COURSE NOTES

TOPIC 1: Introduction to Statistics (Textbook Chapter 1)

Introduction

No doubt you have noticed the large number of facts and figures, often referred to as statistics, that appear in the newspapers and magazines you read, websites you visit, television you watch (especially sporting events), and in grocery stores where you shop. A simple figure is called a statistic. A few examples: • Home and condominium sales declined 6.5% in Charleston, South Carolina in April, 2006 compared to sales in April, 2005. • Tuition and fees for resident undergraduate students at public four year institutions averaged $5,491 for 2005-06, a 7.1%increase over 2004-05. • Approximately 24 million medicare beneficiaries were enrolled in the new prescription drug program as of January, 2006. • The government reported that 138,000 jobs were added to the economy in April, 2006. • The Dow Jones Industrial Average was 11,094.04 on May 30, 2006.

You may think of statistics simply as a collection of numerical information. However, statistics has a much broader meaning.

Learning Objectives
After completing this chapter, you will be able to: 1. Understand why we study statistics. 2. Explain what is meant by descriptive statistics and inferential statistics. 3. Distinguish between a qualitative variable and a quantitative variable. 4. Distinguish between a discrete variable and a continuous variable. 5. Distinguish among nominal, ordinal, interval, and ratio levels of measurement.

Key Contents

What is Statistics?
How do we define the word statistics? We encounter it frequently in our everyday language. It really has two meanings. In the more common usage, statistics refers to numerical information. Examples include the average starting salary of college graduates, the number of deaths due to alcoholism last year, the change in the Dow Jones Industrial Average from yesterday to today. In these examples statistics are a value or a percentage. Other examples include: • The typical automobile in the United States travels 11,099 miles per year, the typical bus 9,353 miles per year, and the typical truck 13,942 miles per year. In Canada the corresponding information is 10,371 miles for automobiles, 19,823 miles for buses, and 7,001 miles for trucks. • The mean time waiting for technical support is 17 minutes. • The mean length of the business cycle since 1945 is 61 months.
The above are all examples of statistics. Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting data to assist in making more effective decisions. A collection of numerical information is called statistics.
We often present statistical information in a graphical form. A graph is often useful for capturing reader attention and to portray a large amount of information. For example, Chart 1–1 shows Frito-Lay volume and market share for the major snack and potato chip categories in supermarkets in the United States. It requires only a quick glance to discover there were nearly 800 million pounds of potato chips sold and that Frito-Lay sold 64 percent of that total. Also note that Frito-Lay has 82 percent of the corn chip market.

CHART 1-1 Frito-Lay Volume and Share of Major Snack Chip Categories in U.S. Supermarkets [pic]

