The Difference between Math and Stats | Nathan Dalaklis

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6 May 201904:22
EducationalLearning
32 Likes 10 Comments

TLDRThe video script delves into the distinction between mathematics and statistics, two disciplines that, despite their shared reliance on numbers and logic, are fundamentally different in their approach and application. The speaker, Nathan, highlights that while mathematics is structured and axiomatic, leading to definitive solutions through proofs, statistics is more experimental and interpretative, aligning with scientific principles where hypotheses are tested and results lead to further inquiry. Statistics is used for making educated guesses and finding correlations, whereas mathematics provides a theoretical framework to understand the properties of statistical models. The script emphasizes the unique value of each field, their differences, and the complementary nature of their relationship, suggesting that while statisticians and mathematicians may not always appreciate each other's work, the tools from both fields are indispensable in certain scenarios. The video concludes by acknowledging the interconnectedness of mathematics and statistics, despite their distinct methodologies.

Takeaways
  • 📐 **Mathematics and Statistics are Different**: While both use numbers and logic, mathematics is more straightforward and structured, whereas statistics involves more interpretation and is more experimental in nature.
  • 🏫 **Separate Departments**: Many colleges and universities have separate departments for mathematics and statistics, indicating the distinction between the two fields.
  • 🔍 **Statistics and the Scientific Method**: Statistics aligns more closely with the scientific method, involving hypothesis testing and experimental results that lead to further questions and analysis.
  • 📉 **Mathematical Rigor**: Mathematics provides a structured path to a solution that is proven true, independent of data limits, whereas statistics often involves estimation and educated guesses.
  • ⚖️ **Bias in Statistics**: In statistics, an estimator might be biased, but this term has a fixed definition in mathematics, and the estimate is not a guaranteed outcome.
  • 🔗 **Correlations and Relationships**: Statistics are used to find correlations between different groups or events, providing insights into their relationships.
  • 📚 **Theoretical vs. Practical**: Mathematical statistics focuses on the properties of distributions and probability in a theoretical sense, while applied statistics can lead to practical algorithms.
  • 🤝 **Interdisciplinary Connections**: Mathematics and statistics are closely related, with each field benefiting from the tools and insights of the other.
  • 🧮 **Generalization and Properties**: Mathematics can be used to generalize statistical models and learn about their essential properties.
  • 🔬 **Feasibility in Mathematical Problems**: Some mathematical problems may not have feasible solutions, but statistical tools can help produce algorithms that approximate the solution.
  • 📈 **Statistical Estimators**: Statistics can provide a good estimate for certain outcomes, such as height or the arrival of a text message, which mathematics cannot predict.
  • 🤔 **Different Ways of Thinking**: The speaker finds mathematics more appealing but acknowledges that both fields require different ways of thinking and are both valuable in understanding the world.
Q & A

  • Why does the speaker feel that statistics is distinct from mathematics?

    -The speaker feels that statistics is distinct from mathematics because it allows for more interpretation and is more experimental in nature, aligning with the principles of the sciences. Unlike the structured path in mathematics, statistics involves making hypotheses, conducting experiments, and interpreting results that can lead to more questions and further statistical experiments.

  • What is the speaker's view on the separation of statistics and mathematics in academic departments?

    -The speaker believes that statistics is separate enough from mathematics to warrant its own place in department titles. They note that several colleges and universities have departments dedicated to each, intentionally separating statistics from other fields of mathematics like topology and combinatorics.

  • How does the speaker describe the approach of mathematics to problem-solving?

    -The speaker describes the approach of mathematics as very structured, with a path to a solution that is declared true through biological proof, independent of the limits of data.

  • According to the speaker, what are some limitations of mathematics?

    -The speaker mentions that mathematics cannot predict certain real-world outcomes, such as how tall someone will be or when their next text message will arrive. These are areas where statistics can provide an educated guess based on past data.

  • How does statistics differ from mathematics in terms of its application to real-world problems?

    -Statistics uses past data to make educated guesses and find correlations between different groups of things or events, providing insight into their relationships. It employs tools like statistical estimators to approximate solutions, which may not be exact but can be very close to the actual value.

  • What is the role of mathematics in understanding statistical models?

    -Mathematics plays a role in generalizing different statistical models and ideas, allowing for a deeper understanding of their essential properties. It provides a theoretical framework that can enhance the study of statistical methods.

  • How can mathematical tools aid in solving problems that may not have feasible solutions?

    -Mathematical tools can be used to create algorithms that approach very close to the solution of a given problem, even when a theoretical solution may not be feasible. This is particularly useful in statistics, where approximations are often acceptable.

  • What is the relationship between the deeper study of statistical methods and mathematics?

    -The deeper one goes into the study of statistical methods, the more mathematics comes into play in the design. This indicates a close relationship between the two fields, despite their differences.

  • Why did the speaker create this video?

    -The speaker created this video in response to a request from a friend who is a teacher. One of their students asked for a video about statistics, prompting the speaker to share their perspective on the differences between mathematics and statistics.

  • What is the speaker's personal preference between mathematics and statistics?

    -The speaker finds mathematics more appealing, although they acknowledge that both fields are distinct and closely related, each with its own set of useful tools.

  • How does the speaker view the relationship between statisticians and mathematicians?

    -The speaker suggests that statisticians and mathematicians may not always enjoy each other's fields, but they recognize that tools from both are useful and come in handy from time to time.

