Statistical Principles for Problem Solving

Problem solving and decision making are important aspects of running a business. One of the tools that can help in solving real world problems is statistics. Descriptive statistics include graphical techniques to organize and summarize data so that they are more easily comprehensible and simple processes to summarize basic parameters of distributions. Inferential statistics helps decision makers solve problems in more complex situations and to draw conclusions about what the data signify rather than merely describing what they look like. Statistics can also be of help in business problem solving through the development of mathematical models. To meaningfully apply statistics to real world data, the researcher needs to do two things: Control the situation so that the research is only measuring what it is supposed to measure and include as many of the relevant factors as possible so that the research fairly emulates the real world experience.

Just as in the rest of life, problem solving and decision making are important aspects of running a business. Changes in the economy, innovations by one's competitors, and new demands and expectations of the marketplace all mean that the organization needs to constantly adjust how it does business in order to stay competitive and gain or maintain its market share. For example, implementation of new technology to become more competitive may require an investment, and the savvy organization needs to determine whether or not the benefits of the investment will exceed its costs. Similarly, if a long-standing business process cannot keep up with expanding customer demands, management must determine whether to try to repair the existing process or develop and implement an entirely new one. If the engineering department proposes a new widget to be added to the product line, it must be determined whether or not potential customers are likely to buy the product as well as whether or not the addition will compete with the existing product line or enhance it.

Many times, decisions such as these are made subjectively based on the insights of experienced managers and other decision makers. This is not necessarily a bad idea: veteran managers can take advantage of years of experience to extrapolate trends in ways that are still not possible through the use of quantitative techniques alone. In addition, in some situations there may be insufficient data to use quantitative techniques, necessitating the use of qualitative forecasting methods. However, not every manager or decision maker has the skills or experience to make such decisions unaided. In addition, real world problems are complex, and insightful managers use every tool at their command to make the best decisions possible. One of these tools is statistics.

Statistics is a branch of mathematics that deals with the analysis and interpretation of data. Mathematical statistics provides the theoretical underpinnings for various applied statistical disciplines, including business statistics, in which data are analyzed to find answers to quantifiable questions. Applied statistics uses these techniques to solve real world problems. In general, there are two types of statistical tools:

• Descriptive statistics helps one describe and summarize data so that they can be more easily understood.
• Inferential statistics is used in the analysis and interpretation of data to make inferences from the data and help in the processes of problem solving and decision making.

Descriptive Techniques: Graphing Techniques

Descriptive statistics helps describe data through graphical techniques that organize and summarize them so that they are more easily comprehensible. Descriptive statistics also include various processes to simply summarize basic parameters of distributions including various techniques to find the "average" or mathematically describe the shape of the distribution. There are many graphing techniques available.

Frequency Distribution

One of the most frequently used methods is the frequency distribution. In this type of graph, data are divided into intervals of typically equal length, thereby decreasing the number of data points on the graph and organizing the data to make them easier to comprehend. Other types of graphing techniques used in descriptive statistics include histograms, Pareto diagrams, scatter plots, and graphs.

Histograms

Histograms are vertical bar charts that graph frequencies of objects within various classes on the y axis against the classes on the x axis.

Pareto Diagrams

Pareto diagrams are vertical bar charts that graph the number and types of defects for a product or service against the order of magnitude (from greatest to least). Pareto charts are often shown with cumulative percentage line graphs to more easily show the total percentage of errors accounted for by various defects.

Scatter Plots

Scatter plots graphically depict two-variable numerical data so that the relationship between the variables can be examined. For example, of one wanted to know the relationship between number of defects observed in a given month and the cost of the loss of quality to the company, these two values (number of defects and concomitant cost) could be graphed on a two-dimensional graph so that one could better understand the relationship.

Examples of a histogram (with frequency distribution), Pareto chart (with cumulative percentage line graph) and scatter plot are shown in Figure 1.

