150 practice questions and 100 flashcards covering core statistics from data collection to inference and regression.
Statistics 101 is one of the most failed first-year university subjects in South Africa and one of the most practically valuable. Tour practice set covers the full first-year statistics curriculum: descriptive statistics, probability, probability distributions (binomial, Poisson, normal), sampling and estimation (confidence intervals), and hypothesis testing (z-tests, t-tests, chi-square).
The most common failure point is understanding p-values and hypothesis testing conceptually. Every explanation in tour set uses plain language first, then the formal definition, because the conceptual understanding is what the exam actually tests, not just the mechanics of the formula.
Statistics 101. Covers the fundamentals of statistics: collecting and summarising data, probability theory, probability distributions, sampling, confidence intervals, hypothesis testing, and regression analysis. Essential for any quantitative field.
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Question 01 of 150
What is the mean of 2, 4, 6, 8, 10?
Explanation: Mean = sum/count = (2+4+6+8+10)/5 = 30/5 = 6.
Question 02 of 150
What is the median of 3, 1, 4, 1, 5, 9, 2?
Explanation: Sort: 1,1,2,3,4,5,9. Middle value (4th of 7) = 3.
Question 03 of 150
What is the mode of 2, 3, 3, 4, 5, 3, 2?
Explanation: Mode: most frequent value. 3 appears 3 times (most). Answer: 3.
Question 04 of 150
What is the range of 10, 3, 7, 15, 2?
Explanation: Range = max − min = 15 − 2 = 13.
Question 05 of 150
What is variance?
Explanation: Variance σ² = Σ(x−μ)²/N (population) or Σ(x−x̄)²/(n−1) (sample). Standard deviation = √variance.
Question 06 of 150
What is standard deviation?
Explanation: SD: most used measure of spread. Small SD: values clustered near mean. Large SD: values widely spread. σ (population), s (sample).
Question 07 of 150
What is the difference between population and sample?
Explanation: Population: all (students in SA). Parameter: population measure (μ, σ). Sample: subset (100 students). Statistic: sample measure (x̄, s). Inference: use sample to estimate population.
Question 08 of 150
What is a normal distribution?
Explanation: Normal: 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD (empirical rule). Used for: height, IQ, measurement errors. Central Limit Theorem: sample means approach normal.
Question 09 of 150
What does the empirical rule state for normal distribution?
Explanation: Empirical rule (68-95-99.7): benchmark for spotting outliers. Value beyond 3σ: very rare (0.3%). Used for quality control, process monitoring.
Question 10 of 150
What is a z-score?
Explanation: z-score: z=2 means 2 SDs above mean. Negative z: below mean. Used to compare values from different distributions. P(Z<1.96) ≈ 0.975.
Question 11 of 150
What is probability?
Explanation: P(A): between 0 and 1. P(A) = favourable outcomes / total outcomes. P(A) + P(not A) = 1.
Question 12 of 150
What is conditional probability?
Explanation: P(A|B): 'probability of A given B.' Bayes' theorem: P(A|B) = P(B|A)·P(A)/P(B). Key in medical testing, spam filters.
Question 13 of 150
What are mutually exclusive events?
Explanation: Mutually exclusive: rolling a 3 OR a 5 on one die. Can't get both. P(A∪B) = P(A)+P(B). Not same as independent!
Question 14 of 150
What are independent events?
Explanation: Independent: flipping heads twice. P(H1 and H2) = 0.5×0.5 = 0.25. Dependent: drawing cards without replacement.
Question 15 of 150
What is the binomial distribution?
Explanation: Binomial: X~B(n,p). P(X=k) = C(n,k)·p^k·(1-p)^(n-k). Mean = np, Variance = np(1-p). Example: number of heads in 10 flips.
Question 16 of 150
What is the Poisson distribution?
Explanation: Poisson: X~Po(λ). P(X=k) = e^(-λ)·λ^k/k!. Mean = Variance = λ. Used for: phone calls per hour, defects per unit, accidents per year.
Question 17 of 150
What is a confidence interval?
Explanation: CI: x̄ ± z*(σ/√n) for mean. 95% CI: z* = 1.96. Wider CI: more confident but less precise. Affected by: sample size (larger n = narrower), confidence level, variability.
Question 18 of 150
What does a 95% confidence interval mean?
