Junior Statistician Qualification Examination cheat sheet

Mean = Σx/n; for grouped data use class midpoints m_i with Mean = Σf_i m_i / Σf_i.

Median position = (n+1)/2 in an ordered list; for grouped data use Median = L + [(n/2 − CF)/f]·w, where L is lower class boundary, CF cumulative frequency below, f class frequency, w class width.

Mode = value with highest frequency; grouped mode = L + [(f_1 − f_0)/(2f_1 − f_0 − f_2)]·w.

Sample variance s² = Σ(x − x̄)²/(n−1); population variance σ² = Σ(x − μ)²/N. Standard deviation is the positive square root.

Coefficient of variation CV = (s/x̄)·100% compares dispersion across units; CV is undefined or unstable when x̄ is near zero.

Quartiles: Q1 at position (n+1)/4, Q3 at 3(n+1)/4; IQR = Q3 − Q1; a common outlier rule flags values below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Pearson skewness ≈ 3(mean − median)/s; positive value means right skew. A histogram with a long right tail indicates mean > median.

Bar charts present categorical frequencies; histograms present continuous class frequencies with contiguous bars; pie charts require parts summing to a whole.

Cumulative frequency ogive plots upper class boundaries against cumulative frequency; the median is read at cumulative frequency n/2.

Axioms: 0 ≤ P(A) ≤ 1; P(S) = 1; for disjoint events P(∪A_i) = ΣP(A_i).

Addition rule: P(A∪B) = P(A) + P(B) − P(A∩B). Conditional: P(A|B) = P(A∩B)/P(B), valid when P(B) > 0.

Independence: P(A∩B) = P(A)P(B), equivalently P(A|B) = P(A).

Bayes: P(A|B) = P(B|A)P(A) / Σ_i P(B|A_i)P(A_i) over a partition.

Binomial: P(X=k) = C(n,k)p^k(1−p)^(n−k), mean np, variance np(1−p); requires fixed n, independent trials, constant p.

Poisson: P(X=k) = e^(−λ)λ^k/k!, mean = variance = λ; limit of binomial when n large and p small with λ = np.

Normal: standardize z = (x − μ)/σ; empirical rule gives about 68%, 95%, 99.7% within 1σ, 2σ, 3σ.

Exponential: f(x) = λe^(−λx), x ≥ 0; mean = 1/λ, variance = 1/λ²; memoryless property.

Expectation is linear: E(aX + bY) = aE(X) + bE(Y); Var(aX + bY) = a²Var(X) + b²Var(Y) + 2abCov(X,Y), with Cov = 0 when independent.

An estimator is unbiased if E(θ̂) = θ; efficiency compares variance among unbiased estimators; MSE = Var(θ̂) + bias².

Standard error of the mean: SE = σ/√n, estimated by s/√n. Standard error shrinks with √n, not n.

Confidence interval for μ with known σ: x̄ ± z_(1−α/2)·σ/√n; use t_(n−1) when σ is estimated from the sample.

CI for a proportion: p̂ ± z_(1−α/2)·√(p̂(1−p̂)/n); the Wald interval is unreliable when np̂ or n(1−p̂) is small.

Margin of error halves only if n is quadrupled; required n = (z·σ/E)² for a target margin E.

CI for a variance uses (n−1)s²/χ²_(n−1) with separate lower and upper chi-square quantiles.

Central limit theorem: for large n the sampling distribution of x̄ is approximately normal regardless of population shape, with mean μ and variance σ²/n.

A 95% confidence interval means that 95% of intervals built this way cover the parameter; it does not state that the parameter has 95% probability of lying in this interval.

Structure: H0 and H1 set before data; α is chosen in advance; the p-value is P(data at least as extreme | H0), not P(H0 | data).

Reject H0 when p-value < α, or equivalently when the test statistic falls in the rejection region.

Type I error = rejecting a true H0, probability α; Type II error = failing to reject a false H0, probability β; power = 1 − β.

One-sample z: z = (x̄ − μ0)/(σ/√n). One-sample t: t = (x̄ − μ0)/(s/√n) on n−1 df.

Two-sample t (pooled): t = (x̄1 − x̄2)/[s_p√(1/n1 + 1/n2)], with s_p² = [(n1−1)s1² + (n2−1)s2²]/(n1+n2−2), df = n1+n2−2.

Paired data: analyze differences d_i with a one-sample t test on d̄ and s_d, df = n−1.

Chi-square goodness of fit: χ² = Σ(O − E)²/E, df = k − 1 − (number of estimated parameters); expected counts should be at least 5.

Power increases with larger effect size, larger n, larger α, and smaller σ. A one-sided test has higher power than a two-sided test at the same α when the direction is correct.

Failing to reject H0 is not evidence that H0 is true; report the CI alongside the test to show the plausible effect range.

Simple linear regression: ŷ = b0 + b1x, with b1 = Sxy/Sxx = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)² and b0 = ȳ − b1x̄.