Descriptive Techniques: Central Tendency

In addition to graphing techniques, descriptive statistics can be used to describe the central tendency and the variability of a sample. Measures of central tendency estimate the midpoint of a distribution. These measures include the median (the number in the middle of the distribution when the data points are arranged in order), the mode (the number occurring most often in the distribution), and the mean (a mathematically derived measure in which the sum of all data in the distribution is divided by the number of data points in the distribution). For example, as shown in Figure 2, for the distribution 2, 3, 3, 7, 9, 14, 17, the mode is 3 (there are two 3s in the distribution, but only one of each of the other numbers), the median is 7 (when the seven numbers in the distribution are arranged numerically, 7 is the number that occurs in the middle), and the mean (or arithmetic mean) is 7.857 (the sum of the seven numbers is 55; 55 / 7 = 7.857).

Descriptive Techniques: Variability

In addition to measures of central tendency, descriptive statistics include measures of variability that summarize how widely dispersed the data are over the distribution. The first of these statistics is the range, which is the difference between the highest and lowest scores. By knowing the range in addition to one of the measures of central tendency, one can better understand the data. For example, two distributions with a mean of 10 would be quite different if the range of one was between 1 and 100 and the range of the other was between 9 and 11. However, although the range does help one better understand the data, it is not a very refined statistic. To better understand the shape of the distribution, one can calculate the standard deviation. This is a mathematically derived index of the degree to which scores differ from the mean of the distribution and describes how far the typical score in a distribution is from the mean of the distribution. The standard deviation is obtained by determining the deviation of each score from the mean (i.e., subtracting the mean from the score), squaring the deviations (i.e., multiplying them by themselves), adding the squared deviations, and dividing this number by the total number of scores. The larger the standard deviation, the farther away it is from the midpoint of the distribution.

Shewhart Control Charts

These two types of descriptive statistics can be combined into a technique called Shewhart control charts that help quality control engineers and managers solve problems regarding quality of a product and whether or not a process is under control. An example of this kind of chart is the X-bar chart (so named for X¾, the mathematical symbol for the arithmetic mean). This is a chart of the means of some characteristic of the product (e.g., acceptability of solder joints) of small random samples taken from the production line over time. As shown in Figure 3, the means are plotted over time on a chart that contains a center line (i.e., the mean for the process) and upper and lower control limits that are three standard deviations on either side of this line. X-bar charts help quality control engineers and managers determine where problem areas lie in a production process. If all the points plotted on the chart fall between the upper and lower control limits, the process is considered to be in control. However, if sample means fall outside the control limits, the process is considered to be out of control and the process is stopped so that an assignable cause can be determined. In addition to X-bar charts that keep track of processes by examining the means of samples, quality control charts include R charts that keep track of the range, p charts track the proportion of defective products, c charts track the number of defects, and s charts that examine sample variance.

Inferential Statistics

Although descriptive statistics are useful in helping one better understand data, inferential statistics help decision makers to problem solve based on data for more complex situations.

Inferential statistics helps decision makers to draw conclusions about what the data signify rather than merely describing what they look like. Through the use of inferential statistics, one can analyze various characteristics of a random sample to determine whether or not any observed differences are due to chance or because of a real difference in the sample from the population. Inferential statistics includes a wide range of tests of statistical significance to help solve problems and make decisions by estimating the degree to which it is probable that the observed results occurred by chance alone or are due to an experimental manipulation or other predetermined factor.

Correlation

Another frequently used statistical tool with real world applications is correlation. This tool allows one to determine the degree to which two events or variables are related. For example, a market researcher may want to know if people with greater disposable income are more likely to buy luxury items. The correlation coefficient is the statistical method for determining this relationship. Correlation may be positive (i.e., as the value of one variable increases the value of the other variable increases), negative (i.e., as the value of one variable increases the value of the other variable decreases), or zero (i.e., the values of the two variables are unrelated). Correlation does not imply causation, however: Just because two variables are related does not mean that one caused the other. Both may be caused by a third, unknown factor.

