cumulative frequency preceding the median class, f is the frequency of the median class, and i is the median class interval. Median The Median of a sample of data organized in a frequency distribution is computed by: 3- 83

84. 1-84 Finding the Median Class To determine the median class for grouped data Construct a cumulative frequency distribution. Divide the total number of data values by 2. Determine which class will contain this value. For example, if n=50, 50/2 = 25, then determine which class will contain the 25 th value. 3- 84

85. 1-85 Example 12 continued 3- 85

86. 1-86 Example 12 continued From the table, L=5, n=10, f=3, i=2, CF=3 3- 86

87. 1-87 The Mode of Grouped Data The modes in example 12 are 6 and 10 and so is bimodal. Mode The Mode for grouped data is approximated by the midpoint of the class with the largest class frequency. 3- 87