BCS Written Statistics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Central Limit Theorem

For independent identically distributed variables with finite mean and variance, the sampling distribution of the sample mean approaches normality as sample size grows, regardless of the population distribution.

Bayes' theorem

Expresses the posterior probability of an event given evidence as the likelihood times the prior divided by the marginal probability of the evidence, enabling updating of beliefs.

Poisson distribution

Counts the number of rare events in a fixed interval when events occur independently at a constant rate; its mean and variance both equal the rate parameter lambda.

Variance of a sum

For independent random variables, the variance of their sum equals the sum of their variances; dependence adds twice the covariance for each pair.

Type I and Type II error

Type I error rejects a true null hypothesis with probability alpha; Type II error fails to reject a false null with probability beta. Power equals one minus beta.

Confidence interval

An interval computed from sample data that contains the true parameter with a stated long-run coverage probability, typically 95 percent, under repeated sampling.

p-value

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

Maximum likelihood estimator

The parameter value that maximizes the likelihood of the observed data; under regularity conditions it is consistent and asymptotically normal with variance from Fisher information.

One-way ANOVA

Partitions total variability into between-group and within-group sums of squares and tests equality of several means using an F ratio with k minus one and N minus k degrees of freedom.

Randomization

Assignment of experimental units to treatments by chance so that unmeasured confounders are balanced across groups in expectation, supporting causal interpretation of treatment effects.

Interaction effect

Occurs when the effect of one factor on the response depends on the level of another factor, meaning treatment effects are not additive across factor combinations.

Tukey's HSD

A post hoc procedure controlling family-wise error for all pairwise comparisons of group means after a significant ANOVA, using the studentized range distribution.

Finite population correction

The factor one minus n over N applied to sampling variance when sampling without replacement from a finite population; it reduces variance as the sample approaches the population size.

Stratified sampling

Divides the population into homogeneous strata and samples independently within each stratum, reducing variance relative to simple random sampling when strata differ in their means.

Cluster sampling

Selects entire groups of units rather than individuals; it is often cheaper but less precise than simple random sampling because units within clusters tend to be similar.

Design effect

The ratio of the variance of an estimator under a complex sampling design to its variance under simple random sampling with the same sample size.

Coefficient of determination

The proportion of total variation in the response explained by the regression model, ranging from zero to one and denoted R squared.

Multicollinearity

Strong linear relationships among predictor variables that inflate standard errors and destabilize coefficient estimates; variance inflation factors above about ten indicate a serious problem.

Logistic regression

Models the log-odds of a binary outcome as a linear function of predictors; exponentiated coefficients give odds ratios for a one-unit change in the predictor.

Principal component analysis

A dimension-reduction method producing orthogonal linear combinations of variables ordered by explained variance, useful for summarizing correlated data and detecting structure.

Stationarity

A time series property where mean, variance, and autocovariance do not change over time; many forecasting models require differencing to achieve approximate stationarity.

ARIMA model

A model combining autoregressive terms, differencing to remove trend, and moving average terms; orders p, d, and q specify each component respectively.

Autocorrelation function

Plots the correlation between a series and its lagged values across lags; it helps identify moving average order and seasonal patterns in time series data.

Exponential smoothing

A forecasting method producing a weighted average of past observations with weights declining geometrically, controlled by a smoothing constant between zero and one.

Total fertility rate

The average number of children a woman would bear over her lifetime if current age-specific fertility rates persisted; computed by summing age-specific rates across reproductive ages.

Life expectancy at birth

The average number of years a newborn would live if current age-specific mortality rates remained constant; a summary measure derived from a life table.

Crude birth rate

The number of live births in a year per thousand mid-year population; it is influenced by the age and sex structure of the population.

Dependency ratio

The ratio of the population aged under fifteen plus those aged sixty-five and over to the working-age population aged fifteen to sixty-four, usually expressed per hundred.

Bootstrap

A resampling method that draws repeated samples with replacement from observed data to estimate standard errors, confidence intervals, and sampling distributions without parametric assumptions.

Cross-validation

A model evaluation technique that splits data into training and testing folds, estimating out-of-sample prediction error and helping detect overfitting.

Monte Carlo simulation

A computational method using repeated random sampling to approximate integrals, distributions, or probabilities; estimate precision improves roughly with the square root of the number of draws.

Reproducible analysis

Practice of documenting data, code, software versions, and random seeds so that an analysis can be exactly rerun and verified by another analyst.