Exploring the Distinction Between Mathematics and Statistics: Debunking Misconceptions

The relationship between mathematics and statistics has long been a subject of debate and misunderstanding. This article aims to clarify these misconceptions by examining a pertinent quote attributed to W. Edwards Deming and providing a comprehensive analysis of the differences and similarities between the two fields.

Understanding the Misleading Quote

The quote, “Statistics is not a subset of mathematics, mathematics is that subset of statistics in which bias and error are arbitrarily assumed to be zero,” was initially attributed to W. Edwards Deming, a prominent statistician, professor, author, and consultant. Deming is well-known for his contributions to quality management and statistics, particularly in industrial settings. However, this quote has been recast and reinterpreted, often in a manner that shifts its intended meaning.

A alternative version suggests that “Statistics is not a branch of applied mathematics; applied mathematics is the branch of statistics in which the variables have no random variation.” This remark emphasizes the difference between the two fields, highlighting the importance of recognizing that statistics involves random variables, a concept that is not inherent in pure mathematics.

The Role of Mathematics in Statistics

While mathematics provides the theoretical foundation for statistical methods, it is important to recognize that statistics is a distinct field with its own methodologies and applications. Mathematicians often focus on theoretical aspects, like theorems and proofs, whereas statisticians apply these theories to real-world data, accounting for variability and potential biases.

According to David Moore, in his article entitled “Should mathematicians teach statistics,” mathematics can serve as a framework for statistical analysis but cannot encapsulate the full scope of statistical inquiry. This perspective underscores the idea that statistics is not merely a subset of mathematics but a discipline in its own right.

Key Distinctions

W. Edwards Deming's quote, correctly understood, emphasizes the need to understand the assumptions and limitations inherent in statistical models. It highlights that statistics cannot simply be reduced to mathematical constructs without acknowledging the real-world complexities and uncertainties. This point is reinforced by Herman Rubin's five rules for applying statistics, which can be summarized into two key principles: the necessity of making assumptions and the importance of questioning those assumptions.

Mathematics as a Fundamental Tool

Despite the significance of statistical methods, it is critical to recognize that they are built on a solid mathematical foundation. Statistical tools and techniques rely on mathematical concepts to explain and model real-world phenomena. Even when dealing with bias and error, these are understood and analyzed through mathematical frameworks. For instance, the Central Limit Theorem, the Cramer-Rao Lower Bound, and the Gauss-Markov Theorem are all fundamental mathematical results that provide the basis for much of statistical analysis.

The Scope and Nature of Statistics

Statistics is not just about applying mathematical tools; it is a discipline that involves understanding variability, randomness, and the inherent uncertainties in data. Variables in statistics are often subject to random variation, which is a key distinction from the deterministic models used in pure mathematics. This random variation requires a more nuanced approach to analysis, emphasizing the importance of statistical inference and decision-making.

A quote by Herman Rubin, often paraphrased as "You must make assumptions and you must not believe your assumptions," encapsulates the essence of statistical practice. Statisticians must formulate assumptions that guide their analysis but must also critically evaluate these assumptions. Failure to do so can lead to flawed conclusions and misinterpretations of data.

A consultant must be cautious when making assumptions on behalf of a client. The client's context and specific needs must inform the assumptions made, ensuring that the statistical analysis remains relevant and accurate.

Conclusion

The discourse on the relationship between mathematics and statistics reflects the evolving nature of both fields. While statistics builds upon mathematical principles, it does so in a context that acknowledges and addresses the complexities of real-world data. The quote by Deming and the insights from Moore and Rubin underscore the importance of understanding the assumptions and limitations inherent in statistical analysis. By recognizing the unique nature of statistics as a discipline in its own right, we can better appreciate the value and challenges of statistical reasoning in various applications.