Explanation: Frequentist interpretation: the CI either contains the true parameter or doesn't. 95% of intervals constructed tour way will contain it. NOT: 95% chance the parameter is in tour specific interval.
Question 19 of 150
What is hypothesis testing?
Explanation: H₀ (null hypothesis): no effect/difference. H₁ (alternative): effect exists. Calculate test statistic → p-value. If p < α: reject H₀. Never 'accept' H₀.
Question 20 of 150
What is a p-value?
Explanation: p-value: p < 0.05: statistically significant (reject H₀ at 5% level). p = 0.03: if H₀ true, only 3% chance of seeing results tour extreme. Lower p = stronger evidence against H₀.
Question 21 of 150
What is a Type I error?
Explanation: Type I error (α): false positive. e.g., concluding treatment works when it doesn't. α = 0.05: 5% chance of Type I. Reduce α → increase β (Type II). Trade-off.
Question 22 of 150
What is a Type II error?
Explanation: Type II error (β): false negative. Missing a real effect. Power = 1-β: probability of correctly detecting effect. Increase n → more power → lower β.
Question 23 of 150
What is statistical power?
Explanation: Power = 1 - β. Affected by: sample size (larger n = more power), effect size (larger effect easier to detect), significance level (higher α = more power but more Type I errors).
Question 24 of 150
What is a t-test?
Explanation: One-sample t: compare sample mean to hypothesised value. Independent samples t: compare two group means. Paired t: compare same subjects in two conditions. Assumes normality.
Question 25 of 150
What is ANOVA?
Explanation: ANOVA: H₀: all group means equal. F = variance between groups / variance within groups. Large F → reject H₀. If significant: post-hoc tests (Tukey, Bonferroni) identify which groups differ.
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Question 26 of 150
What is the chi-square test?
Explanation: Chi-square: χ² = Σ(O-E)²/E. Goodness-of-fit: does distribution match expected? Independence: are variables related? Requires expected count ≥ 5 per cell.
Question 27 of 150
What is correlation?
Explanation: Pearson r: +1 perfect positive, 0 no linear relationship, -1 perfect negative. r = 0.7: strong positive. Correlation ≠ causation. Scatterplot before concluding.
Question 28 of 150
What is simple linear regression?
Explanation: Regression: find best-fit line minimising sum of squared residuals (OLS). b = slope (change in y per unit x). a = intercept. R²: proportion of variance explained.
Question 29 of 150
What is R² (R-squared) in regression?
Explanation: R²: R²=0.75 means 75% of y variance explained by x. Higher is better. R² = r² for simple regression. Adjusted R²: penalises for additional predictors (multiple regression).
Question 30 of 150
What is an outlier?
Explanation: Outlier: Z-score >3 or <-3, or beyond 1.5×IQR from quartiles. Investigate: data entry error (remove), genuine extreme value (keep, use robust methods), influential point in regression.
Explanation: Box plot: box spans Q1-Q3 (IQR). Line = median. Whiskers extend to min/max (or 1.5×IQR). Dots beyond whiskers: outliers. Useful for comparing distributions.
Question 33 of 150
What is a histogram?
Explanation: Histogram: bins (class intervals) on x-axis, frequency/relative frequency on y-axis. Shape: symmetric, right-skewed, left-skewed, bimodal. Shows: distribution shape, modality, spread.
Question 34 of 150
What is sampling error?
Explanation: Sampling error: random variability. SE of mean = σ/√n. Larger n → smaller SE. Different from sampling bias (systematic error due to non-random sampling).
Question 35 of 150
What is a sampling distribution?
Explanation: Sampling distribution of x̄: mean = μ, SD = σ/√n. CLT: regardless of population distribution, sampling distribution of x̄ approaches normal as n increases (usually n≥30).
Question 36 of 150
What is the Central Limit Theorem?
Explanation: CLT: critical foundation for inference. If n≥30: can use z-test even if population not normal. Explains why normal distribution appears everywhere.
Question 37 of 150
What is a left-skewed distribution?
Explanation: Left-skewed (negative skew): tail pulls mean left. Mean < Median. e.g., exam scores where most students score high but a few fail badly. Use median as better central tendency measure.
Question 38 of 150
What is a right-skewed distribution?
Explanation: Right-skewed (positive skew): tail pulls mean right. Mean > Median. e.g., income, house prices, reaction times. Median better measure of centre than mean for skewed data.
