Math and statistics articles by category

This page categorizes my informal math and statistics articles and blog posts. See this page for informal articles on other topics such as programming. See also journal articles and technical reports.


Preventing and detecting errors
Interpreting and misinterpreting probabilities
Clinical trials
Mathematical statistics
Random number generation
Pure math



Tukey tallying
Honeybee geneology
Coping with exponential growth
Selection bias and bombers
Canonical example of Bayes' theorem in detail
Linear interpolation
Quadratic interpolation
Sales tax included
Six degrees of Paul Erdős
Connecting Fibonacci and geometric sequences
Divisibility rules in hexadecimal
Golden ratio and special angles
Silver ratio
Three rules of thumb Approximations with Pythagorean triangles
Best rational approximation
There isn’t a googol of anything

Preventing and detecting errors

Reproducible research
In praise of tedious proofs
Errors in math papers
Coloring numbers
Proofs of false statements
Probability that a study result is true
Literate programming and statistics
Irreproducible analysis
Programming the last mile
Complementary validation
Troubleshooting Sweave
Preventing an unpleasant Sweave surprise

Interpreting and misinterpreting probabilities

Plausible reasoning
How loud is the evidence?
What is a confidence interval?
Probability that a number is prime
Learning is not the same as gaining information
Laws of large numbers and small numbers
The law of small numbers
Probability and information
Example of the law of small numbers
What a probability means
Irrelevant uncertainty
False positives for medical papers
False positives for medical tests
Most published research results are false
Musicians, drunks, and Oliver Cromwell


C. S. Lewis on modeling
Thick tails
Metabolism and power laws
Time spent in coffee shops
Networks and power laws
Why heights are normally distributed
Why heights are not normally distributed
How to linearize data for regression
Approximate problems and approximate solutions
Wine, Beer, and Statistics
What happens when you add a new teller? (queuing theory)
Adult heights and mixture distributions
The probability that Shakespeare wrote a play
Hard disk array failure probabilities
Are men better than women at chess?
Small advantages show up in the extremes
Ignorance doesn't change reality
Server utilization: Joel on queuing
Cost-benefit analysis versus benefit-only analysis

Clinical trials

Stopping trials of ineffective drugs earlier
Three ways of tuning an adaptively randomized trial
Galen and clinical trials
Dose-finding: why start at the lowest dose?
Innovation II (scurvy trial from 1601)
Most published research results are false
False positives for medical papers
Population drift
Randomized trials of parachute use
Small effective sample size does not mean uninformative
Early evidence-based medicine

Mathematical statistics

Clickable chart of distribution relationships
Upper and lower bounds for normal probabilities
Student-t distribution as a mixture of normals
Sums of uniform random variables
Conjugate prior relationships
Robust priors
Determining parameters from quantiles
Quantifying the error in the central limit theorem
Four characterizations of the normal distribution
Random inequalities I, II, III, IV, V, VI, VII, VIII
Estimating Cauchy distribution location parameter
Inverse gamma distribution
Negative binomial distribution
Relating two definitions of expectation
Four types of errors
Stochastic independence
Normal approximation details:
Poisson approximation to binomial
Wilson-Hilferty approximation for the Poisson CDF
Camp-Paulson approximation for the binomial CDF
Relative error in normal approximations
Stable distributions
Unbiased estimators can be terrible
Bias, consistency, and efficiency
Criticisms of significance testing
Canonical examples from robust statistics
Example of efficiency for mean vs. median
Estimating reporting rates

Random number generation

Pitfalls in Random Number Generation
Simple random number generation
Random number generator controversy
Random number generation in C++ TR1
Rolling dice for normal samples


Approximating a solution that doesn't exist
Making a singular matrix non-singular
Ten surprises from numerical linear algebra
Microsoft equation editor improved
Soft maximum and how to compute it
IEEE arithmetic exceptions in C++, Python
Computing standard deviation
Computing regression coefficients
Fibonacci numbers at work (numerical integration)
Five kinds of subscripts in R
Computing the inverse of the normal CDF
Approximate problems and approximate solutions
How to calculate binomial probabilities
How to calculate correlation accurately
Calculating percentiles in memory-bound applications
Avoiding overflow, underflow, and loss of precision
Overflow and loss of precision
What to make “u” in integration by parts
Integration and pragmatism
Quick TeX to graphic utility
LaTeX on Windows
Everything begins with “p”
Mercator projection
Asymptotic notation
R for Programmers coming from other languages
Step size for numerical differential equations
The error function and the normal distribution
Working with distributions in Mathematica, R/S-PLUS, Excel, and Python
Random number generation in C++
Innovation IV (Tukey on efficiency)
Index of tail weight
Delta method
When the normal approximation to the Student t isn't good enough
Double exponential integration
Floating point numbers are a leaky abstraction
Convex optimization
Code for the error function and normal CDF
Anatomy of a floating point number

Pure math

Modes of convergence
Breastfeeding, the golden ratio, and rational approximation
Finite differences
A knight’s random walk
How to differentiate a non-differentiable function (intro to distribution theory)
Navier-Stokes equations
Splitting a convex set through its center
Log concave functions
The gamma function
Binomial coefficient generalizations
Multi-index notation
Computing binomial coefficients
Upper and lower bounds on binomial coefficients
Selecting with replacement (Stanley's symbol)
Stanley's twelvefold way (combinatorics notes)
Jenga mathematics
What is the cosine of a matrix?
Gilbreath's conjecture
Orthogonal polynomials
Chebyshev polynomials
Hypergeometric functions
Functions of regular variation
Determining whether a number is a square
How many numbers are squares mod m
Fast exponentiation
Linear congruences
Quadratic congruences
Constructive proof of the Chinese remainder theorem
Old math books
The smoothest curve through a set of points
Quasi-random sequences in art and integration
Fractional derivatives
Means and inequalities
Polynomial interpolation error
Sledgehammer technique for trig integrals
Spherical trigonometry
Why care about spherical trig?
The pqr theorem
A surprising theorem in complex variables
Perfect numbers: even, odd
Fractional integration
Diagram of Bessel functions
Gamma function identities
Special function diagram