Practice and prepare
Practice for what you need today
Strengthen one standard, practice choosing methods, or rehearse a complete Test Run.
Choose how to practice →Module 1
Descriptive Statistics
Vocabulary, graphs, and numerical summaries — the language every statistical argument is built on.
L-DS-1
Statistical Vocabulary and Sampling
No prerequisites — Entry point Learning Outcome
Use appropriate statistical vocabulary and identify the characteristics and limitations of common sampling methods.
Prerequisites
None — entry point
Content
- Population vs. sample; parameter vs. statistic (\(\mu\) vs. \(\bar{x}\), \(\sigma\) vs. \(s\))
- Qualitative (nominal/ordinal) vs. quantitative (discrete/continuous) variables
- Sampling methods: simple random, stratified, systematic, cluster, multistage, convenience
- Bias in sampling and its effect on validity of conclusions
- Survey design principles
⚠ Common Pitfalls
- Confusing population parameter with sample statistic (e.g., treating \(\bar{x}\) as if it were \(\mu\))
- Misidentifying variable type (e.g., treating an ordinal scale as quantitative)
- Mistaking convenience sampling for random sampling
Standards
L-DS-2
Data Visualization
Requires: L-DS-1 Learning Outcome
Select, construct, and interpret appropriate graphs for different types of data, and identify misleading representations.
Prerequisites
L-DS-1
Content
- Frequency distributions: absolute, relative (\(\frac{f}{n}\)), and cumulative frequencies
- Histograms (quantitative data) vs. bar charts (qualitative data)
- Pie charts, stem-and-leaf plots, scatter plots, and time series plots
- Graph misrepresentation: truncated axes, inconsistent scales, misleading displays
⚠ Common Pitfalls
- Using a histogram for categorical data (a bar chart is correct)
- Confusing cumulative relative frequency with relative frequency
- Misreading histogram intervals as discrete categories
Standards
L-DS-3
Central Tendency Measures
Requires: L-DS-2 Learning Outcome
Calculate and interpret mean, median, and mode, and select the measure most appropriate for the distribution's shape.
Prerequisites
L-DS-2
Content
- Mean \(\bar{x} = \frac{\sum x_i}{n}\): calculation, properties, sensitivity to outliers
- Median: calculation procedure (sort first, then locate), use with skewed data
- Mode: unimodal, bimodal, multimodal; mode for qualitative data
- Relationship between mean, median, and mode as a function of skewness
⚠ Common Pitfalls
- Computing the median without first sorting the data
- Applying the mean to heavily skewed data where the median is more representative
- Forgetting that the mean is pulled toward the tail in a skewed distribution
Standards
L-DS-4
Variability and Spread
Requires: L-DS-3 Learning Outcome
Calculate and interpret range, variance, standard deviation, and interquartile range, and identify the appropriate measure for a given context.
Prerequisites
L-DS-3
Content
- Range as a basic but unstable measure of spread
- Population variance \(\sigma^2\) vs. sample variance \(s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}\) — the \(n-1\) denominator
- Population SD \(\sigma\) vs. sample SD \(s\)
- Quartiles (Q1, Q2, Q3), IQR \(=\) Q3 \(-\) Q1, and the five-number summary
- Coefficient of variation; identifying outliers using \(1.5 \times \text{IQR}\) fences
⚠ Common Pitfalls
- Using \(n\) instead of \(n-1\) in the sample variance formula
- Computing IQR correctly but misidentifying Q1 vs. Q3 with an odd-sized dataset
- Confusing variance \(s^2\) with standard deviation \(s\) (failing to take the square root)
Standards
L-DS-5
Position and Distribution Shape
Requires: L-DS-4 Learning Outcome
Use percentiles, z-scores, and the Empirical Rule to interpret positions within a distribution and describe its shape.
Prerequisites
L-DS-4
Content
- Percentiles, quartiles, and deciles — locating and interpreting
- Z-score (standardization): \(z = \frac{x - \mu}{\sigma}\); interpretation in standard deviations from the mean
- Skewness: right-skewed (tail extends right), left-skewed, symmetric distributions
- Empirical Rule (68–95–99.7) for approximately normal distributions
⚠ Common Pitfalls
- Confusing z-score with percentile rank (z-score measures distance in SDs, not proportion below)
- Applying the Empirical Rule to non-normal (e.g., heavily skewed) distributions
- Misinterpreting "right-skewed" — the tail points right, not the peak
Module 2
Probability Foundations
The language and rules of probability — sample spaces, event operations, conditional probability, and combinatorics.