Question 39 of 150
What is the difference between discrete and continuous data?
Explanation: Discrete: finite or countably infinite values. Count data. Continuous: uncountably infinite values in a range. Measurement data. Different distributions apply to each.
Question 40 of 150
What is the null hypothesis?
Explanation: H₀: μ = 0, or μ₁ = μ₂, or no relationship. We test whether data provides enough evidence to reject H₀. We never 'prove' H₀ — we fail to reject it.
Question 41 of 150
What is one-tail vs two-tail test?
Explanation: Two-tail: H₁: μ ≠ μ₀. Critical regions in both tails. One-tail: H₁: μ > μ₀ or μ < μ₀. All critical region in one tail. One-tail more powerful when direction known in advance.
Explanation: Multiple regression: controls for confounders. b₁: effect of x₁ holding x₂ constant. Adjusted R²: accounts for number of predictors. Assumptions: linearity, independence, homoscedasticity, normality.
Question 44 of 150
What is the law of large numbers?
Explanation: Law of large numbers: n→∞, x̄→μ. Practical: larger samples more reliable. Not same as CLT (which says shape of sampling distribution approaches normal).
Question 45 of 150
What is a frequency distribution?
Explanation: Frequency distribution: class, frequency, relative frequency (%), cumulative frequency. Foundation for histogram. Shows data spread.
Question 46 of 150
What is weighted mean?
Explanation: Weighted mean = Σ(wᵢxᵢ)/Σwᵢ. GPA: 3-credit course weighted more than 1-credit. Portfolio return: weighted by investment size.
Question 47 of 150
What is a percentile?
Explanation: Percentile: 90th percentile on a test = scored better than 90% of test-takers. Q1=25th percentile, median=50th, Q3=75th percentile.
Question 48 of 150
What is Bayes' theorem?
Explanation: Bayes: medical testing. P(disease|positive test) = P(positive test|disease)×P(disease)/P(positive test). Even accurate tests can have high false positive rate if disease is rare.
Question 49 of 150
What is the coefficient of variation (CV)?
Explanation: CV = (s/x̄) × 100%. CV = 10%: SD is 10% of mean. Compare variability of stock portfolios with different average returns. Unitless.
Question 50 of 150
What is regression to the mean?
Explanation: Regression to mean: very tall parents tend to have children closer to average height. Extreme sports performances often followed by more average ones. Important in interpreting 'improvement.'
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Question 51 of 150
What is the F-test in ANOVA?
Explanation: F = MSB/MSW. MSB (mean square between): variance due to group differences. MSW (mean square within): variance within groups. F-distribution used. If F > critical value: reject H₀.
Question 52 of 150
What is logistic regression?
Explanation: Logistic regression: predict probability. Output: log-odds, converted to probability via sigmoid. Used for: disease diagnosis, spam detection, credit default. Coefficients: log-odds ratios.
Question 53 of 150
What is a non-parametric test?
Explanation: Non-parametric: Mann-Whitney U (t-test alternative), Kruskal-Wallis (ANOVA alternative), Spearman correlation (Pearson alternative), Wilcoxon signed-rank. Use when normality assumption violated.
Question 54 of 150
What is Spearman's rank correlation?
Explanation: Spearman ρ: ranks data first, then calculates Pearson on ranks. Use when: data ordinal, or not normally distributed, or outliers present. Same interpretation as Pearson r.
Question 55 of 150
What is a Type I error rate in multiple comparisons?
Explanation: Multiple comparisons: 20 tests at α=0.05 → 64% chance of at least one false positive. Bonferroni: α/k (k=number of tests). More conservative → less power.
Question 56 of 150
What is a confidence level vs significance level?
Explanation: Confidence level = 1 - α. 95% confidence corresponds to α = 0.05. Higher confidence level → wider CI. Choosing α is a research decision before data collection.
Question 57 of 150
What is a probability density function (PDF)?
Explanation: PDF: for continuous variables. P(a
Question 58 of 150
What is the expected value?
Explanation: E(X): for discrete: Σx·P(x). For continuous: ∫x·f(x)dx. Properties: E(aX+b) = aE(X)+b, E(X+Y) = E(X)+E(Y). For n trials: E(success) = np.
Question 59 of 150
What is stratified sampling?
