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).