Managerial Statistics Module 1

Transcription

1 Title : REVIEW OF ELEMENTARY STATISTICAL TOPICS Period : 4 hours I. Objectives At the end of the lesson, students are expected to: 1. define statistics and describe its fundamental principles 2. identify the areas of statistical activities and study 3. illustrate how data are collected, organized, and interpreted to support decision making 4. illustrate how statistical graphs are selected and designed for presentation purposes 5. explain the importance of statistics in managerial decision-making II. Subject Matter 1. Topics a. Introduction b. Major Areas of Statistics c. Statistical Methods d. Statistical Data e. Variables f. Statistical Studies g. Statistical Graphics 2. Educational Resource(s) a. Black, K. (2008). Business Statistics for Contemporary Decision Making, 5 th Edition, John Wiley & Sons: USA b. Lind, D., Marchal, W, Wathen, S. (2008). Statistical Techniques in Business & Economics, 13 th Edition, NY MacGraw-Hill Irwin: New York c. Willemse, I. (2004). Statistical Methods for Business and Business Calculations, 2 nd Edition, Juta & Co.: South Africa d. e. f Materials a. Lecture notes b. PowerPoint presentation c. PCs/Calculators III. Learning Procedures and Strategies 1. Preparatory Activity a. Introduction: Information is an essential element in management and much of it is in statistical form. Managers need statistics to help set targets and objectives to Page 1 of 18

2 monitor performance against standards, to exercise control and to assist in making decisions. Everybody uses statistics to compare process in shops, to look at the performance of new cars, to buy a house and so on; managers use statistics in similar ways and for a range of activities, from buying policy to quality control, market research, investment decisions, forward planning and union bargaining. Statistics is a vital part of decision-making because it can narrow the area of disagreement and can help define a problem once it has been recognized to exist. Its systematic collection of data distinguishes statistics from other kinds of information and makes it of particular value to management. Managers need information about what has happened in the past, what is happening now, and what is expected to happen in the future. Statistics is an aid to the exercise of management control. Managers need statistical information to help them set targets and objectives, to monitor performance against set standards and in deciding what to do when the performance has been compared with these standards, and they need to know if objectives have been achieved and if the best results have been obtained from the available money, people and equipment. Managers need to understand statistical data, to be able to collect statistical information if it is not available and to apply statistical techniques to their work. They need to be able to quantify and summarize business-related data from a range of sources, to tackle business problems requiring statistical analysis and to present reports and recommendations arising from the analysis of statistical data. While statistical methods can be used in many fields of work and research, this module concentrates on the use of these methods in decision making. 2. Lesson Proper A. What is statistics? Statistics is concerned with the development and application of processes, methods, and techniques for collecting, organizing, presenting, analyzing, and interpreting quantitative data to aid decision making. 1. Collection of data. A structure of statistical investigation is based on a systematic collection of data. These data are classified as internal and external data. Internal data are obtained from internal records while external data are collected from external agencies which can either be primary or secondary data. (Note: pooling internal and external data requires an appropriate statistical methodology to ensure that merging leads to unbiased estimates) 2. Organization of data. The volume of collected data is a large quantity of figures which requires organization. Organizing data involves editing and checking for omissions, for irrelevant answers or for wrong computations before selecting and designing the appropriate graph for presentation. Page 2 of 18

