NCEA Level 3 Statistics cheat sheet

Random variable X has mean E(X)=Σx·P(x) and variance Var(X)=E(X²)−[E(X)]², where E(X²)=Σx²·P(x); standard deviation is the square root of the variance.

Binomial conditions: fixed number of trials n, each trial independent, constant probability p of success, two outcomes; then P(X=r)=C(n,r)·p^r·(1−p)^(n−r), E(X)=np, Var(X)=np(1−p).

Normal distribution: convert to the standard normal using z=(x−μ)/σ, then P(X<x)=P(Z<z) read from tables or software; use inverse normal to find an x-value from a given probability.

Continuity correction for a discrete variable approximated by a normal: replace P(X≤k) with P(X<k+0.5) and P(X≥k) with P(X>k−0.5) before standardising.

Inverse normal with a lower tail gives x=μ+z·σ; for an upper tail (area to the right) find z for the complementary area, or negate z, before converting back.

Combining independent random variables: E(aX+bY)=aE(X)+bE(Y); Var(aX+bY)=a²Var(X)+b²Var(Y) only when X and Y are independent (subtraction also adds variances).

Standard error of the sample mean is σ/√n; for a proportion it is √(p(1−p)/n), and using the estimated p̂ in place of p gives the estimated standard error.

Confidence interval for a mean with known σ: x̄ ± z·σ/√n; the margin of error halves only if n is quadrupled, since margin is proportional to 1/√n.

For a difference between two independent means the standard error is √(σ₁²/n₁ + σ₂²/n₂); pool variances only when a common σ is justified by the context.

Hypothesis test structure: state H₀ and Hₐ in terms of the population parameter, compute the test statistic, find the P-value, then compare with the significance level α and conclude in context.

One-sample z-test statistic for a mean: z=(x̄−μ₀)/(σ/√n); two-sided tests use 2× the one-tail area beyond the observed statistic.

P-value is the probability, assuming H₀ is true, of a result at least as extreme as observed in the direction of Hₐ; it is not the probability that H₀ is true.

Reject H₀ when P-value < α; failing to reject does not prove H₀ - it means the sample did not provide sufficient evidence against it.

Type I error: rejecting a true H₀ (probability α). Type II error: failing to reject a false H₀. Decreasing α reduces Type I risk but increases Type II risk for a fixed n.

Paired comparison: analyse the differences of matched pairs as a single sample, testing H₀: μ_d=0 with the mean and standard deviation of the differences.

Chi-square goodness of fit statistic: χ²=Σ(O−E)²/E with degrees of freedom = (number of categories − 1), valid when expected counts are at least 5.

Chi-square test of independence uses an r×c table with df=(r−1)(c−1); expected count for a cell is (row total × column total)/grand total.

Significance level α is fixed before testing; report the P-value and interpret it in context, not just as 'accept' or 'reject'.

Bivariate data: describe scatter plots for direction (positive/negative), strength (strong/weak), form (linear/nonlinear) and unusual points before calculating anything.

Correlation coefficient r lies between −1 and 1; it measures the strength of linear association only, and is not affected by which variable is explanatory.

r² (coefficient of determination) gives the proportion of variation in the response variable explained by the linear regression on the explanatory variable.

Least-squares regression line ŷ=b₀+b₁x with b₁=r·(s_y/s_x) and b₀=ȳ−b₁x̄; the line always passes through the point (x̄, ȳ).

Interpret slope as the predicted change in y for a one-unit increase in x; interpret intercept only when x=0 is within the observed data range.

Extrapolation beyond the observed range of x is unreliable; correlation does not imply causation, so a strong r may reflect a lurking variable.

Residual = observed y − predicted ŷ; a residual plot with random scatter around zero supports a linear model, while curvature or a funnel shape suggests the model is inappropriate.

Influential points change the regression line markedly; removing them should be justified and their effect on r and the line reported.

Time series: identify trend, seasonal pattern and irregular components; use moving means to smooth, and index numbers with a base period to compare values over time.

Sampling methods: simple random, systematic, stratified and cluster; describe how each is carried out and identify the bias each may introduce.

Sampling bias arises when the sample is not representative; non-sampling errors include measurement error, non-response and question wording effects.

A census measures every unit in the population and avoids sampling error but is costly and may increase non-sampling error.

Sample size affects the precision of estimates, not their expected value: larger n reduces the standard error but does not remove bias.

The central limit theorem: for large n the distribution of the sample mean is approximately normal with mean μ and standard error σ/√n, regardless of the population shape.

Design an experiment with control and treatment groups, random allocation and, where possible, blinding and blocking to reduce confounding.

In an observational study, associations can be described but causal claims require the explanatory variable to be randomly assigned.

Margin of error for a proportion is z·√(p̂(1−p̂)/n); the largest margin occurs at p̂=0.5, so n=z²/(4m²) approximates the sample size needed for margin m.

Probability rules: P(A∪B)=P(A)+P(B)−P(A∩B); for mutually exclusive events P(A∩B)=0; P(A')=1−P(A).

Conditional probability P(A|B)=P(A∩B)/P(B); events are independent when P(A∩B)=P(A)·P(B), equivalently P(A|B)=P(A).

A two-way table can be converted to probabilities by dividing cell counts by the grand total; marginal probabilities use row or column totals.

Tree diagrams multiply along branches and add across branches for mutually exclusive paths; label each branch with a conditional probability.

Venn diagrams and contingency tables are interchangeable representations of the same two-event probability information.

Expected value of a game or decision is Σ(outcome × probability); a decision is favourable when expected net gain is positive, but expected value ignores risk and variability.

Risk and relative risk: relative risk = risk in exposed group / risk in unexposed group; a value of 1 indicates no difference in risk.

Report statistical findings with the estimate, its uncertainty (confidence interval or standard error) and the sample size, then interpret in the context of the problem.

Round final answers to an appropriate number of significant figures (usually 3) and keep unrounded values in intermediate calculations.

When comparing groups, describe shape, centre and spread, and note that a difference in means may be explained by variability as well as a real effect.

Use units and clearly defined variables in every written interpretation; a conclusion must refer back to the original research question.

Distinguish statistical significance from practical importance: a very small P-value in a large sample may correspond to a trivial effect size.

Effect size or the difference relative to the standard deviation helps judge practical importance when samples are large.