Introduction to Statistical Thinking

The book included a description of some elements of statistics, element that we thought are simple enough to be explained as part of an introductory course to statistics and are the minimum that is required for any person that is involved in academic activities of any field in which the analysis of data is required. Now, as you finish the book, it is as good time as any to say some words regarding the elements of statistics that are missing from this book.

One element is more of the same. The statistical models that were presented are as simple as a model can get. A typical application will require more complex models. Each of these models may require specific methods for estimation and testing. The characteristics of inference, e.g. significance or confidence levels, rely on assumptions that the models are assumed to possess. The user should be familiar with computational tools that can be used for the analysis of these more complex models. Familiarity with the probabilistic assumptions is required in order to be able to interpret the computer output, to diagnose possible divergence from the assumptions and to assess the severity of the possible effect of such divergence on the validity of the findings.

Statistical tools can be used for tasks other than estimation and hypothesis testing. For example, one may use statistics for prediction. In many applications it is important to assess what the values of future observations may be and in what range of values are they likely to occur. Statistical tools such as regression are natural in this context. However, the required task is not testing or estimating the values of parameters, but the prediction of future values of the response.

A different role of statistics is in the design stage. We hinted in that direction when we talked about in an earlier chapter about the selection of a sample size in order to assure a confidence interval with a given accuracy. In most applications, the selection of the sample size emerges in the context of hypothesis testing and the criteria for selection is the minimal power of the test, a minimal probability to detect a true finding. Yet, statistical design is much more than the determination of the sample size. Statistics may have a crucial input in the decision of how to collect the data. With an eye on the requirements for the final analysis, an experienced statistician can make sure that data that is collected is indeed appropriate for that final analysis. Too often is the case where researcher steps into the statistician’s office with data that he or she collected and asks, when it is already too late, for help in the analysis of data that cannot provide a satisfactory answer to the research question the researcher tried to address. It may be said, with some exaggeration, that good statisticians are required for the final analysis only in the case where the initial planning was poor.

Last, but not least, is the theoretical mathematical theory of statistics. We tried to introduce as little as possible of the relevant mathematics in this course. However, if one seriously intends to learn and understand statistics then one must become familiar with the relevant mathematical theory. Clearly, deep knowledge in the mathematical theory of probability is required. But apart from that, there is a rich and rapidly growing body of research that deals with the mathematical aspects of data analysis. One cannot be a good statistician unless one becomes familiar with the important aspects of this theory.

I should have started the book with the famous quotation: “Lies, damned lies, and statistics”. Instead, I am using it to end the book. Statistics can be used and can be misused. Learning statistics can give you the tools to tell the difference between the two. My goal in writing the book is achieved if reading it will mark for you the beginning of the process of learning statistics and not the end of the process.

In the second part of the book we have learned many subjects. Most of these subjects, especially for those that had no previous exposure to statistics, were unfamiliar. In this forum we would like to ask you to share with us the difficulties that you encountered.

What was the topic that was most difficult for you to grasp? In your opinion, what was the source of the difficulty?

When forming your answer to this question we will appreciate if you could elaborate and give details of what the problem was. Pointing to deficiencies in the learning material and confusing explanations will help us improve the presentation for the future editions of this book.