L-PR-1
Basic Probability Concepts
Requires: L-DS-1 Learning Outcome
Calculate probabilities of simple, compound, and complementary events using the fundamental rules of probability.
Prerequisites
L-DS-1
Content
- Sample space and events; classical, empirical, and subjective probability
- Probability rules: \(0 \leq P(A) \leq 1\); \(P(S) = 1\); Addition Rule; Complement Rule: \(P(A') = 1 - P(A)\)
- Compound events: union \(A \cup B\) and intersection \(A \cap B\)
- Mutually exclusive events; Venn diagrams as a reasoning tool
⚠ Common Pitfalls
- Confusing mutually exclusive events with independent events (two distinct concepts)
- Applying \(P(A \cup B) = P(A) + P(B)\) when events overlap (missing the intersection)
- Treating \(P(A \cap B) = 0\) whenever events appear unrelated
Standards
L-PR-2
Conditional Probability
Requires: L-PR-1 Learning Outcome
Apply conditional probability and the multiplication rule to solve problems involving dependent and independent events.
Prerequisites
L-PR-1
Content
- Conditional probability: \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\)
- General multiplication rule: \(P(A \cap B) = P(B) \cdot P(A \mid B)\)
- Independent events: definition \(P(A \mid B) = P(A)\) and test \(P(A \cap B) = P(A) \cdot P(B)\)
- Tree diagrams for multi-stage experiments
⚠ Common Pitfalls
- Confusing \(P(A \mid B)\) with \(P(B \mid A)\) — the direction of conditioning matters
- Applying the multiplication rule for independent events to dependent events
- Confusing independence with mutual exclusivity — mutually exclusive events with positive probability are always dependent
Standards
L-PR-3
Counting Techniques
Requires: L-PR-1 Learning Outcome
Use the Fundamental Counting Principle, permutations, and combinations to count outcomes and compute probabilities.
Prerequisites
L-PR-1
Content
- Fundamental Counting Principle for sequential, independent choices
- Factorials: \(n! = n \cdot (n-1) \cdots 1\); convention \(0! = 1\)
- Permutations (order matters): \(_nP_r = \frac{n!}{(n-r)!}\)
- Combinations (order irrelevant): \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\)
- Distinguishing permutations from combinations in context
- Applications to computing probabilities over equally-likely sample spaces
⚠ Common Pitfalls
- Using a permutation formula when order does not matter
- Applying the Fundamental Counting Principle when choices are not independent
- Forgetting that \(\binom{n}{0} = \binom{n}{n} = 1\)
Standards
Module 3
Probability Distributions
Random variables, probability models, and named distributions — the bridge from probability rules to statistical inference.
L-PR-4
Discrete Random Variables
Requires: L-PR-1 Learning Outcome
Construct and interpret probability distributions for discrete random variables, and compute expected value and variance.
Prerequisites
L-PR-1
Content
- Random variables: discrete vs. continuous; notation \(X\), \(P(X = x)\)
- Probability mass function (PMF) and its properties (\(\sum P(X = x) = 1\))
- Expected value: \(E(X) = \mu = \sum x \cdot P(X = x)\)
- Variance: \(\text{Var}(X) = E(X^2) - [E(X)]^2\); standard deviation \(\sigma_X\)
- Cumulative distribution function (CDF)
⚠ Common Pitfalls
- Confusing \(E(X)\) (the long-run average) with the most likely value (mode)
- Computing \(E(X^2)\) as \([E(X)]^2\) — these are equal only in degenerate cases
- Forgetting that variance is always non-negative
Standards
L-PR-5
Binomial Distribution
Requires: L-PR-4, L-PR-3 Learning Outcome
Apply the binomial distribution to calculate probabilities and describe the shape and spread of a binomial random variable.
