BCS Written Statistics key terms

A rule that assigns to each possible value of a discrete random variable the probability that the variable takes that value. All assigned probabilities are non-negative and sum to one.

A non-negative function whose integral over any interval gives the probability that a continuous random variable falls in that interval. The total area under its curve equals one.

A function giving the probability that a random variable is less than or equal to a stated value. It is non-decreasing, runs from zero to one, and works for both discrete and continuous variables.

A result stating that the sum or mean of many independent, identically distributed variables with finite variance tends toward a normal distribution as sample size grows, regardless of the population shape.

A symmetric, bell-shaped continuous distribution fully described by its mean and variance. Many test statistics follow it approximately, which underpins a large share of inference procedures.

A normally distributed quantity rescaled by subtracting its mean and dividing by its standard deviation, giving mean zero and variance one. Tables of its cumulative probabilities are widely used.

The distribution of the number of successes in a fixed number of independent trials, each with the same success probability. Its mean is the product of trial count and success probability.

A distribution for counts of rare events occurring independently in a fixed interval of time, space, or exposure. Its mean and variance are equal, which gives a quick check of fit.

A single number computed from sample data and used as the best guess of an unknown population quantity. Its quality is judged by bias, variance, and consistency.

A range of plausible values for an unknown population quantity, built so that a stated percentage of intervals from repeated sampling would contain the true value.

An estimator whose expected value over repeated sampling equals the true population quantity. Lack of bias does not by itself guarantee small sampling error.

The standard deviation of an estimator's sampling distribution, measuring how much the estimate varies from sample to sample. It typically shrinks as sample size increases.

A default claim about a population, usually of no effect or no difference, that is retained unless sample evidence is strong enough to contradict it. It drives the logic of significance testing.

The probability, assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as the one observed. It is not the probability that the null hypothesis is true.

Rejecting a true null hypothesis, that is, declaring an effect when none exists. Its probability is controlled by the chosen significance level.

Failing to reject a false null hypothesis, that is, missing a real effect. Its probability is related inversely to power and shrinks with larger samples or bigger true effects.

The probability of correctly rejecting a false null hypothesis. It rises with larger samples, larger true effects, or less variable data, and is often set at 0.80 in planning.

The expected squared difference between an estimator and the true parameter value. It decomposes into squared bias plus variance, so it balances both sources of inaccuracy.

The assignment of experimental units to treatments by a chance mechanism, so that systematic differences among units are spread across groups and bias is avoided.

Independent repetition of a treatment on several experimental units, allowing random error to be estimated and increasing the precision of comparisons between treatment means.

A procedure that splits total variation in a response into parts attributable to treatment factors and to random error, then compares their mean squares with an F test.

An experiment in which similar units are first grouped into blocks and treatments are then randomly assigned within each block, removing block-to-block variability from the error term.

A design in which all experimental units are treated alike before assignment and treatments are allocated purely at random across the whole set of units.

A study in which two or more factors are varied together so that every combination of their levels is observed, permitting estimation of main effects and interactions in one run.

A situation where the effect of one factor on the response depends on the level of another factor, so the factors cannot be interpreted separately.

A selection method in which every possible sample of a given size has an equal chance of being drawn, giving each population unit an equal inclusion probability.

Dividing the population into internally similar groups called strata and drawing independent samples from each, often improving precision and guaranteeing representation of every stratum.

Selecting whole groups of units, such as villages or schools, rather than individual units, which cuts fieldwork cost but usually raises variance because units within a cluster resemble each other.

The list or map of population units from which a sample is actually drawn. Errors or gaps in it cause coverage problems that no sample size can repair.

The ratio of the variance of an estimate under a complex design to its variance under simple random sampling of the same size. Values above one signal a loss of precision.

Inaccuracy in survey results arising from sources other than which units were selected, such as nonresponse, faulty measurement, data entry mistakes, or interviewer effects.

A fitting method that chooses regression coefficients minimizing the sum of squared differences between observed responses and fitted values, yielding the best linear unbiased estimates under standard assumptions.

The proportion of total variation in the response explained by a regression model. It ranges from zero to one and always rises when extra predictors are added, even useless ones.

A version of the explained-variation measure that penalizes the addition of predictors, so it can fall when a new variable contributes little. It is preferred for comparing models with different numbers of terms.

Strong linear association among explanatory variables in a regression, which inflates the standard errors of coefficients and makes individual effects hard to disentangle.

A property of a time series whose mean, variance, and autocorrelation structure do not change over time. Most forecasting models assume it, often after differencing the data.

The correlation between a time series and a lagged copy of itself, describing how strongly successive observations resemble one another. It is a key diagnostic for model choice and adequacy.

Replacing each observation by the change from the previous one, a transformation used to remove trend and make a series approximately stationary before modeling.

Averaging each observation with neighboring values to dampen short-term fluctuations and reveal trend or seasonal pattern. The wider the window, the smoother the resulting series.

A numerical factor expressing how far a given period, such as a month or quarter, typically lies above or below the overall trend level, used to adjust and forecast seasonal series.

The number of live births in a year per 1,000 people in the mid-year population. It is simple to compute but influenced by the population's age and sex structure.

The average number of children a woman would bear over her lifetime if current age-specific fertility rates persisted. It summarizes fertility without distortion from age structure.

The average number of additional years a person would live if current age-specific death rates stayed constant. Often quoted at birth, it is a standard summary of mortality conditions.

A technique that reweights rates to a common reference age structure so that populations with different age compositions can be compared fairly on mortality or other outcomes.

Calculating the number of observations needed to achieve a target precision or power before data collection, using assumed effect sizes, variability, and desired error levels.

A cross-tabulation of counts classified by two or more categorical variables. Association between the variables is usually assessed with a chi-square test of independence.

A coded file that reads raw data, performs every transformation and calculation, and produces the reported results, so that another analyst can rerun it and obtain identical output.

The systematic checking and correction of raw data for missing values, impossible entries, duplicates, and inconsistent coding before any statistical analysis is undertaken.

A program offering ready-made routines for data management, estimation, testing, and graphics, such as R, Stata, SPSS, or SAS, used to carry out analyses reproducibly.