Type of Statistics

The study of statistics is usually divided into two categories: descriptive statistics and inferential statistics.
Descriptive Statistics
The definition of statistics given earlier referred to “Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting data to assist in making more effective decisions.” This facet of statistics is usually referred to as descriptive statistics. Descriptive statistics is a method of organizing, summarizing, and presenting data in an informative way.
For instance, the United States government reports the population of the United States was 179,323,000 in 1960; 203,302,000 in 1970; 226,542,000 in 1980; 248,709,000 in 1990, and 265,000,000 in 2000. This information is descriptive statistics. It is descriptive statistics if we calculate the percentage growth from one decade to the next. However, it would not be descriptive statistics if we used these to estimate the population of the United States in the year 2010 or the percentage growth from 2000 to 2010. Why? Because these statistics are not being used to summarize past populations but to estimate future populations. The following are some other examples of descriptive statistics. • There are a total of 42,796 miles of interstate highways in the United States. The interstate system represents only 1 percent of the nation’s total roads but carries more than 20 percent of the traffic. The longest is I-90, which stretches from Boston to Seattle, a distance of 3,081 miles. The shortest is I-878 in New York City, which is 0.70 of a mile in length. Alaska does not have any interstate highways, Texas has the most interstate miles at 3,232, and New York has the most interstate routes with 28. • According to the Bureau of Labor Statistics, the average hourly earnings of production workers were $17.73 for January 2006. Masses of unorganized data—such as the census of population, the weekly earnings of thousands of computer programmers, and the individual responses of 2,000 registered voters regarding their choice for president of the United States—are of little value as is. However, statistical techniques are available to organize this type of data into a meaningful form. Data can be organized into a frequency distribution. A grouping of data into mutually exclusive classes showing the number of observations in each class.
Inferential Statistics
The second type of statistics is inferential statisticsThe methods used to estimate a property of a population on the basis of a sample.—also called statistical inference. Our main concern regarding inferential statistics is finding something about a population from a sample taken from that population. For example, a recent survey showed only 46 percent of high school seniors can solve problems involving fractions, decimals, and percentages. And only 77 percent of high school seniors correctly totaled the cost of salad, a burger, fries, and a cola on a restaurant menu. Since these are inferences about a population (all high school seniors) based on sample data, we refer to them as inferential statistics. You might think of inferential statistics as a “best guess” of a population value based on sample information.
Note the words population and sample in the definition of inferential statistics. We often make reference to the population living in the United States or the 1.31 billion population of China. However, in statistics the word population has a broader meaning. A population is the entire set of individuals or objects of interest or the measurements obtained from all individuals or objects of interest. may consist of individuals—such as all the students enrolled at Utah State University, all the students in Accounting 201, or all the CEOs from the Fortune 500 companies. A population may also consist of objects, such as all the Cobra G/T tires produced at Cooper Tire and Rubber Company in the Findlay, Ohio, plant; the accounts receivable at the end of October for Lorrange Plastics, Inc.; or auto claims filed in the first quarter of 2006 at the Northeast Regional Office of State Farm Insurance. The measurement of interest might be the scores on the first examination of all students in Accounting 201, the tread wear of the Cooper Tires, the dollar amount of Lorrange Plastics’s accounts receivable, or the amount of auto insurance claims at State Farm. Thus, a population in the statistical sense does not always refer to people.
To infer something about a population, we usually take a sample from the population. A sample is a portion, or part, of the population of interest.

Types of Variables

Qualitative variable
There are two basic types of variables: (1) qualitative and (2) quantitative (see Chart 1–2). When the characteristic being studied is nonnumeric, it is called a qualitative variable. Qualitative variable is a nominal-scale variable that is coded to assume only one of two possible outcomes. For example, a person is considered either employed or unemployed. or an attribute. Examples of qualitative variables are gender, religious affiliation, type of automobile owned, state of birth, and eye color. When the data are qualitative, we are usually interested in how many or what proportion fall in each category. For example, what percent of the population has blue eyes? What percent of the total number of cars sold last month were SUVs? Qualitative data are often summarized in charts and bar graphs.
CHART 1-2 Summary of the Types of Variables

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Quantitative variable
When the variable studied can be reported numerically, the variable is called a quantitative variable. Examples of quantitative variables are the balance in your checking account, the ages of company presidents, the life of an automobile battery (such as 42 months), and the number of children in a family.
Quantitative variables are either discrete or continuous. Discrete variables can assume only certain values, and there are “gaps” between the values. Examples of discrete variables are the number of bedrooms in a house (1, 2, 3, 4, etc.), the number of cars arriving at Exit 25 on I-4 in Florida near Walt Disney World in an hour (326, 421, etc.), and the number of students in each section of a statistics course (25 in section A, 42 in section B, and 18 in section C). We count, for example, the number of cars arriving at Exit 25 on I-4, and we count the number of statistics students in each section. Notice that a home can have 3 or 4 bedrooms, but it cannot have 3.56 bedrooms. Thus, there is a “gap” between possible values. Typically, discrete variables result from counting.
Observations of a continuous variable can assume any value within a specific range. Examples of continuous variables are the air pressure in a tire and the weight of a shipment of tomatoes. Other examples are the amount of raisin bran in a box and the duration of flights from Orlando to San Diego. Grade point average (GPA) is a continuous variable. We could report the GPA of a particular student as 3.2576952. The usual practice is to round to 3 places—3.258. Typically, continuous variables result from measuring.

Level of Measurement

Data can be classified according to levels of measurement. The level of measurement of the data dictates the calculations that can be done to summarize and present the data. It will also determine the statistical tests that should be performed. For example, there are six colors of candies in a bag of M&M’s. Suppose we assign brown a value of 1, yellow 2, blue 3, orange 4, green 5, and red 6. From a bag of candies, we add the assigned color values and divide by the number of candies and report that the mean color is 3.56. Does this mean that the average color is blue or orange? Of course not! As a second example, in a high school track meet there are eight competitors in the 400 meter run. We report the order of finish and that the mean finish is 4.5. What does the mean finish tell us? Nothing! In both of these instances, we have not properly used the level of measurement.