The least-squares line passes through (x̄, ȳ); residuals e_i = y_i − ŷ_i have mean zero and are orthogonal to x.

Coefficient of determination R² = SSR/SST = 1 − SSE/SST, the proportion of variation in y explained by the model; 0 ≤ R² ≤ 1.

Pearson correlation r = Sxy/√(Sxx·Syy), with −1 ≤ r ≤ 1; r² = R² in simple linear regression. Correlation measures linear association only.

Test for zero slope: t = b1/SE(b1) on n − 2 df; equivalent to testing H0: ρ = 0 for Pearson r.

Prediction interval for a new observation is wider than the confidence interval for the mean response at the same x.

Standard error of estimate s_e = √[SSE/(n−2)]; residual plots reveal non-linearity and unequal variance.

Spearman rank correlation applies Pearson's formula to ranks and detects monotone, not necessarily linear, association.

Simple random sampling: every subset of size n is equally likely; requires a sampling frame.

Systematic sampling selects every k-th unit after a random start, k ≈ N/n; periodic frames can bias results.

Stratified sampling divides the population into strata and samples within each; it reduces variance when strata are internally homogeneous.

Cluster sampling samples whole groups; it is usually less precise than simple random sampling but cheaper per unit.

Design principles: randomization, replication, and blocking. Blocking controls known nuisance variation.

Completely randomized design compares treatments by random assignment; randomized complete block design assigns each treatment once per block.

Factorial design with two factors at a and b levels has ab treatment combinations; an interaction means the effect of one factor depends on the level of the other.

Sample size for a mean: n = (z·σ/E)²; for a proportion: n = z²p(1−p)/E², using p = 0.5 when p is unknown.

Nonresponse and undercoverage create bias that a larger sample does not fix; increasing n mainly reduces random sampling error.

Sign test uses the count of positive differences against Binomial(n, 0.5), ignoring zeros or dropping them from n.

Wilcoxon signed-rank test ranks absolute differences and compares the sum of positive ranks to its null distribution; assumes symmetry of differences.

Mann-Whitney U (Wilcoxon rank-sum) compares two independent samples; equivalent to the rank-sum test with normal approximation for large n.

Kruskal-Wallis test extends the rank-sum test to k independent groups; statistic approximately χ² with k − 1 df.

Friedman test is the nonparametric analog of a randomized block design, ranking within blocks across treatments.

Chi-square test of independence: χ² = Σ(O − E)²/E with E_ij = (row total × column total)/n, df = (r−1)(c−1).

Fisher's exact test is used for 2×2 tables with small expected counts; it computes exact hypergeometric probabilities.

McNemar's test handles paired binary data, using only the discordant pairs b and c, with χ² = (b − c)²/(b + c).

Relative risk and odds ratio: RR = [a/(a+b)]/[c/(c+d)]; OR = (a·d)/(b·c). OR approximates RR when the outcome is rare.

Additive decomposition: Y_t = T_t + S_t + C_t + I_t; multiplicative: Y_t = T_t × S_t × C_t × I_t.

Simple moving average of order k smooths a series; centered moving average of order 4 removes quarterly seasonal effects.

Seasonal index by ratio-to-moving-average: divide each observation by the centered moving average, then average by season and adjust so indices sum to k (or 400%).

Exponential smoothing: Ŝ_t = αY_t + (1−α)Ŝ_(t−1); larger α gives more weight to recent observations.

Laspeyres price index = (Σp_t q_0 / Σp_0 q_0) × 100; uses base-period quantities and tends to overstate price rises.

Paasche price index = (Σp_t q_t / Σp_0 q_t) × 100; uses current quantities and tends to understate price rises.

Fisher ideal index = √(Laspeyres × Paasche); it satisfies the time-reversal and factor-reversal tests.

Chain index links period-to-period relatives: multiply successive relatives, or rebase by dividing each index by the new base value and multiplying by 100.

Deflation converts nominal to real values: real value = nominal value / price index × 100.

Data types: numeric (continuous, discrete), categorical (nominal, ordinal), and dates. Type errors silently corrupt means and model fits.

Missing values need an explicit rule: complete-case analysis, mean or model imputation, or multiple imputation. Mean imputation shrinks variance.

Long vs wide format: long format stacks repeated observations with an ID column; wide format puts repeated measures in separate columns.

Reproducibility requires a fixed random seed, a recorded software and package version, and a scripted, not manual, workflow.

Floating-point arithmetic is inexact: compare with a tolerance such as |a − b| < 1e-8 rather than exact equality.

Outlier screening uses the IQR rule or z-scores; flag rather than delete, and document any exclusion.

Recoding categorical variables: dummy coding uses k − 1 indicators for k levels to avoid perfect collinearity with an intercept.

Sorting before computing cumulative sums, ranks, or lags matters; a lag operation depends on correct chronological ordering.

Validation of a data set checks row counts, ranges, duplicate keys, and cross-field consistency before analysis.