3 3. Presentation of data. The collected data are presented in a tabular or graphic form. This systematic order and graphical presentation helps in the further analysis of the data. 4. Analysis of data. Analysis of data is the summarizing of trends and patterns observed in the data, the review of major differentials, and the computation of indexes which are used in determining relationships among variables used in a study. 5. Interpretation of data. Interpretation uses the results of the data analysis to make inferences relevant to the research questions. It seeks a broader meaning of the research relations obtained, as well as their implications, thus linking them to other relevant available knowledge. Major Areas Statistical activities are usually classified into the two major areas mentioned previously: theoretical or mathematical, statistics and applied statistics. 1. Theoretical statistics is concerned with the mathematical elements of the subject with lemmas, theorems, and proofs and in general with the mathematical foundations of statistical methodology. In this area, mathematician develops new theories that will provide new methods with which to attack practical problems. 2. In applied statistics, for a statistician and a decision maker, statistics is a means to an end. The decision maker faces problems and select from the available statistical methods those best fitted to the job at hand. The applied statistician may be asked by the decision maker to participate in designing a sample survey or experiment, or the statistician may be consulted about sampling inspection schemes for statistical quality control. Statistical Methods 1. Statistical methods can be divided into four areas, depending on the type of problems they are used to solve. The first area is descriptive statistics. Methods of organizing, summarizing, and presenting numerical data fall into this area of statistics. Descriptive statistics deals with pictorial and graphic summary of data and numerical summaries. 2. The second area of statistical study is probability. Probability problems arise when a statistician takes a sample from a population and wishes to make statements about the likelihood of the sample s having certain characteristics. A population is a set, or collecting, of items of interest in a statistical study. A sample is a subset of items that have been selected from the population. Example of a probability problem: Page 3 of 18

4 The manager of a retail store knows that 40% of the customers who enter the store will purchase one or more items and 60% will leave without making a purchase. What is the probability that in a sample of ten customers, five will purchase one or more items? Note that in this problem statement the population consists of all the company s customers. Also note that the purchaser/non-purchaser breakdown in the population is given. The sample is the ten customers selected at random, and the question asks what the sample will look like. 3. The third area of statistical study is called statistical inference. Problems involving statistical inference arise when a statistician takes a sample from a population and whishes to make statement about the population s characteristics from the information contained in the sample. A typical inference problem follows: The manager of employee benefits in the personnel department of a large corporation questioned 100 people selected from a work force of 7000 concerning their opinions of a proposed change in the company s medical insurance plan. The medical insurance salesperson claimed, At least 80% of your work force will favor this change. However, the manager s sample of 100 showed that only 70 people in the sample favored the change. Is the insurance salesperson s claim true? If not, what proportion of the total work force favors the change? In this problem the company s work force is the population, and the 100 people whose opinions were recorded from the sample. Here, however, we are given the sample results, and we ask question about what the population looks like. This situation is directly opposite to the situation presented in probability problems. In a probability problem the problem solver knows what the population looks like and raises questions about sample results that are likely to occur. In statistical inference situation the problem solver knows what the sample result looks like and raises question about the population from which the sample came. The fourth area of statistics covered in this module is called quantitative techniques. Methods covered under this area are used to solve a wide range of statistical problems from economic forecasting to deciding how large a production run a manager should order. The four areas of statistics described previously build on one another. For instance, probability methods would not be understood by someone who did not know descriptive statistics. Also, problems in statistical inference cannot be solved by someone who does not understand probability. And the techniques employed to solve special types of statistical problems often use the methods of inference. Types and Sources of Statistical Data Page 4 of 18

5 Statistical data often come from statistical studies, business transaction, trade associations, or government agencies. Statistical data can be classified into two broad categories known as quantitative or numerical data and qualitative or categorical data. Quantitative or numerical data are data that are measured on a scale. For example, the area in square feet or the selling price of a condominium is quantitative data. Qualitative or categorical data/variable are observation that can be classified into a single category or a set of categories. Example of quantitative data and some possible categories are sex male or female; education college degree or no college degree; and marital status married or not married. Variables A variable is a symbol (e.g., x, Y, or σ) that represents any of a specified set of values. For example, suppose we let the variable x represents the percentage of defective units in a shipment of widgets. Since x is a percentage, the variable x could take on any value between 0 and 100. Variables can be classified as categorical (aka, qualitative) or quantitative (aka, numerical). 1. Categorical variables take on values that are names or labels. The color of a ball (e.g., red, green, blue) or the breed of a dog (e.g., collie, shepherd, terrier) would be examples of categorical variables. 2. Quantitative variables are numerical. They represent a measurable quantity. For example, when we speak of the population of a city, we are talking about the number of people in the city - a measurable attribute of the city. Therefore, population would be a quantitative variable. Quantitative variables can be further classified as discrete or continuous. If a variable can take on any value between two specified values, it is called a continuous variable; otherwise, it is called a discrete variable. Some examples will clarify the difference between discrete and continuous variables. a. Suppose the fire department mandates that all fire fighters must weigh between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire fighter's weight could take on any value between 150 and 250 pounds. b. Suppose we flip a coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. Page 5 of 18

