**Contents**:

- What is Calculus Based Statistics?
- Elementary Statistics vs. Calculus Based Statistics
- Calculus Based Statistics: Examples
- Typical Contents for Calculus Based Statistics
- Should I Take Calculus or Non-Calculus Based Statistics?
- Calculus Definitions and How To Articles

## What is Calculus Based Statistics?

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 statistics 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.

## Calculus Based Statistics: Examples

**The following table lists some examples of where you could (or must) apply calculus:**

Some areas of statistics are difficult or impossible to evaluate without calculus.

Some areas of statistics are difficult or impossible to evaluate without calculus.

## 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.

## Definitions and How To Articles

The following how to articles cover the basics of calculus: the tools you will likely need to get you through an introductory calculus based statistics class.

- Area of a Bounded Region
- Area Under a Curve (Excel)
- Average Value of a Function
- Basic Operations on Functions
- Check the Continuity of a Function
- Decompose a Composite Function
- Eliminate exponents
- How to Find Intercepts
- Intersection of Two Lines
- How to Enter Data into a List TI89
- Optimization Problems
- Quadratic Formula
- Related Rates
- Relation vs Function
- Second Derivatives (Test, Finding)
- Sum of a Convergent Geometric Series
- Symbols and Equations (How to Read Them)
- Vertical Tangents

## Calculus based Statistics: Definitions

These full definitions are some of the topics you may come across in a calculus based statistics class:

Jump to A B C D E F G H I J K L M N O P Q R S T U V W X, Y, Z

## A

- Abel’s Test
- Absolute Minimum
- Absolutely Convergent
- Alternating Series Test
- Anisotropic
- Asymptote
- Automatic Differentiation

## B

## C

- Calculus of Variations
- Cartesian Form
- Chain Rule
- Chebyshev Polynomials
- Chebyshev’s Sum Inequality
- Closed Form Solution
- Closed Interval
- Collider Variable
- Common Ratio & Common Difference
- Complex Numbers / Plane
- Concave Up and Down
- Constant Term
- Continued Sum
- Conditional Convergence
- Convergence of Random Variables
- Critical Numbers
- Curly d
- Curvilinear
- Cylindrical Coordinates

## D

- Del Operator (Nabla operator)
- Definite Integral
- Deleted Neighborhood
- Difference Quotient
- Differentiable
- Differential Approximation
- Differential Operator
- Differentiate Definition (“Take the Derivative”)
- Divergent Series
- Domain and Range of a function
- Double Integral

## E

- Einstein Summation (Notation)
- Elliptic Integral
- Empirical Rule & Research
- End Behavior
- Epsilon
- Essential Discontinuity
- Euclidean Space
- Euler’s Number (e)
- Exponential Growth and Decay
- Exterior Calculus
- Explicit and Implicit Solution
- Exponential Model
- Extrema of a Function
- Extreme Values of a Polynomial

## F

- Finite and Infinite Sets
- Finite Calculus (Calculus of Finite Differences)
- First Derivative Test
- Fluxion
- Fourier Analysis
- Fourth Derivative

## G

- Gaussian Distribution, Quadrature
- General Solution (Diffeq)
- Geometric Series
- Global Minimum
- Global Maximum
- Gradient

## Calculus Based Statistics Definitions: H to N

## H

## I

- Ill-conditioned
- Imaginary Numbers
- Implicit Differentiation
- Index Number
- Indeterminate Expression
- Infinitesimal
- Inflection Point
- Initial Value / Condition
- Integral Bounds
- Integral Kernel (Symbol)
- Integrand
- Interval of Convergence
- Intermediate Value Theorem

## J

## K

## L

- Lagrange Interpolating Polynomial
- Lambda Calculus
- Law of Large Numbers / Law of Averages
- Liebniz Notation
- Limit of Product Quotient
- Line Segment, Equivalent
- Linear Operator
- Linear Term
- Local Maximum
- Local Minimum
- Logistic Growth
- Lower Bound, Greatest Lower Bound (GLB) — Infimum
- Linearization & Linear Approximation
- Linearity of Differentiation
- Linearly Independent Solutions
- Logarithm

## M

- MacLaurin Series
- Many to One
- Mean Value Theorem
- Min-Max Theorem
- Moment
- Monotonic Sequence, Series, Function
- Multiplicative Calculus
- Multivariate

## N

- Newton Notation
- Non empty set
- Nonstandard Calculus
- Non-Newtonian Calculus
- n-tuple
- Numerical Integration
- Normal Line
- nth Degree Taylor Polynomial

## Calculus Based Statistics Definitions: O to Z

## O

## P

- Parabola
- Parameterize a Function
- Particular Solution (Diffeq)
- Polar Coordinates
- Power Rule
- Prime Notation in Differentiation
- Product Rule
- Propositional Calculus

## Q

## R

- Ratio Test
- Real Analysis
- Relatively Prime (Coprime, Mutually Prime)
- Removable Discontinuity
- Riemann Sums
- RTH Moment of a Distribution

## S

- Saddle Point
- Secant Method
- Second Derivative Test
- Sequence and Series
- Sequence of Partial Sums
- Shear Mapping
- Simple Closed Curve
- Single Variable Calculus
- Slope / Slope Field
- Spherical Coordinates
- Standard Form
- Stationary Point
- Step Discontinuity
- Subfactorial
- Symmetric Polynomial

## T

- Tangent Line
- Tangent Space
- Taylor Series
- Tensor Definition
- Topological Space
- Transfinite Numbers
- Transformations

## U

## V

## W

## X, Y, Z

## Calculus based Statistics: 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