Explanation: Stratified: gender, age group, region strata. Each stratum sampled separately. More precise than simple random sampling when strata differ. Proportional or equal allocation.
Question 60 of 150
What is cluster sampling?
Explanation: Cluster: geographically based (cities, schools). Select clusters randomly, survey entire cluster. Less precise than stratified but cheaper when population spread over large area.
Question 61 of 150
What is systematic sampling?
Explanation: Systematic: k = N/n. If N=1000, n=100: k=10. Random start between 1-10, then every 10th. Efficient. Risk: periodicity in population aligns with interval.
Question 62 of 150
What is sampling bias?
Explanation: Sampling bias: voluntary response (internet polls — extreme opinions over-represented), convenience (sample students from one class), non-response (certain groups don't respond).
Question 63 of 150
What is an observational study vs an experiment?
Explanation: Experiment: random assignment to treatment/control → confounders balanced → causation inferred. Observational: no manipulation → confounders remain → only association established.
Question 64 of 150
What is a confounding variable?
Explanation: Confounding: ice cream sales and drownings both increase in summer → confounded by temperature (hot weather). Solution: randomisation in experiments, statistical control in observational studies.
Question 65 of 150
What is a box-and-whisker plot?
Explanation: Box plot: box = IQR (Q1-Q3). Line inside box = median. Whiskers extend to min/max or 1.5×IQR. Points beyond whiskers = outliers. Useful for comparing groups.
Explanation: Multiplication rule: P(A∩B) = P(A)·P(B|A). Independent: P(A∩B) = P(A)·P(B). e.g., P(H∩H) = 0.5×0.5 = 0.25 for two coin flips.
Question 69 of 150
What is a five-number summary?
Explanation: Five-number summary: basis for box plot. Min, Q1 (25th %), Median (50th %), Q3 (75th %), Max. Captures centre, spread, and extremes.
Question 70 of 150
What is kurtosis?
Explanation: Kurtosis: leptokurtic (excess kurtosis > 0): more extreme values, peaked. Platykurtic (< 0): fewer extremes, flat. Normal (mesokurtic) = 0. Important in finance (extreme events).
Question 71 of 150
What is skewness?
Explanation: Skewness: positive (right-skewed): income, house prices. Negative (left-skewed): age at retirement. If skewed: mean pulled toward tail, use median for central tendency.
Question 72 of 150
What is a lurking variable?
Explanation: Lurking variable: not measured but affects results. Both shoe size and reading ability increase with age (lurking variable). Nicholas Cage films and pool drownings (both increase over summer).
Question 73 of 150
What is effect size?
Explanation: Effect size: Cohen's d: small=0.2, medium=0.5, large=0.8. Statistical significance depends on n. Large n → significant even trivially small effect. Effect size: practical importance.
Question 74 of 150
What is the law of total probability?
Explanation: Total probability: partition sample space into mutually exclusive events B₁,...,Bₙ. P(A) = P(A|B₁)P(B₁) + P(A|B₂)P(B₂) + ... Foundation for Bayes' theorem.
Question 75 of 150
What is the difference between mean deviation and standard deviation?
Explanation: Mean deviation = Σ|x-x̄|/n. SD = √(Σ(x-x̄)²/n). SD penalises large deviations more (squaring). SD is mathematically more tractable, used in most statistical tests.
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Question 76 of 150
What is a two-sample t-test?
Explanation: Independent samples t: H₀: μ₁ = μ₂. Test statistic: (x̄₁-x̄₂)/SE. Welch's t-test: doesn't assume equal variances (use when in doubt). Effect size: Cohen's d.
Question 77 of 150
What is the p-value interpretation error?
Explanation: p-value misconception: p=0.03 does NOT mean 97% chance H₀ is false. It means: if H₀ were true, 3% chance of getting results tour extreme or more. Don't confuse p(data|H₀) with p(H₀|data).
Question 78 of 150
What is a residual in regression?
Explanation: Residual = actual - predicted. Residual plot: should be random scatter around 0. Patterns indicate: non-linearity, heteroscedasticity, omitted variables.
Question 79 of 150
What is the geometric mean?
Explanation: Geometric mean = (x₁×x₂×...×xₙ)^(1/n). For growth rates: average of 20%, 30%, -10% = (1.2×1.3×0.9)^(1/3) - 1 = 10.6%. Arithmetic mean overstates.