Mathematical Models

Another way that statistics can be of help in business problem solving is through the development of mathematical models. Regression is a statistical technique used to develop a mathematical model for use in predicting one variable from the knowledge of another variable. There are a number of regression techniques available for models to be used in problem solving ranging from simple linear regression to more advanced techniques that allow the analyst to use both multiple independent and dependent variables. No matter the technique, the resulting regression equation is a mathematical model of a real world situation. This can be invaluable for forecasting and learning more about the interaction of variables in the real world. The use of models is particularly helpful for problem solving in complex situations that cannot be solved intuitively. Models are very useful tools that create a representation that can be examined and manipulated to include relevant variables and relationships in the decision making process to reflect various alternate solutions to a problem. However, models need to be validated at each step and consequently adjusted as necessary and revalidated in order to optimize their use in decision making. This process is illustrated in Figure 4.

Applications

Inferential statistics allow one to problem solve by determining the answers to real world questions. The goal of research is to describe, explain, and predict behavior. For example, the marketing department may need solve the problem of which of two proposed new company logos will be most memorable and will have the most positive image in the minds of prospective customers, or the engineering department may need to determine which of two graphical user interfaces is more user friendly. Inferential statistics can help answer such questions.

Research Requirements

To solve these or other real world problems, the researcher needs to do two things: Control the situation so that the research is only measuring what it is supposed to measure and include as many of the relevant factors as possible so that the research fairly emulates the real world experience. As shown in Figure 5, in experimental research, a stimulus (e.g., a new graphical user interface) is presented to the research subjects (e.g., potential customers) and a response is observed and recorded (e.g., which interface they liked better). Researchers look at the independent variable (i.e., the stimulus or experimental condition that is hypothesized to affect behavior -- such as the type of graphical user interface) and the dependent variable (i.e., the observed effect on behavior caused by the independent variable -- such as the user's response to the interfaces). However, real world problems are rarely this clean, and one must also consider extraneous variables -- variables that affect the outcome of the experiment (i.e., whether or not the people questioned like the new interface) that have nothing to do with the independent variable itself. For example, if a person is in a hurry to get somewhere else or has something else on his or her mind, he or she may not give the alternative interfaces all the consideration that is needed to tell which is better. Although it is impossible to control for every possible extraneous variable, a well-designed experiment controls for as many of the extraneous variables as possible.

Laboratory Research

Laboratory research allows one the most control over variables. However, it often is far-removed from real life and the actual problems that need to be solved. Although controlling variables is important, it is just as important to conduct research in a setting as close to the real world as possible. For example, users will typically use a user interface in less than ideal conditions, such as when they are tired or distracted, not just when they are at their best. In order to be able to extrapolate research results to the real world and have them be meaningful, it is important to design an experiment that not only controls extraneous variables, but is as realistic as possible.

Field Simulation & Experimentation

In addition to laboratory research, there are several research techniques that can be used to investigate real world business problems. A more realistic approach to real world research is offered by simulation. This technique allows the researcher to bring in more real world variables while still controlling many of the extraneous variables. For example, people could be asked to do a set number of tasks with the new graphical user interface in an environment that simulates they way will actually would use it in the real world (e.g., while doing other tasks or being exposed to various distractions). Another approach to real world research is the field experiment in which people are given the product to try at work under the actual conditions in which they would use it. However, although this approach has the advantage of being more realistic, it has the concomitant disadvantage of giving the researcher less control over extraneous variables.

Field Study & Survey Research

The laboratory experiment, field experiment, and simulation give the experimenter some control over the variables. However, it is not always possible to do research in such settings. There are other approaches to studying business problems that trade more realism for less control. One of these approaches is the field study. This approach examines how people behave in the real world. For example, if both interface designs are already on the market, the researcher could observe what type of people bought each design and use this information to determine how to target the marketing of the product. The field study can also be combined with another research technique called survey research. Subjects could be interviewed by a member of the research team or asked to fill out a questionnaire regarding their preferences, reactions, habits, or other questions of interest to the researcher. Although it is possible in theory to develop a very detailed research instrument that could be used to collect all the data that the researcher needed for analysis, in practice, such detailed instruments are often more lengthy than the potential research subject's attention span. Survey research and interview techniques have the additional drawback of not being based on observation. Because of this fact, there is no way to know whether or not the subject is telling the truth for any number of reasons (e.g., s/he did not have much time to answer the questionnaire, was not really interested in helping in the research, did not like the company, wanted to please the researcher).