There are actually four levels of measurement: nominal, ordinal, interval, and ratio. The lowest, or the most primitive, measurement is the nominal level. The highest, or the level that gives us the most information about the observation, is the ratio level of measurement.

Nominal Level

For the nominal level of measurement observations of a qualitative variable can only be classified and counted. There is no particular order to the labels. The classification of the six colors of M&M’s milk chocolate candies is an example of the nominal level of measurement. We simply classify the candies by color. There is no natural order. That is, we could report the brown candies first, the orange first, or any of the colors first. Gender is another example of the nominal level of measurement. Suppose we count the number of students entering a football game with a student ID and report how many are men and how many are women. We could report either the men or the women first. For the nominal level the only measurement involved consists of counts. Table 1–1 shows a breakdown of the sources of the world oil supply. The variable of interest is the country or region. This is a nominal-level variable because we record the information by source of the oil supply and there is no natural order. Do not be distracted by the fact that we summarize the variable by reporting the number of barrels produced per day.

Table 1-1 Source of World Oil Supply for 2004.

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Table 1–1 shows the essential feature of the nominal scale of measurement: There is no particular order to the categories.

In order to process data on oil production, gender, employment by industry, and so forth, the categories are often numerically coded 1, 2, 3, and so on, with 1 representing OPEC, 2 representing OECD, for example. This facilitates counting by the computer. However, because we have assigned numbers to the various categories, this does not give us license to manipulate the numbers. For example, 1 + 2 does not equal 3, that is, OPEC + OEDC does not equal former U.S.S.R. To summarize, the nominal-level data have the following properties: 1. Data categories are represented by labels or names. 2. Even when the labels are numerically coded, the data categories have no logical order.

Ordinal-Level Data

The next higher level of data is the ordinal level. Table 1–2 lists the student ratings of Professor James Brunner in an Introduction to Finance course. Each student in the class answered the question “Overall, how did you rate the instructor in this class?” The variable rating illustrates the use of the ordinal scale of measurement. One classification is “higher” or “better” than the next one. That is, “Superior” is better than “Good,” “Good” is better than “Average,” and so on. However, we are not able to distinguish the magnitude of the differences between groups. Is the difference between “Superior” and “Good” the same as the difference between “Poor” and “Inferior”? We cannot tell. If we substitute a 5 for “Superior” and a 4 for “Good,” we can conclude that the rating of “Superior” is better than the rating of “Good,” but we cannot add a ranking of “Superior” and a ranking of “Good,” with the result being meaningful. Further we cannot conclude that a rating of “Good” (rating is 4) is necessarily twice as high as a “Poor” (rating is 2). We can only conclude that a rating of “Good” is better than a rating of “Poor.” We cannot conclude how much better the rating is.
Table 1-2 Rating of a Finance Professor
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In summary the properties of the ordinal level of data are: 1. Data classifications are represented by sets of labels or names (high, medium, low) that have relative values. 2. Because of the relative values, the data classified can be ranked or ordered.
Interval-Level Data
The interval level of measurement is the next highest level. It includes all the characteristics of the ordinal level, but, in addition, the difference between values is a constant size. An example of the interval level of measurement is temperature. Suppose the high temperatures on three consecutive winter days in Boston are 28, 31, and 20 degrees Fahrenheit. These temperatures can be easily ranked, but we can also determine the difference between temperatures. This is possible because 1 degree Fahrenheit represents a constant unit of measurement. Equal differences between two temperatures are the same, regardless of their position on the scale. That is, the difference between 10 degrees Fahrenheit and 15 degrees is 5, the difference between 50 and 55 degrees is also 5 degrees. It is also important to note that 0 is just a point on the scale. It does not represent the absence of the condition. Zero degrees Fahrenheit does not represent the absence of heat, just that it is cold! In fact 0 degrees Fahrenheit is about–18 degrees on the Celsius scale.

Another example of the interval scale of measurement is women’s dress sizes. Listed below is information on several dimensions of a standard U.S. women’s dress.
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