Notation & glossary

STAT 45203 · the symbols and terms in one place

Instructor-authored reference. Original definitions; nothing reproduced from any text. Math is live text (MathML), not images.

Notation

Symbol	Read as	Meaning
$x_1, \dots, x_n$	the data	a sample of $n$ observations
$x_{(1)} \le \dots \le x_{(n)}$	order statistics	the sample sorted; $x_{(1)}$ is the min, $x_{(n)}$ the max
$\bar{x}$	x-bar	the sample mean
$\tilde{x}$ or $m$	the median	the middle value (average of the two middle values if $n$ is even)
$\hat{F}(x)$	F-hat of x	the empirical CDF: the fraction of the sample $\le x$
$R_i$	the rank of $x_i$	its position when the data are sorted (average of positions for ties)
$\theta$	theta	a population parameter (the thing we want to know)
$\hat{\theta}$	theta-hat	an estimate of $\theta$ computed from the sample
$\theta^*$	theta-star	a bootstrap value of the statistic, from one resample
$B$	—	the number of resamples (bootstrap) or shuffles (permutation)
$\text{SE}$	standard error	the standard deviation of a statistic’s sampling distribution
$\alpha$	alpha	a significance level (e.g. 0.05); the target Type I error rate
$\text{MAD}$	mad	median absolute deviation, $\operatorname{median}(|x_i - \tilde{x}|)$

Glossary

Assumption-light method. A method that stays useful under weaker assumptions than the normal-theory default — resampling, rank-based, or robust.

Bootstrap. Resampling from the sample itself, with replacement and the same size, to approximate how a statistic varies from sample to sample.

Bootstrap distribution. The collection of a statistic recomputed on many bootstrap resamples. Its spread estimates the statistic’s standard error; it is centered at the sample statistic, not the true parameter.

Breakdown point. The fraction of observations that can be made arbitrarily extreme before a summary becomes useless. The median’s breakdown point is 50%; the mean’s is 0%.

ECDF (empirical CDF). The step function $\hat{F}(x)$ giving the fraction of the sample at or below each value — a distribution-free picture of the whole sample.

Exchangeability. The assumption that, under the null, the labels on the observations could be swapped without changing the distribution — the engine of a permutation test.

Influence / leverage. Leverage is how unusual a point’s predictor value is; influence is how much a single point changes a fit. A high-leverage point with a large residual is highly influential.

Order statistic. The sample sorted into position: $x_{(1)}$ (minimum) up to $x_{(n)}$ (maximum).

Percentile interval. A bootstrap confidence interval read off as the middle $1-\alpha$ of the bootstrap distribution (e.g. the 2.5th to 97.5th percentiles).

Permutation test. A test that builds a null distribution by reshuffling labels among the observed values — holding the values fixed, varying only the assignment.

Randomization test. A permutation-style test justified by the actual random assignment in an experiment; inference is to the units that were randomized.

Rank. An observation’s position in the sorted data. Rank-based methods use ranks instead of raw values, trading exact spacing for resistance to outliers.

Robust / resistant summary. A summary (median, trimmed mean, MAD) that changes little when a small fraction of the data is extreme.

Sampling distribution. The distribution of a statistic across hypothetical repeated samples from the population — usually unseen; the bootstrap approximates its spread.

Stochastically larger. Group A is stochastically larger than B if, at every threshold, A is more likely to exceed it — what the rank-sum test detects (not, in general, a difference in means).

Trimmed mean. The mean after discarding a fixed fraction of the smallest and largest values — a compromise between the mean and the median.

Type I error / power. Type I error is rejecting a true null (target $\alpha$); power is rejecting a false null. Both are things you can measure with a simulation study.

Wilcoxon signed-rank / rank-sum. Rank-based tests for paired differences (signed-rank) and two independent groups (rank-sum / Mann–Whitney).