6 However, it could not be any number between 0 and plus infinity. We could not, for example, get 2.5 heads. Therefore, the number of heads must be a discrete variable. Univariate vs. Bivariate Data Statistical data is often classified according to the number of variables being studied. a. Univariate data. When one conducts a study that looks at only one variable, the study works with univariate data. Suppose, for example, that a survey is conducted to estimate the average weight of high school students. Since the study works with one variable (weight), the study is said to be working with univariate data. b. Bivariate data. When a study is conducted to examine the relationship between two variables, the study is working with bivariate data. To study the relationships between the height and weight of high school students is an example of bivariate data since the study works with two variables (height and weight). Two Broad Categories of Statistical Studies 1. Observational study. In an observational study the investigator examines variables of interest by using observed or historical data. The investigator does not directly control or determine which subjects or units receive treatments that are thought to have an effect on the variables of interest in the study. For example, an investor in real estate may examine the relationship between the selling price and the area in square feet of condominiums. The investigator does not directly control the area of condominiums, and the data are collected from historical sales records. Thus the study is an observational study. 2. Experimental study. In an experimental study the investigator directly controls or determines which subjects or experimental units or material receive treatments that are thought to have an effect on variables of interest. For example, in a water pollution study an analyst may impartially or randomly select several containers of water (experimental units or material) that have been directly assigned to be filtered by one of four types of filters (treatments). Then he may measure pollution counts (variable of interest) to determine which filter(s) result in the purest water. The analyst directly controls or determines the type of filter used to filter each container of water, so the study is an experimental study. Random selection or assignment of subjects or experimental material to treatments guards against bias, such as induced bias. Page 6 of 18

7 B. Introduction to Statistical Graphics Graphics can be used as an effective method of visual communication. Statistical graphics forms discuss in this module are line charts, bar or column charts, grouped bar charts, combination charts, and pie charts. 1. Line charts use lines between data points to depict the magnitude of data for two items or for one item over time. The heights of the line allow the user to compare magnitudes easily. For example, the amounts of retail sales at a new shopping mall for the years 1984, 1985, 1986, and 1987 were $5, $6, $8, and $9 million, respectively. These data are presented in the form of a line chart in figure 1.1 the magnitudes of the figures can be compared very easily in this chart. Figure 1.1. Line Chart Depicting Retail Sales at a Shopping Mall over Time 2. Bar charts are used to depict the magnitude of data for different qualitative categories or over time. The lengths or heights of the bars allow the user to compare the magnitude easily. For example, the expenditures proposed for a recent federal budget for benefit payments to individuals, defense, interest, grants to states and localities, and other operations as qualitative categories were $410, $280, $150, $100, and $60 billion, respectively. These data are presented in the form of a bar chart in figure 1.2. The magnitudes of the proposed expenditures can be compared very easily in this chart. Page 7 of 18

8 Figure 1.2. Bar Chart Depicting Proposed Federal Expenditures For a second example, the amounts of retail sales at a new shopping mall which were presented in Figure 1.1 are depicted in the form of a bar or column chart in Figure 1.3. The magnitudes of the sales figures can be compared very easily in this chart. Figure 1.3. Bar Chart for retail Sales over Time 3. Grouped bar charts can be used to depict the magnitudes of two or more grouped data items for different qualitative categories or over time. For example, the retail sales amounts at the new shopping mall were projected in 1983 to be $6, $6.5, $7.5, and $9.5 million for the years 1984, 1985, 1986, and 1987, respectively. The projected and actual sales amounts are depicted in a grouped bar chart in figure 1.4. Comparisons of magnitudes of the projected and actual sales amounts over time can be made easily with this chart. Page 8 of 18