Question 80 of 150
What is Simpson's paradox?
Explanation: Simpson's paradox: UC Berkeley admissions — women accepted at lower overall rate but higher rate in most departments (different departments applied to). Always examine subgroups.
Question 81 of 150
What is heteroscedasticity?
Explanation: Heteroscedasticity: residuals fan out (or in) as x increases. e.g., income variance increases with age. Detect with residual plot. Fix: weighted least squares, log transformation.
Explanation: Interaction: fertiliser effect on crop yield depends on water level. Include x₁×x₂ term. Significant interaction: effects of x₁ and x₂ can't be interpreted independently.
Question 84 of 150
What is overfitting?
Explanation: Overfitting: too complex model. High R² in sample, poor predictions out of sample. Detect: test set performance. Fix: cross-validation, regularisation (ridge/lasso), simpler model.
Question 85 of 150
What is cross-validation?
Explanation: k-fold CV: split data into k folds. Train on k-1, test on 1. Rotate k times. Average test error = CV estimate. LOOCV: leave one out. Better estimate than single train-test split.
Question 86 of 150
What is a paired t-test?
Explanation: Paired t: H₀: mean difference = 0. More powerful than independent t when within-subject variability is less than between-subject variability. e.g., blood pressure before/after medication.
Question 87 of 150
What is a Likert scale?
Explanation: Likert: 1=Strongly disagree to 5=Strongly agree. Technically ordinal. Many researchers treat as interval (calculate means). More rigorous: use ordinal methods (median, Mann-Whitney).
Question 88 of 150
What is the difference between correlation and causation?
Explanation: Correlation ≠ causation. Both shoe size and reading ability correlate with age (confound). Establish causation: randomised experiment, temporal order, plausible mechanism, no confounding.
Question 89 of 150
What is a stem-and-leaf plot?
Explanation: Stem-and-leaf: 23 → stem 2, leaf 3. Shows shape like histogram but retains original values. Back-to-back: compare two distributions.
Question 90 of 150
What is the coefficient of determination?
Explanation: R² = r² (simple regression). R² = 1: perfect prediction. R² = 0: X explains nothing. Adjusted R²: penalises extra predictors in multiple regression.
Question 91 of 150
What is a sampling frame?
Explanation: Sampling frame: telephone directory (excludes non-landline households). Student list (excludes dropouts). Quality of sample depends on quality of frame.
Question 92 of 150
What is a cumulative frequency distribution?
Explanation: Cumulative frequency: how many observations are below a given value. Ogive: line graph of cumulative relative frequency. Used to find: median, percentiles, proportions.
Question 93 of 150
What is a random variable?
Explanation: Random variable: X = sum of two dice. P(X=7) = 6/36. Expected value E(X) = Σx·P(x). Discrete: PMF. Continuous: PDF and CDF.
Question 94 of 150
What is the complement rule?
Explanation: Complement: P(Aᶜ) = 1 - P(A). Often easier: P(at least one success) = 1 - P(no success). P(at least one 6 in 4 rolls) = 1 - (5/6)⁴.
Question 95 of 150
What is a two-way table (contingency table)?
Explanation: Contingency table: rows = one variable, columns = another. Joint frequencies, marginal totals. Chi-square test of independence uses contingency table.
Question 96 of 150
What is the standard error of the mean?
Explanation: SE = σ/√n. Doubles n → SE decreases by factor of √2. SE is SD of sampling distribution. Used in CI: x̄ ± z·SE. Smaller SE → more precise estimate.
Question 97 of 150
What is a quartile?
Explanation: Quartiles: Q1, Q2, Q3. IQR = Q3-Q1. Box plot whiskers extend to min/max (within 1.5 IQR). Same as 25th, 50th, 75th percentiles.
Question 98 of 150
What is the difference between descriptive and inferential statistics?
Explanation: Descriptive: what does tour data show? Summarise. Inferential: what can we conclude about the population? Test hypotheses, estimate parameters.
Question 99 of 150
What is experimental design?
Explanation: Experimental design: Random assignment (prevents confounding), control group (baseline), replication (multiple observations), blinding (prevent bias — single/double-blind).
Question 100 of 150
What is the t-distribution?
