# 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.

## Categories

Elementary

Preventing and detecting errors

Interpreting and misinterpreting probabilities

Modeling

Clinical trials

Mathematical statistics

Random number generation

Practicalities

Pure math

## Elementary

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 ErdÅ‘s

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

## Modeling

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

Bias

Four types of errors

Stochastic independence

Normal approximation details:

beta

binomial

gamma

Poisson

Student-t

Logistic

Cosine

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

## Practicalities

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