9 Figure 1.4. Grouped Bar Chart for Projected and Actual Retail Sales over Time 4. Combination charts use lines and bars to depict the magnitudes of two or more data items for different categories or for different times. For example, from 1983 through 1987 a software company had sales of $20, $28, $36, $64 and $74 thousand, respectively. A statistical method known as regression analysis was used to estimate sales for 1983 through 1990 to be $18, $29, $42, $56, $74, $94, $117, and $141 thousand, respectively. The actual and estimated sales amounts are depicted in a combination chart in figure 1.5. The magnitudes of these values can be compared easily in this figure. Figure 1.5. Combination Chart Depicting Actual and Estimated Sales over Time Page 9 of 18

10 5. Pie chart can be used effectively to depict the proportions or percentages of a total quantity that correspond to several qualitative categories (usually five or fewer). Each category is depicted as a wedge of a circle, or a piece of a pie. The angle (in degrees) of each wedge is equal to the category s proportion multiplied by 360. For example, a study of the balance sheet of 510 firms found that the average proportions of total assets for the cash and securities, receivables, inventories, and long-term assets account were 8%, 19%, 24%, and 49%, respectively (Stowe, Watson and Robertson, 1980). Figure 1.6. Pie Charts for Percentages of Assets and Liabilities The average proportion of total liabilities designated as accounts payable, other current debt, long-term debt, and equity accounts were 11%, 15%, 27%, and 47%, respectively. The proportions of total assets and total liabilities for the account categories are depicted in the pie charts of figure 1.6. Only five forms of statistical graphics have been presented in this module because the purpose is simply to introduce statistical graphics. 1. Frequency Distribution Numerical information comes in the form of raw data such as a list of numbers. Data in this form is worthless. It provides little or no information about the situation investigated. Clearly, the data must be processed in some way in order to be useful in managerial terms. Tabulating data in the form of frequency distribution increases the ability to detect patterns and meaning. Page 10 of 18

11 A frequency distribution is a tabular summary of a set of data that shows the frequency or number of data items that fall in each of several distinct classes. A frequency distribution is also known as a frequency table. Summarizing and Presenting Data on Frequency Tables There are a wide variety of ways to summarize and present data. Most of the common methods will be summarized here, along with the usual conventions and terms for each. Frequency Tables A frequency table lists in one column the data categories or classes and in another column the corresponding frequencies. A common way to summarize or present data is with a standard frequency table. Frequency refers to the number of times each category occurs in the original data. Frequency tables for group and ungrouped data are as follows: Grade Frequency 9 (freshmen) (sophomores) (juniors) (seniors) 20 Test Score Frequency Often, the category column will have continuous data and hence be presented via a range of values. In such a case, terms used to identify the class limits, class boundaries, class widths, and class marks must be well understood. For the following examples, use the data above (20088 Algebra Diagnostic score distribution). Class limits are the largest or smallest numbers which can actually belong to each class. Page 11 of 18

12 For this example, the class limits are as displayed above in the left table column. For the largest class they are 120 and 139. Each class has a lower class limit and an upper class limit. Class boundaries are the numbers which separate classes. They are equally spaced halfway between neighboring class limits. For this example, the boundaries would be -0.5, 19.5, 39.5, 59.5, 79.5, 99.5, 119.5, and Note that is another name for and identical with Class marks are the midpoints of the classes. For this example, the class marks are 9.5, 29.5, 49.5,... It may be necessary to utilize class marks to find the mean and standard deviation, etc. of data summarized in a frequency table. Class width is the difference between two class boundaries (or corresponding class limits). For this example, the class width is Following are guidelines for constructing frequency tables. 1. The classes must be "mutually exclusive" no element can belong to more than one class. 2. Even if the frequency is zero, include each and every class. 3. Make all classes the same width. (However, open ended classes may be inevitable.) 4. Target between 5 and 20 classes, depending on the range and number of data points. 5. Keep the limits as simple and as convenient as possible (multiple of width?). If limits are not immediately obvious based on the data, try to find an appropriate width by rounding up the range divided by the number of classes. Your lower limit should be either the lowest score, or a convenient value slightly less. Avoid irrelevant decimal places. Large data sets justify having more classes. One published guide is: number of classes = 1 + log2n. This gives you 5 classes for small data sets of 12 to 22 elements and 10 classes for larger data sets of 362 to 724 elements. The seven classes used above for 50 elements is right on target. It is not uncommon to omit empty classes be alert for such guideline violations! Omitted classes do not change the class width, but can be a real source of confusion! Relative frequency tables contain the relative frequency instead of absolute frequency. Page 12 of 18