Terms & Concepts

Data: (sing. datum) In statistics, data are quantifiable observations or measurements that are used as the basis of scientific research.

Descriptive Statistics: A subset of mathematical statistics that describes and summarizes data.

Inferential Statistics: A subset of mathematical statistics used in the analysis and interpretation of data. Inferential statistics are used to make inferences such as drawing conclusions about a population from a sample and in decision making.

Mathematical Statistics: A branch of mathematics that deals with the analysis and interpretation of data. Mathematical statistics provide the theoretical underpinnings for various applied statistical disciplines, including business statistics, in which data are analyzed to find answers to quantifiable questions.

Model: A representation of a situation, system, or subsystem. Conceptual models are mental images that describe the situation or system. Mathematical or computer models are mathematical representations of the system or situation being studied.

Regression: A statistical technique used to develop a mathematical model for use in predicting one variable from the knowledge of another variable.

Sample: A subset of a population. A random sample is a sample that is chosen at random from the larger population with the assumption that such samples tend to reflect the characteristics of the larger population.

Statistical Significance: The degree to which an observed outcome is unlikely to have occurred due to chance.

Statistics: A branch of mathematics that deals with the analysis and interpretation of data. Mathematical statistics provide the theoretical underpinnings for various applied statistical disciplines, including business statistics, in which data are analyzed to find answers to quantifiable questions. Applied statistics uses these techniques to solve real world problems.

Variable: An object in a research study that can have more than one value. Independent variables are stimuli that are manipulated in order to determine their effect on the dependent variables (response). Extraneous variables are variables that affect the response but that are not related to the question under investigation in the study.

Bibliography

Aliev, R. A., PedrycZ, W. W., & Huseynov, O. H. (2012). Decision theory with imprecise probabilities. International Journal of Information Technology & Decision Making, 11(2), 271-306. Retrieved November 27, 2013 from EBSCO Online Database Business Source Premier. http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=75255353

Black, K. (2006). Business statistics for contemporary decision making (4th ed.). New York: John Wiley & Sons.

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Suggested Reading

Aparisi, F., Avendaño, G., & Sanz, J. (2006). Techniques to interpret T2 control chart signals. IIE Transactions, 38(8), 647-657. Retrieved August 28, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=20917671&site=ehost-live

Bardia, S. C., Shweta, K., & Garima, B. (2011). Inferential statistics as a measure of judging the short-term solvency: An empirical study of five pharmaceutical companies in India. IUP Journal Of Accounting Research & Audit Practices, 10(1), 69-80. Retrieved November 27, 2013 from EBSCO Online Database Business Source Premier. http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=57489825

Bowerman, B. L. & O'Connel, R. T. (2005). Business statistics in practice (4th ed.). Columbus, OH: Irwin/McGraw-Hill.

Groebner, D. F., Shannon, P. W., Fry, P. C., & and Smith, K. D. (2003). Business statistics: A decision-making approach (6th ed.). Upper Saddle River, NJ: Prentice Hall.

Levine, D. M., Krehbiel, T. C., & Berenson, M. L. (2005). Business statistics: First course (4th ed.). Upper Saddle River, NJ: Prentice Hall.

Ormerod, T., MacGregor, J., Chronicle, E., Dewald, A., & Chu, Y. (2013). Act first, think later: The presence and absence of inferential planning in problem solving. Memory & Cognition, 41(7), 1096-1108. Retrieved November 27, 2013 from EBSCO Online Database Business Source Premier. http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=90290452

Taplin, R. (2007). Enhancing statistical education by using role-plays of consultations. Journal of the Royal Statistical Society, 170(2), 267-300. Retrieved August 28, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=24651036&site=ehost-live