NZ Scholarship Statistics flashcards

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

Standard error of a mean

The standard deviation of the sampling distribution of the sample mean, equal to the population standard deviation divided by the square root of the sample size.

Central limit theorem

For sufficiently large samples, the distribution of the sample mean is approximately normal regardless of the shape of the population distribution, enabling z and t inference.

Confidence interval interpretation

A 95% confidence interval is generated by a procedure that captures the true parameter in 95% of repeated samples; it is not a probability statement about a fixed parameter.

Margin of error scaling

The margin of error is roughly two standard errors for 95% confidence and shrinks proportionally to one over the square root of n, so quadrupling n halves it.

Random assignment versus sampling

Random assignment of treatments supports causal conclusions about the units studied; random sampling supports generalisation from the sample to the wider population.

p-value definition

The probability, computed assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as that observed in the data.

Type I and Type II errors

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

Statistical versus practical significance

A statistically significant result may be trivially small in context; report effect size and interval width because very large samples make tiny effects significant.

Bonferroni correction

A multiple-comparison adjustment that divides the significance level alpha by the number of tests performed, controlling familywise error at the cost of reduced power.

Chi-square goodness of fit

A test comparing observed category counts with expected counts under a hypothesised distribution; expected counts below about five make the approximation unreliable.

Correlation coefficient properties

Pearson r measures only the strength and direction of linear association, lies between minus one and one, is unaffected by rescaling, and is sensitive to outliers.

Conditional probability formula

The probability of event A given event B equals the probability of both events divided by the probability of B, provided B has nonzero probability.

Independence of events

Events A and B are independent when the probability of both equals the product of their probabilities, equivalently when the conditional probability of A given B equals A's probability.

Bayes' theorem

Expresses the posterior probability of a hypothesis given evidence as the likelihood times the prior, divided by the total probability of the evidence.

Base-rate neglect

Ignoring prevalence when interpreting test results; a highly sensitive and specific test for a rare condition still produces mostly false positives among positive results.

Variance of a linear combination

For aX plus bY, the variance is a squared times Var(X) plus b squared times Var(Y) plus twice ab times the covariance, which vanishes under independence.

Binomial distribution conditions

Fixed number of independent trials, two outcomes per trial, and constant success probability; the mean is np and the variance is np times one minus p.

Geometric distribution memorylessness

The number of trials until the first success has mean one over p, and past failures do not alter the probability of success on future trials.

Normal standardisation

Converting a value to a z-score by subtracting the mean and dividing by the standard deviation, allowing use of standard normal tables and symmetry.

Reporting in context

Answers should name the variable, units, direction of effect, and uncertainty; a bare numerical result without interpretation loses communication credit.

Stating assumptions explicitly

Justify a method by naming its requirements, such as independence, random sampling, or approximate normality, and describe how violations would affect conclusions.

Randomisation test logic

Shuffling group labels many times builds the distribution of a statistic under the null hypothesis, against which the observed statistic is compared.

Bootstrap resampling

Repeatedly resampling the observed data with replacement estimates the sampling variability of a statistic without relying on parametric distributional assumptions.

Prediction versus confidence intervals

A prediction interval for an individual future observation is wider than a confidence interval for the mean response because it adds residual variability.

Simpson's paradox

An association observed in aggregated data can reverse direction when the data are split by a lurking grouping variable that differs in composition across groups.

Relative versus absolute risk

A large relative reduction from a very small baseline risk may correspond to a negligible absolute change, so both framings should be reported.

Regression to the mean

Extreme measurements tend to be followed by less extreme ones on remeasurement, so before-and-after comparisons without controls overstate apparent intervention effects.

Confounder versus mediator

A confounder distorts the exposure-outcome relationship and should be adjusted for; a mediator lies on the causal pathway, so adjusting for it can remove the effect.

Graphical integrity checks

Verify that axes start at zero where appropriate, scales are consistent, and area or volume encodings are not used to exaggerate differences between categories.

Interquartile range and outliers

The IQR is the upper quartile minus the lower quartile; values beyond 1.5 IQR below Q1 or above Q3 are conventionally flagged as outliers.

Residual plot diagnostics

Curvature in residuals suggests a missing nonlinear term, fanning indicates non-constant variance, and random scatter supports the fitted linear model.

Least-squares regression line

The line minimising the sum of squared residuals has slope r times sy over sx and intercept ybar minus slope times xbar, passing through the point of means.