Explanation: t-distribution: df (degrees of freedom) = n-1. Heavier tails = more probability in extremes. As df → ∞, t → z. For n<30 or σ unknown: use t instead of z.
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Question 101 of 150
What is a point estimate?
Explanation: Point estimate: precise but uncertain (no error bound). Interval estimate (CI): range that likely contains parameter. Both needed: point for best guess, CI for uncertainty.
Question 102 of 150
What is statistical significance vs practical significance?
Explanation: With large n, tiny effects become statistically significant (p < 0.05) but may be practically meaningless. Always report effect size alongside p-value.
Question 103 of 150
What is the Law of Complementary Events?
Explanation: Complement: P(getting at least one head in 3 flips) = 1 - P(no heads) = 1 - (0.5)³ = 0.875. Often easier to calculate complement.
Question 104 of 150
What is the median vs mean in skewed distributions?
Explanation: Use median for skewed data or when outliers present. Use mean for symmetric distributions. Income: median household income is more meaningful than mean (skewed right by billionaires).
Question 105 of 150
What is a bimodal distribution?
Explanation: Bimodal: two modes/peaks. e.g., heights of mixed male and female population. Suggests: mixture of two distributions. Statistical tests assuming normality may be invalid.
Question 106 of 150
What is an underfitted model?
Explanation: Underfitting (high bias): linear model for non-linear data. Poor R². Under-learns. Fix: more complex model, add variables, polynomial terms, interaction effects.
Question 107 of 150
What is probability sampling?
Explanation: Probability sampling: SRS, stratified, cluster, systematic. Non-probability: convenience, purposive, snowball. Only probability sampling supports valid statistical inference.
Question 108 of 150
What is an empirical distribution?
Explanation: Empirical distribution: actual data distribution. Empirical CDF (eCDF): step function from data. Compare to theoretical distribution (normal, t) — if they match, theoretical applies.
Question 109 of 150
What is a confidence interval for a proportion?
Explanation: Proportion CI: 95%: p̂ ± 1.96√(p̂(1-p̂)/n). Margin of error E = z*√(p̂(1-p̂)/n). Required n = (z*/E)² × p̂(1-p̂).
Question 110 of 150
What is the difference between a census and a survey?
Explanation: Census: complete enumeration (expensive, time-consuming). Survey: sample-based estimate. Census Bureau: every 10 years. Surveys: faster, cheaper, used for ongoing monitoring.
Question 111 of 150
What is a lurking variable?
Explanation: Lurking variable: both chocolate consumption and Nobel prizes correlate with wealth/development. Control for wealth: chocolate-Nobel link disappears. Always think about what you haven't measured.
Question 112 of 150
What is the normal approximation to the binomial?
Explanation: Normal approx to binomial: use when np≥5 and n(1-p)≥5. Continuity correction: P(X≤k) ≈ P(Z≤(k+0.5-np)/√(np(1-p))).
Question 113 of 150
What is a data distribution's kurtosis excess?
Explanation: Excess kurtosis = kurtosis - 3. Normal: excess = 0. Leptokurtic: >0 (fat tails). Platykurtic: <0 (thin tails). Used in finance for tail risk assessment.
Question 114 of 150
What is regression toward the mean with an example?
Explanation: e.g., tallest players in a game often shorter the next game (random variation component). Kids of very tall parents closer to average height. Explains why 'Sports Illustrated Cover Jinx' appears to exist.
Question 115 of 150
What is a p-value threshold and its limitations?
Explanation: p=0.05 convention (Fisher): arbitrary. p=0.049 ≠ real effect; p=0.051 ≠ no effect. ASA statement (2016): don't use p>0.05 to mean 'no association.' Report effect sizes and CIs.
Question 116 of 150
What is a one-way ANOVA assumption?
Explanation: ANOVA assumptions: 1) Independent observations. 2) Normal distribution within each group. 3) Homogeneity of variance (Levene's test). Violations: use Welch ANOVA or Kruskal-Wallis.
Question 117 of 150
What is the rule of thumb for sample size in hypothesis testing?
Explanation: Sample size depends on: desired power (usually 0.80), effect size, α level. Use power analysis (G*Power) for proper calculation. n=30 is a rough rule for CLT, not hypothesis testing.
Question 118 of 150
What is Chebyshev's inequality?
