Statistical Dispersion & Testing

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Abstractions about describing distributions and testing their location, variation, dependence, and fit. They include summary and rank statistics, standardized scores, variance measures, diagnostic plots, asymptotic behavior, time-series tests, and procedures for comparing samples or models.

44 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• 68–95–99.7 rule — The normal-distribution rule that about 68%, 95%, and 99.7% of probability lies within one, two, and three standard deviations of the mean.
• Anderson–Darling test — Test a sample’s agreement with a specified continuous distribution by integrating squared empirical-CDF deviations with extra weight in the tails.
• Asymptotic theory (statistics) — The large-sample framework that studies limiting distributions, consistency and efficiency of estimators and tests as sample size tends to infinity.
• Barnardisation — A statistical-disclosure-control method that pseudo-randomly perturbs nonzero interior table counts by plus one, zero or minus one according to a fixed probability rule before recomputing totals.
• Coefficient of variation — A dimensionless relative-dispersion statistic equal to standard deviation divided by mean, interpreted only where the measurement scale and nonzero mean make the ratio meaningful.
• Correlation ratio — An effect-size measure equal to the square root of between-category variance divided by total variance, detecting nonlinear mean association.
• Count data — Observations taking nonnegative integer values because they record event or object counts rather than ranks or arbitrary numeric labels.
• F-test of equality of variances — A parametric hypothesis test that compares two independent normal-population variances using the ratio of their sample variances.
• Fano factor — A count-dispersion ratio equal to variance divided by mean, with one marking a Poisson baseline.
• Five-number summary — Compress a univariate ordered dataset into minimum, first quartile, median, third quartile, and maximum, exposing center, spread, skew, and extremes while remaining dependent on the chosen quartile convention.
• Folded-t and half-t distributions — Nonnegative distributions obtained by taking the absolute value of a Student-t variate, with the half-t arising from a centered symmetric t distribution restricted or folded at zero.
• Generalized entropy index — A parameterized family of decomposable inequality measures computed from powers or logarithms of each observation's ratio to the population mean.
• Higher-order statistics — Statistics based on third- or higher-order moments, cumulants or spectra that characterize distributional shape and nonlinear dependence beyond mean and covariance.
• Hoover index — An inequality measure equal to the share of total income or another resource that would need redistribution to achieve equal per-capita shares.
• Inverse probability weighting — An estimation method that weights observed units by the inverse probability of their observed sampling, treatment or response status to reconstruct a target population or intervention distribution.
• Kendall rank correlation coefficient — A rank-association statistic based on the excess of concordant over discordant observation pairs.
• KPSS test — A time-series test whose null hypothesis is level or trend stationarity and whose alternative is a unit-root process.
• Lexis ratio — A dispersion ratio comparing observed variation in grouped binomial proportions with the variation expected under one common success probability.
• Location parameter — A distribution parameter whose change translates the probability law along its sample space without changing its shape.
• Logrank test — A nonparametric hypothesis test comparing the event-time distributions of groups by accumulating observed-minus-expected events across ordered failure times while accounting for right censoring.
• Mean integrated squared error — The expected integrated squared difference between a functional estimator and its unknown target, commonly used as global density-estimation risk.
• Midhinge — Summarize distributional location by averaging the first and third quartiles, placing the center halfway between the hinges while keeping the interquartile spread analytically separate.
• Nemenyi test — A rank-based post-hoc multiple-comparison procedure that identifies pairs of treatments whose average ranks differ beyond a familywise-error-controlled critical distance after repeated-block comparison.
• Noncentral distribution — A distribution family for a statistic under a shifted alternative, indexed by a noncentrality parameter in addition to the central family parameters.
• Normal probability plot — A quantile plot comparing ordered observations with expected normal quantiles so approximate normality appears linear and systematic departures reveal skew, tails, mixtures or outliers.
• Pearson correlation coefficient — The unitless covariance of two variables divided by the product of their standard deviations, measuring linear association from minus one to one.
• Point-biserial correlation coefficient — The Pearson correlation between one continuous variable and a genuinely dichotomous variable, expressible through group means, proportions and overall standard deviation.
• Polychoric correlation — An estimate of the correlation between two latent normally distributed continuous variables inferred from their observed ordinal categories through threshold models.
• Quantile — A distribution cut point at or below which a specified cumulative probability lies, defined through a generalized inverse when the distribution has jumps or flat regions.
• Quartile — One of the three cut points corresponding approximately to the 25^th, 50^th and 75^th percentiles, dividing ordered data or a distribution into four equal-probability parts.
• Seven-number summary — A descriptive-statistics summary using seven ordered quantiles to show center, spread and tail structure more finely than a five-number summary.
• Sign test — Test a paired-difference or one-sample median null by reducing non-tied observations to positive and negative signs and evaluating the positive count against its exact binomial distribution under a declared null probability, usually one half.
• Standard normal table — A tabulation of cumulative probabilities or tail areas for standardized normal-distribution values.
• Standard score — A normalized value equal to an observation’s deviation from a reference mean divided by the reference standard deviation.
• Statistical thinking — A mode of reasoning that treats outcomes as products of interconnected processes containing variation and uses data, uncertainty and context to guide learning and improvement.
• Studentization — Dividing a sample statistic by a sample-based estimate of its standard deviation.
• Summary statistics — Compact numerical measures that describe central tendency, dispersion, distribution shape, or relationships in a set of observations.
• Two-way analysis of variance — An analysis-of-variance model estimating two categorical factors’ main effects and their interaction on a continuous response.
• U-statistic — A statistic formed by averaging a symmetric kernel over all fixed-size subsets of a sample, yielding an unbiased estimator of its corresponding population functional.
• Uncorrelatedness — A second-moment relation in which two random variables or vectors have zero covariance, excluding linear association without generally implying independence.
• Unevenly spaced time series — A time series represented by observation-time and value pairs whose successive observation intervals are not constant.
• Variance — The expected squared deviation of a random variable from its mean, measuring dispersion in squared units and equaling its second central moment.
• Variation ratio — A nominal-data dispersion measure equal to the proportion of observations outside the modal category.
• Z-test — A hypothesis test whose null statistic follows, exactly or approximately, a standard normal distribution after centering and scaling by a known or consistently estimated standard error.