13 Relative frequencies can be expressed either as percentages or their decimal fraction equivalents. Cumulative frequency tables contain frequencies which are cumulative for subsequent classes. In a cumulative frequency table, the words less than usually also appear in the left column. Different Ways of Presenting a Frequency Table One-Way Tables When a table presents data for one, and only one, categorical variable, it is called a one-way table. A one-way table is the tabular equivalent of a bar chart. Like a bar chart, a one-way table displays categorical data in the form of frequency counts and/or relative frequencies. One-Way Frequency Tables When a one-way table shows frequency counts for a particular category of a categorical variable, it is called a one-way frequency table. Below, the bar chart and the frequency table display the same data. Both show frequency counts, representing travel choices of 10 travel agency clients. Choice USA Europe Asia Frequency Relative Frequency Tables When a one-way table shows relative frequencies for particular categories of a categorical variable, it is called a one-way relative frequency table. Each of the tables below summarizes data from the bar chart above. Both tables are relative frequency tables. The table on the left shows relative frequencies as a proportion, and the table on the right shows relative frequencies as a percentage. Choice USA Europe Asia Proportion Choice USA Europe Asia Percentage Two-Way Tables A two-way table (also called a contingency table) is a useful tool for examining relationships between categorical variables. The entries in the cells Page 13 of 18

14 of a two-way table can be frequency counts or relative frequencies (just like a one-way table). Two-Way Frequency Tables Below is a two-way table which shows the favorite leisure activities for 50 adults - 20 men and 30 women. Because entries in the table are frequency counts, the table is a two-way frequency table. Dance Sports TV Total Men Women Total Entries in the "Total" row and "Total" column are called marginal frequencies or the marginal distribution. Entries in the body of the table are called joint frequencies. Looking only at the marginal frequencies in the Total row, one might conclude that the three activities had roughly equal appeal. Yet, the joint frequencies show a strong preference for dance among women; and little interest in dance among men. Two-Way Relative Frequency Tables Dance Sports TV Total Men Women Total Relative frequencies can also be displayed in two-way tables. The above table shows preferences for leisure activities in the form of relative frequencies. The relative frequencies in the body of the table are called conditional frequencies or the conditional distribution. Two-way tables can show relative frequencies for the whole table, for rows, or for columns. The above shows relative frequencies for the whole table. Below, the table on the left shows relative frequencies for rows; and the table on the right shows relative frequencies for columns. Dance Sports TV Total Men Women Total Relative Frequency of Row Dance Sports TV Total Men Women Total Relative Frequency of Column Each type of relative frequency table makes a different contribution to understanding the relationship between gender and preferences for leisure Page 14 of 18

15 activities. For example, "Relative Frequency for Rows" table most clearly shows the probability that each gender will prefer a particular leisure activity. For instance, it is easy to see that the probability that a man will prefer dance is 10%; the probability that a woman will prefer dance is 53%; the probability that a man will prefer sports is 50%; and so on. Such relationships are often easier to detect when they are displayed graphically in a segmented bar chart. A segmented bar chart has one bar for each level of a categorical variable. Each bar is divided into "segments", such that the length of each segment indicates proportion or percentage of observations in a second variable. The above segmented bar chart uses data from the "Relative Frequency for Rows" table above. It shows that women have a strong preference for dance; while men seldom make dance their first choice. Men are most likely to prefer sports, but the degree of preference for sports over TV is not great. Histograms The term histogram comes from the Greek words meaning web and write. As such it is a way to untangle data. Another name for a histogram is a bar graph or bar chart, although some texts differentiate between the two. In a histogram the vertical axis has the frequency, while the horizontal axis has the intervals. No gaps are allowed between the bars. The distribution of the data: normal, skewed left, skewed right, should be fairly obvious from a bar graph. Histograms are quite commonly used to visually display frequency and relative frequency charts. Again, some texts indicate that a bar graph is used for categorical data and allows gaps between the bars. Illustrated below are a bar graph and the accompanying TI-83+ settings for the US presidential inauguration data. A histogram is a graphic presentation of a frequency distribution and is constructed by erecting bars or rectangles on the class intervals. Page 15 of 18

