## 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...

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