Explanation: Chebyshev: k=2 → at least 75% within 2 SDs (vs 95% for normal). k=3 → at least 89% within 3 SDs. Applies to ANY distribution, unlike empirical rule (normal only).
Question 119 of 150
What is a t-test for one sample?
Explanation: One-sample t: df=n-1. e.g., Is the mean weight of bags different from 500g? H₀: μ=500. Calculate t, compare to t-critical or p-value.
Question 120 of 150
What is the Wilcoxon signed-rank test?
Explanation: Wilcoxon signed-rank: use when paired t-test assumptions not met. Ranks |differences|. Assigns signs. Test statistic W. Use when differences not normally distributed.
Question 121 of 150
What is a histogram bin width?
Explanation: Bin width: balance between noise and smoothness. Sturges' rule: k = 1+log₂(n) bins. Scott's rule: bin width = 3.5s/n^(1/3). Freedman-Diaconis: 2×IQR/n^(1/3). Trial and error works too.
Question 122 of 150
What is the Mann-Whitney U test?
Explanation: Mann-Whitney U (Wilcoxon rank-sum): no normality assumption. Ranks all observations, compares rank sums. Use when t-test assumptions violated with two independent groups.
Question 123 of 150
What is a 99% confidence interval compared to 95%?
Explanation: 99% CI uses z*=2.576 vs z*=1.96 for 95%. 99% CI is wider. Trade-off: more confidence = less precision. Typically report 95% CI; use 99% for high-stakes decisions.
Question 124 of 150
What is covariance?
Explanation: Cov(X,Y) = Σ(xᵢ-x̄)(yᵢ-ȳ)/(n-1). Units: product of X and Y units (hard to interpret). Pearson r = Cov(X,Y)/(SD_x × SD_y) — standardised version.
Question 125 of 150
What is the Kolmogorov-Smirnov test?
Explanation: KS test: tests whether sample comes from specified distribution. KS statistic: max difference between empirical and theoretical CDF. Also: two-sample KS for comparing two distributions.
Explanation: Bonferroni: 10 tests at α=0.05 → each test at α=0.005. Conservative (may miss real effects). Alternative: Benjamini-Hochberg (controls false discovery rate, less conservative).
Question 129 of 150
What is a power analysis?
Explanation: Power analysis: specify α (usually 0.05), power (usually 0.80), effect size (Cohen's d for t-test). Calculate minimum n. Underpowered studies waste resources and miss real effects.
Question 130 of 150
What is a confidence interval for a difference in means?
Explanation: Two-sample CI: if 95% CI for μ₁-μ₂ = (2, 8): statistically significant (doesn't include 0), practical: difference of 2-8 units. Contains 0: cannot reject H₀.
Question 131 of 150
What is the difference between a sample statistic and a population parameter?
Explanation: Parameters: population characteristics, usually unknown. Statistics: sample estimates of parameters. Greek letters for parameters (μ, σ, π). Roman for statistics (x̄, s, p̂).
Question 132 of 150
What is sampling without replacement?
Explanation: Without replacement: common in surveys. Changes probabilities with each draw. Hypergeometric distribution (vs binomial with replacement). FPC (finite population correction) applies.
Question 133 of 150
What is a frequency polygon?
Explanation: Frequency polygon: connect tops of histogram bars at midpoints with line. Useful for overlaying multiple distributions. Smooth version: kernel density estimate (KDE).
Question 134 of 150
What is a stem-and-leaf plot advantage over histogram?
Explanation: Stem-and-leaf: can identify exact values (23, 27, 29 from stem 2). Histogram only shows frequency in bin. Back-to-back stem-and-leaf: compare two groups easily.
Question 135 of 150
What is measurement scale?
Explanation: Scales: Nominal (gender, colour). Ordinal (Likert, finish position). Interval (temperature °C — 0°C not absence of heat). Ratio (height, weight — 0 means none). Determines valid statistics.
Question 136 of 150
What is a Q-Q plot?
Explanation: Q-Q plot: if data normal, points fall on y=x diagonal. Systematic deviation: distribution not normal. Tails off diagonal: heavy/light tails. Curve: skewness.
Question 137 of 150
What is a discrete probability distribution?
Explanation: Discrete PMF: P(X=x) ≥ 0, ΣP(X=x) = 1. Examples: Binomial (n trials, k successes), Poisson (events per interval), Geometric (trials until first success).
