**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**(Columbia, 2021).

*consumers*of statistics and calculus-based statistics is more suited for people who want to*create*statistics## 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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