16 Illustration: F r e q u e n c y Number of absences Figure 1.7. Histogram of Employee Absences at Western Electronics A relative frequency histogram has the same shape and horizontal scale as a histogram, but the vertical scale is now the relative frequency. A Pareto chart is a bar graph for qualitative data. The bars in a pareto chart should be arranged in descending order of frequency, from left to right. Frequency polygons are similar to histograms, but use line segments to connect the points. When constructing a frequency polygon, the class marks should be used on the horizontal scale. The graph should also be extended to the left and right so that it begins and ends with a frequency of 0. Cumulative frequency polygons, also known as ogives, are also commonly encountered. The line in an ogive (pronouced "oh-jive") will always have nonnegative slope. Note: All the figures presented in this module were originally plotted with the aid of Excel Spreadsheet and SPSS softwares. Page 16 of 18

17 c. Ending Activity a. Summary The term statistics is often used as a synonym for data. However in the modern sense of the word, statistics involves the development and application of methods and techniques for collecting, analyzing, and interpreting data to aid decision making. Four areas of statistics are important in the application of statistics to aid business and economic decision making. These areas are descriptive statistics, probability, statistical inference, and statistical techniques. Decision makers are interested in statistical method because these methods can help them avoid making those inept decisions and can help them make intelligent, reliable decision. Thus, they are most interested in the application of statistics to their particular problems, but they must also have a good foundation in the theory of probability and statistics in order to use these applications wisely. IV. Evaluation/Assessment a. Self-test 1: Test your understanding of this lesson. 1. Tabular and graphical summaries of data sets are used to organize, summarize, and present data in a form that provides information that is useful for making decisions or inferences. Statistical methods that accomplish these tasks fit into the category of: a. inferential statistics b. probability c. descriptive statistics d. quantitative approach 2. Which of the following statements are true? (1) All variables can be classified as quantitative or categorical variables. (2) Categorical variables can be continuous variables. (3) Quantitative variables can be discrete variables. a. 1 only b. 2 only c. 3 only d. 1 and 2 e. 1 and 3 3. Fill in the missing words to the quote: Statistical methods may be described as methods for drawing conclusions about based on computed from a. statistics, samples, populations b. populations, parameters, samples c. statistics, parameters, samples Page 17 of 18

18 d. parameters, statistics, populations e. populations, statistics, samples 4. Statistical data are often in the form of raw, unorganized numerical values. a. True b. False 5. Which of the following statements are true? (1) area in square feet is a quantitative datum (2) selling price of a condominium is a qualitative datum (3) sex (male or female) is a qualitative datum (4) education (college degree or no college degree) is a categorical datum a. 1 only b. 2 only c. 4 only d. 2, 3, and 4 e. 1, 3, and 4 b. Problem Set 1 1. An automobile dealership had sales of $5, $13, $21, $49, and $68 million in 2004 to Present the data in a line chart. 2. A company has experienced the following financial results for earnings before interest and taxes (EBIT) and profits (EBIT and profits are in millions of dollars): Year EBIT Profits Present the data in a grouped bar chart. 3. A franchise chain of restaurants for 1983 through 1987 had 4, 8, 16, 26, and 82 restaurants, respectively. A statistical model estimated the number of restaurants for 1983 through 1990 to be 4, 8, 16, 33, 140, 281, and 586. Present the data in a combination chart. 4. A study of the asset/liability structures of expenditure and revenue data as pie charts. Assets for cash, liquid securities, investment securities, loans, and other assets to be 0.18, 0.03, 0.14, 0.59, and 0.06, respectively (Simonson, Stowe, and Watson, 198.). The proportions of liabilities and capital for demand deposits, purchased funds, core deposits, other liabilities, and equity were 0.29, 0.40, 0.21, 0.05, and 0.06, respectively. Present these data in pie charts. Page 18 of 18