NZ Scholarship Statistics key terms

A numerical quantity whose value depends on the outcome of a random process, so it takes values with associated probabilities. Discrete versions take countable values; continuous versions take any value in an interval.

A complete specification of the possible values of a random variable together with their probabilities (discrete) or probability density (continuous). It describes long-run behaviour rather than any single outcome.

For a continuous random variable, a non-negative function whose integral over an interval gives the probability of falling in that interval. Density values are not themselves probabilities and may exceed one.

The function giving the probability that a random variable is less than or equal to a given value. It is non-decreasing, runs from 0 to 1, and works for both discrete and continuous variables.

The probability-weighted average of a random variable over many repetitions. For a discrete variable it is the sum of each value multiplied by its probability; it need not be a possible value.

The expected squared deviation of a random variable from its mean, measuring spread. Its square root is the standard deviation, which is expressed in the original units.

The square root of variance, giving spread in the same units as the variable. It is the typical distance of observations from the mean under roughly symmetric distributions.

The standard deviation of the sampling distribution of an estimator. It quantifies how much an estimate would vary between repeated samples and shrinks as sample size grows.

The probability distribution of a statistic across all possible samples of a fixed size from a population. It underpins inference because it describes how an estimate varies from sample to sample.

The result that the sampling distribution of a sample mean or sum approaches normality as sample size increases, regardless of the population shape, provided variance is finite. It justifies large-sample normal methods.

A symmetric bell-shaped continuous distribution defined by mean and standard deviation. It is the limiting form of many sample means and underlies z-based inference and the empirical 68-95-99.7 spread rule.

Converting a value to a z-score by subtracting the mean and dividing by the standard deviation. This places values from different distributions on a common scale for comparison and probability lookup.

A value below which a specified proportion of a distribution or dataset lies. The median is the 0.5 quantile and percentiles are quantiles expressed in hundredths.

Two events are independent when the occurrence of one does not change the probability of the other, so joint probability equals the product of the individual probabilities. For variables, joint behaviour factorises similarly.

The probability of one event given that another has occurred, written P(A|B) and computed as the joint probability divided by the probability of the conditioning event. It underlies Bayes' theorem and tree diagrams.

A rule for reversing conditional probabilities: the posterior probability of a hypothesis given data is proportional to the likelihood of the data times the prior probability of the hypothesis.

Events that cannot both occur together, so their joint probability is zero and the probability of either is the sum of their probabilities. Mutually exclusive events with positive probability are not independent.

The distribution of the number of successes in a fixed number of independent trials, each with the same success probability. It is discrete, with mean np and variance np(1-p).

A discrete distribution for counts of events occurring randomly and independently in a fixed interval, parameterised by the mean rate. Its mean and variance are equal, which is a useful diagnostic.

A formal procedure that compares observed data with what a null hypothesis predicts, using a test statistic and its distribution to decide whether the evidence is strong enough to reject the null.

The default claim that a hypothesis test assesses, typically stating no effect, no difference, or a specific parameter value. It is rejected only when data are unlikely under it.

The claim that contradicts the null hypothesis and represents what a researcher suspects, such as a difference or an association. It may be one-sided or two-sided and shapes the rejection region.

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

The threshold probability, chosen before testing and usually denoted alpha, below which a p-value leads to rejecting the null hypothesis. It bounds the Type I error rate.

Rejecting a true null hypothesis, a false positive. Its probability is controlled by the significance level, and reducing it usually increases the risk of a Type II error.

Failing to reject a false null hypothesis, a false negative. Its probability relates inversely to power and grows when effects are small, variability is high, or samples are small.

The probability that a hypothesis test correctly rejects a false null hypothesis. It increases with larger effect size, larger sample size, lower variability, and a more lenient significance level.

A range of plausible values for a population parameter, constructed so that a stated percentage of such intervals from repeated samples would contain the true value. Width reflects precision.

A resampling method that builds a sampling distribution by repeatedly resampling with replacement from the observed data. It estimates standard errors and intervals without assuming a parametric form.

A resampling procedure that repeatedly shuffles or reassigns labels to simulate the distribution of a statistic under the null hypothesis. It gives a p-value without distributional assumptions.

A method for modelling the relationship between a response variable and one or more explanatory variables, typically by fitting a line or curve that minimises squared deviations.

The estimation principle that chooses parameter values minimising the sum of squared residuals. In simple linear regression this yields the line passing through the means with slope based on covariance and variance.

The difference between an observed response value and the value predicted by a fitted model. Residual plots are used to check linearity, constant variance, and unusual points.

The proportion of variation in the response explained by a regression model, denoted R-squared. It ranges from 0 to 1 and does not by itself prove causation or model adequacy.

A standardised measure between -1 and 1 of the strength and direction of linear association between two variables. It is symmetric and unaffected by changes of units but sensitive to outliers.

An observation whose removal substantially changes a fitted model's estimates. It often combines high leverage in the explanatory variable with a large residual.

A variable associated with both the explanatory and response variables that distorts the apparent relationship between them. Randomisation in experiments protects against confounding, but observational studies do not.

A study in which treatments are assigned to units by chance, so treatment groups are comparable on average and confounding is controlled. This supports causal conclusions about the treatment effect.

A study in which researchers record variables without assigning treatments. Associations found may reflect confounding or reverse causation, so causal claims are generally not justified.

A relationship in which a change in one variable genuinely produces a change in another. Establishing it requires strong design such as randomisation, not merely statistical association or correlation.