Calculus-Based Statistics

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Calculus-Based statistics takes the four core concepts of calculus (Continuity, Limits, Definite integral, Derivative) and applies them to statistical theory. Essentially, non-calculus based statistics is for consumers of statistics and calculus-based statistics is more suited for people who want to create statistics (Columbia, 2021).

Elementary Statistics vs. Calculus-Based Statistics

There are many similarities and some important differences between Elementary Statistics and Calculus-Based Statistics. Whichever class you take, you’re going to cover the same core concepts including ANOVA, Confidence intervals, Correlation, Regression and Statistical Inference.

The main difference is in how these topics are approached. For example, in a basic stats class, confidence intervals are introduced as a way to describe the distribution of parameters (i.e. how spread out your estimated results are). The focus is on how to make and interpret these intervals. With a calculus based approach, functions are derived for these intervals. In a non-calc class, you might study the survival function for a certain species. In a calculus-based course, you might take that same survival function and integrate it to show that the average lifetime for that species is the area under the curve.

In non-calculus statistics classes the focus is on using statistics. Calculus-based statistics is more about creating the statistics (for others to consume). It is generally a more rigorous class that will help you to:

• Create statistics from scratch for any data type,
• Understand where many statistical rules and assumptions come from,
• Extend basic tests and procedures to non-standard situations.

Typical Contents for Calculus-Based Statistics

Many of the following concepts are found in a calculus-based statistics class. They are either not covered at all in an elementary statistics class or are skimmed over:

• Finite and Infinite Sets,
• Increasing and decreasing functions,
• Moment-generating functions,
• Probability axioms,
• Continuity,
• Limit of functions,
• Derivatives (e.g. the chain rule and implicit differentiation),
• Summation notation,
• Integrals (e.g. Riemann integrals, improper integrals),
• Extrema of functions,
• Sequence and series (including power series, Taylor series and monotonic series),
• Tests for series convergence.

On the other hand, you probably won’t see these advanced calculus concepts in a calculus-based statistics class or an elementary statistics class. In fact, you probably won’t come across them at all unless you dive into the realm of mathematical statistics:

• Differential forms: integrands for complicated domains.
• Essential discontinuities: discontinuities that jump wildly as they get closer to the limit.
• Exterior calculus: a high dimensional extension of calculus.
• Holomorphic Functions: functions that are infinitely differentiable.
• Non-Newtonian Calculus: a family of non-linear calculi.
• Ornstein-Uhlenbeck Process:a differential equation that models the motion of a particle under friction.
• Punctured Disks: A flat disk with a pinprick. One example of a myriad of shapes that can be integrated in calculus, but that you won’t see in statistics.
• Tangent Spaces: a generalization of a two-dimensional curve tangent line to manifolds.
• Tetration Functions: iterated exponentiation.

Should I Take Calculus or Non-Calculus-Based Statistics?

Which course you take largely depends on what your future goals are.

Few disciplines really need calculus-based statistics. Those that do include economics, mathematical statistics and many research-heavy fields. In addition, some academic disciplines encourage (or require) a calculus background in addition to statistics. For example, statistics and calculus are highly sought after skills by research-active life scientists at the University of Arizona (Watkins, 2010).

As elementary statistics focuses more on data analysis, it’s well-suited to pre-med students, social science majors, and business majors.

For other disciplines, it’s a grey area. For example, data scientists can get away with not knowing calculus for many positions but many career opportunities will require you to use calculus to explore data.

References

Columbia University. (2021). Statistics. Retrieved January 4, 2021 from: http://bulletin.columbia.edu/columbia-college/departments-instruction/statistics/
DeGroot, M. & Schervish, J. (2019). Probability and Statistics (Classic Version), 4th edition.
Pearson.
Gemignani, M. (2004). Calculus and Statistics. Dover.
Watkins, J. (2010). On a Calculus-based Statistics Course for Life Science Students. CBE Life Sci Educ. 2010 Fall; 9(3): 298–310

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