1. What is Statistics? GOALS ONE Understand why we study statistics. TWO
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. 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.
What is Meant by Statistics?

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

4. 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.
EXAMPLE 2: According to Consumer Reports, General Electric washing machine owners reported 9 problems per 100 machines during The statistic 9 describes the number of problems out of every 100 machines.
Types of Statistics

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

6. Types of Statistics (examples of inferential statistics)
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 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 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.
Types of Statistics (examples of inferential statistics)

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

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

9. Quantitative variables can be classified as either Discrete or Continuous.
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).
Types of Variables

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

11. Summary of Types of Variables

12. There are four levels of data
Nominal
Ordinal
Interval
Ratio
Levels of Measurement

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

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

15. Ordinal level: involves data arranged in some order, but the differences between data values cannot be determined or are meaningless.
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.
Levels of Measurement

16. Interval level Temperature on the Fahrenheit scale.
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.
Temperature on the Fahrenheit scale.
Levels of Measurement

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

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

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

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

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

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

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

25. Constructing a frequency distribution20
Drawing conclusions
Presenting data (graph) 15
Organizing data (frequency distribution) 10 5
Collecting raw data
1.5
3.5
5.5
7.5
9.5
11.5
13.5
Constructing a frequency distribution

26. Class Midpoint: A point that divides a class into two equal parts
Class Midpoint: A point that divides a class into two equal parts. This is the average of the upper and lower class limits.
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.
Class Frequency: The number of observations in each class.
Definitions

27. 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.
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.
EXAMPLE 1

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

29. Step Two: Determine the class interval or width using the formula
H – L k
33.8 – 10.3 5
i > =
= 4.7
where H=highest value, L=lowest value
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. Step Three: Set the individual class limits and
Steps Four and Five: Tally and count the number of items in each class.
EXAMPLE 1 continued

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

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

33. Graphic Presentation of a Frequency Distribution
The three commonly used graphic forms are Histograms, Frequency Polygons, and a Cumulative 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.
Graphic Presentation of a Frequency Distribution

34. Histogram for Hours Spent Studying
midpoint
Histogram for Hours Spent Studying

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

36. Frequency Polygon for Hours Spent Studying

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.
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. Cumulative frequency table
Cumulative Frequency Table for Hours Spent Studying
Cumulative frequency table

39. Cumulative Frequency Distribution For Hours Studying

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

41. Example 3 continued

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

43. Bar Chart for the Unemployment Data

44. A Pie Chart is useful for displaying a relative frequency distribution
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.
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

45. Pie Chart for Running Shoes

46. Describing Data: Numerical Measures
3- 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

47. Describing Data: Numerical Measures
3- 47
Describing Data: Numerical Measures
FOUR
Compute and interpret the range, the mean deviation, the variance, and the standard deviation of ungrouped data.
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

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

49. µ is the population mean N is the total number of observations.
3- 49
For ungrouped data, the Population Mean is the sum of all the population values divided by the total number of population values:
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

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

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

52. A statistic is a measurable characteristic of a sample.
3- 52
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
Example 2

53. Properties of the Arithmetic Mean
3- 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 of the Arithmetic Mean

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

55. 3- 55
The Weighted Mean of a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula:
Weighted Mean

56. 3- 56
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 $ Compute the weighted mean of the price of the drinks.
Example 4

57. 3- 57
The Median is the midpoint of the values after they have been ordered from the smallest to the largest.
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.
The Median

58. The median (continued)
3- 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)

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

60. Properties of the Median
3- 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

61. 3- 61
The Mode is another measure of location and represents the value of the observation that appears most frequently.
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.
The Mode: Example 6

62. The Relative Positions of the Mean, Median, and Mode
3- 62
Symmetric distribution: A distribution having the same shape on either side of the center
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

63. Skewness is the measurement of the lack of symmetry of the distribution.
The coefficient of skewness can range from 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
Median X sk - = 3

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

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

66. 3- 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

67. 3- 67
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:
The geometric mean is used to average percents, indexes, and relatives.
Geometric Mean

68. 3- 68
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.
Example 7

69. Geometric Mean continued
3- 69
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.
Geometric Mean continued

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

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

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

73. Variance and standard deviation
3- 73
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 formula:
Variance and standard deviation

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

75. Sample variance and standard deviation
3- 75
Sample variance (s2)
Sample standard deviation (s)
Sample variance and standard deviation

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

77. Coefficient of variation
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 ( ) s
Median X sk - = 3
= -.035
Example 2 revisited

78. where k is any constant greater than 1.
3- 78
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

79. Interpretation and Uses of the Standard Deviation
3- 79
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

80. Interpretation and Uses of the Standard Deviation
3- 80
68%
95%
99.7%
m-3s
m-2s
m-1s m
m+1s
m+2s
m+ 3s
Interpretation and Uses of the Standard Deviation

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

82. 3- 82
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.
Example 12

83. The Median of Grouped Data
3- 83
The Median of a sample of data organized in a frequency distribution is computed by:
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.
The Median of Grouped Data

84. Finding the Median Class
3- 84
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 25th value.
Finding the Median Class

85. 3- 85
Example 12 continued

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

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