  • What does the speaker suggest about the value of approximations in statistics?

    -The speaker suggests that in many cases, approximations in statistics, which are derived from mathematical estimators and algorithms, can be good enough, even when they are not exact.

Outlines
00:00
📊 Understanding the Distinction Between Mathematics and Statistics

The speaker, Nathan, begins by sharing his personal perspective on the difference between mathematics and statistics. Despite his proficiency in math, he finds statistics more challenging due to its interpretative nature. He argues that while both fields use numbers and logic, mathematics is straightforward with a clear path to solutions, whereas statistics involves more interpretation and is experimental, aligning with scientific principles. Nathan emphasizes that statistics and mathematics are distinct, with separate departments in educational institutions, and each has its own unique approach to understanding the world. He also acknowledges the interrelation between the two, noting how mathematical tools can refine statistical models and how statistical methods can aid in solving mathematical problems.

Mindmap
Keywords
💡Mathematics
Mathematics is a field that involves the use of logical reasoning and quantitative calculation to solve problems and prove statements. It is characterized by its structured path to solutions and the use of axioms. In the video, it is contrasted with statistics, highlighting its concreteness and the definitive nature of its proofs. An example from the script is the mention of 'topology' and 'combinatoric algebra', which are specific branches of mathematics.
💡Statistics
Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. It is experimental and more open to interpretation compared to mathematics. The video discusses how statistics involves making hypotheses, conducting experiments, and drawing conclusions based on data. It is also used to make educated guesses and find correlations, as illustrated by the script's mention of 'statistical estimator'.
💡Axioms
Axioms are basic principles or statements that are accepted as true without proof, often serving as the foundation for deductive reasoning and the construction of a formal system of logic or mathematics. The script refers to working from a 'set of axioms' as a fundamental aspect of mathematical reasoning.
💡Interpretation
Interpretation in the context of the video refers to the act of assigning meaning to data or results, which is more prevalent in statistics than in mathematics. The speaker mentions that statistics 'felt like there was a lot more room left up to interpretation', emphasizing the subjective nature of statistical analysis.
💡Sciences
The sciences are a broad group of disciplines that study the natural and physical world through observation and experimentation. The video aligns statistics more closely with the sciences due to its experimental nature and the process of forming hypotheses and conducting experiments, which is a common practice in scientific research.
💡
💡Hypotheses
A hypothesis is a proposed explanation for a phenomenon, which can be tested through experimentation. The script discusses how in statistics, hypotheses are made and experiments conducted, which either confirm or refute these hypotheses, leading to further questions and statistical experiments.
💡Estimator
In statistics, an estimator is a rule or method used to estimate the value of a parameter based on observed data. The video uses the term 'statistical estimator' to illustrate how statistics can provide an approximation or estimate of a value, such as predicting height or the arrival of a text message, which is inherently uncertain.
💡Bias
Bias in statistics refers to a systematic error in the estimation of a parameter, leading to results that consistently deviate from the actual value. The script mentions the concept of 'bias' in the context of an estimator, highlighting the importance of understanding whether an estimator is biased or unbiased.
💡Correlations
Correlation in statistics is a measure of the degree to which two or more variables are linearly related. It helps to understand the strength and direction of the relationship between variables. The video discusses how statistics is used to find correlations between different groups of things or events.
💡Distributions
A distribution in statistics refers to the way data points are spread over a range of values. The video mentions 'distributions' in the context of mathematical statistics, where the properties of distributions and probability are studied in a theoretical manner.
💡Algorithms
An algorithm is a set of rules or a procedure for solving problems, especially in mathematics and computer science. The script refers to algorithms as tools that can be used to approximate solutions to mathematical problems, particularly when exact solutions are not feasible.
Highlights

The speaker excels in math and enjoys the concreteness of working from a set of axioms, while statistics feels more open to interpretation.

Statistics is often thought of as a math subject, but the speaker argues that it should be considered a separate field due to its different approach and applications.

Statistics is experimental at heart and aligns more with scientific principles, while math has a structured path to a proven solution.

Math provides a biological proof, while statistics relies on past data and educated guesses using statistical estimators.

Statistics can estimate things that math cannot, such as predicting height or when a text message will arrive.

Math can be used to analyze whether a statistical estimator is biased or not, but the estimate is still just an approximation.

Statistics is used to find correlations between different groups or events, while math studies the properties of distributions and probability in a theoretical sense.

Despite being different, math and statistics are closely related and can complement each other in certain situations.

Mathematicians can generalize statistical models and ideas to learn more about their essential properties.

Theoretically solved mathematical problems may not provide feasible solutions, but statistical tools can be used to create algorithms that approximate the solution.

As one delves deeper into statistical methods, the role of mathematics in their design becomes more apparent.

Statisticians and mathematicians may not always enjoy each other's fields, but they can find the tools from both disciplines useful at times.

The video was inspired by a question from a student, showing the speaker's commitment to education and addressing common misconceptions.

The speaker provides a nuanced perspective on the differences between math and stats, while acknowledging their interrelatedness.

The video aims to clarify why math and stats are considered distinct yet related fields, based on the speaker's personal experience and expertise.

The speaker encourages viewers to like and subscribe for more mathematics content, showing his passion for sharing knowledge.

The video concludes with a reminder that while math and stats have different methods and applications, they both contribute valuable insights into understanding the world.

Transcripts

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