Question 138 of 150
What is skewness and how is it measured?
Explanation: Skewness: 0 = symmetric, >1 or <-1 = highly skewed. Right-skewed (positive): income, house prices. Left-skewed (negative): exam scores (most pass, few fail badly).
Question 139 of 150
What is the geometric distribution?
Explanation: Geometric: P(X=k) = (1-p)^(k-1)×p. Mean = 1/p. e.g., Number of coin flips until first head (p=0.5): mean = 2 flips. Memoryless property.
Question 140 of 150
What is the hypergeometric distribution?
Explanation: Hypergeometric: sampling without replacement. P(X=k) = C(K,k)×C(N-K,n-k)/C(N,n). e.g., Probability of 2 defective in sample of 5 from batch of 20 with 4 defective.
Question 141 of 150
What is the relationship between sample size and margin of error?
Explanation: ME ∝ 1/√n. Double n → ME reduced by √2 ≈ 1.41. Quadruple n → ME halved. Diminishing returns: large n needed for small improvement. Budget vs precision trade-off.
Question 142 of 150
What is a normal probability plot?
Explanation: Normal probability plot (normal Q-Q plot): deviations from diagonal indicate non-normality. S-shape: heavy tails. Curves: skewness. Points beyond bounds: outliers.
Question 143 of 150
What is homoscedasticity?
Explanation: Homoscedasticity: residuals have constant spread. Residual plot: random cloud around 0. Heteroscedastic: fan or funnel shape. Violation: standard errors biased, tests invalid.
Explanation: Expected frequency: chi-square compares O (observed) to E (expected). Large |O-E|: variables not independent. Assumption: E ≥ 5 for each cell (else: Fisher's exact test).
Question 146 of 150
What is a uniform distribution?
Explanation: Uniform: U(a,b). Every value between a and b equally likely. Mean = (a+b)/2. Variance = (b-a)²/12. Continuous: rolling a fair die approximates discrete uniform.
Question 147 of 150
What is the exponential distribution?
Explanation: Exponential: time until next event. Mean = 1/λ. Memoryless: P(X>s+t|X>s) = P(X>t). Used for: wait times, equipment lifetime, radioactive decay.
Question 148 of 150
What is a Type III error?
Explanation: Type III (informal): testing wrong hypothesis. e.g., right answer to wrong question. Not universally defined but highlights importance of formulating correct H₀ and H₁.
Question 149 of 150
What is bootstrapping in statistics?
Explanation: Bootstrap: resample n observations with replacement B times (B=1000-10000). Calculate statistic each time. Distribution of statistics = bootstrap sampling distribution. CIs, SEs without distributional assumptions.
Question 150 of 150
What is the difference between descriptive and inferential statistics?
Explanation: Descriptive: mean, SD, graphs — describe what you have. Inferential: hypothesis tests, CIs — generalise beyond your data to population.
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The mean is the arithmetic average (sum ÷ count) and is affected by outliers. The median is the middle value when data is ordered and is not affected by outliers. For skewed distributions, the median is often a better measure of centre.
The p-value is the probability of observing data at least as extreme as your sample, assuming the null hypothesis is true. If the p-value ≤ significance level (usually 0.05), you reject the null hypothesis. A p-value does not measure the probability that H₀ is true.
Standard deviation measures the average distance of data values from the mean. A low standard deviation means data is clustered near the mean; a high standard deviation means data is spread out. It is the square root of variance.
In a normal distribution: 68% of data falls within ±1 standard deviation of the mean, 95% within ±2 standard deviations, and 99.7% within ±3 standard deviations. Tour is also called the empirical rule.
A Type I error is a false positive — rejecting a true null hypothesis. Its probability equals α (the significance level). A Type II error is a false negative — failing to reject a false null hypothesis. Its probability is β. Statistical power is 1 − β.
The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as sample size increases, regardless of the original population distribution. Tour is why we can use normal-based tests even when the population is not normal.
Correlation measures the strength and direction of a linear relationship between two variables (r ranges from -1 to +1). Causation means one variable directly causes changes in another. Correlation does not imply causation — confounding variables or reverse causation may explain a relationship.
A 95% confidence interval means that if the same study were repeated 100 times, approximately 95 of the resulting intervals would contain the true population parameter. It does not mean there is a 95% probability that the true value lies in any specific interval.