Prerequisites
L-PR-4 L-PR-3
Content
- Four BINS conditions: Binary outcomes, Independent trials, fixed Number, constant Success probability
- Binomial formula: \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)
- Mean: \(\mu = np\); variance: \(\sigma^2 = np(1-p)\); SD: \(\sigma = \sqrt{np(1-p)}\)
- Cumulative probabilities: "at most," "at least," "more than," "fewer than"
⚠ Common Pitfalls
- Applying the formula when trials are not independent or \(p\) changes between trials
- Confusing "exactly \(k\)" with "at least \(k\)" — point vs. cumulative probability
- Omitting the combination coefficient \(\binom{n}{k}\) from the formula
Standards
L-PR-6
Normal Distribution
Requires: L-PR-4, L-DS-5 Learning Outcome
Use the normal distribution and z-score transformation to find probabilities and values for normally distributed variables.
Prerequisites
L-PR-4 L-DS-5
Content
- Properties of the normal distribution: bell-shaped, symmetric about \(\mu\), asymptotic tails
- Standard normal \(Z \sim N(0, 1)\); z-table gives left-tail area \(P(Z < a)\)
- Finding probabilities: \(P(Z < a)\), \(P(Z > a) = 1 - P(Z < a)\), \(P(a < Z < b)\)
- Standardizing: \(z = \frac{x - \mu}{\sigma}\) to transform \(X \sim N(\mu, \sigma^2)\) to \(Z\)
- Finding values given a probability (inverse normal / quantile lookup)
⚠ Common Pitfalls
- Reading the z-table as \(P(Z > a)\) instead of \(P(Z < a)\) — most tables give left-tail area
- Forgetting to standardize \(X\) to \(Z\) before using the table
- Adding two left-tail areas instead of subtracting when finding \(P(a < Z < b)\)
Standards
Module 4
Statistical Inference
The Central Limit Theorem bridges probability to inference — the engine behind every confidence interval and hypothesis test.
L-INF-1
Sampling Distributions and the Central Limit Theorem
Requires: L-PR-6, L-DS-4 Learning Outcome
Apply the Central Limit Theorem to describe the sampling distribution of the sample mean and compute related probabilities.
Prerequisites
L-PR-6 L-DS-4
Content
- Sampling distribution of \(\bar{x}\): mean \(\mu_{\bar{x}} = \mu\) and standard error \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\)
- Central Limit Theorem: for large \(n\), \(\bar{x}\) is approximately normal regardless of population shape
- Effect of sample size on the shape and spread of the sampling distribution
- Finite population correction (overview)
- Computing probabilities for sample means: \(z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}\)
⚠ Common Pitfalls
- Confusing population standard deviation \(\sigma\) with standard error \(\sigma/\sqrt{n}\)
- Applying the CLT when \(n < 30\) and the population is heavily skewed
- Forgetting that the CLT describes \(\bar{x}\) across many samples, not individual observations
Standards
L-INF-2
Confidence Intervals for a Population Mean (Large Sample)
Requires: L-INF-1 Learning Outcome
Construct and interpret confidence intervals for a population mean using the z-distribution when \(n \geq 30\).
Prerequisites
L-INF-1
Content
- Point estimate vs. interval estimate; margin of error \(E = z^* \cdot \frac{\sigma}{\sqrt{n}}\)
- CI formula: \(\bar{x} \pm z^* \cdot \frac{\sigma}{\sqrt{n}}\) (use \(s\) when \(\sigma\) unknown and \(n \geq 30\))
- Critical z-values: \(z^* = 1.645\) (90%), \(z^* = 1.96\) (95%), \(z^* = 2.576\) (99%)
- Correct interpretation: "We are 95% confident that \(\mu\) falls between [L, U]"
- Sample size determination: \(n = \left(\frac{z^* \cdot \sigma}{E}\right)^2\)
⚠ Common Pitfalls
- Interpreting a single CI as "\(P(\mu\) is in this interval\() = 0.95\)" — the confidence level applies to the method, not one fixed interval
- Expecting a higher confidence level to also give more precision (wider interval)
- Using the z-distribution when \(n < 30\) and \(\sigma\) is unknown
Standards
L-INF-3
Confidence Intervals for a Population Mean (σ Unknown)
Requires: L-INF-2 Learning Outcome
Construct and interpret confidence intervals for a population mean using the t-distribution when \(\sigma\) is unknown and estimated by \(s\), at any sample size.
Prerequisites
L-INF-2
Content
- t-distribution: bell-shaped, heavier tails than normal, indexed by degrees of freedom \(df = n - 1\)
- Critical t-values from t-table; t approaches z as df increases
- CI formula: \(\bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}\)
- Conditions: approximately normal population, \(\sigma\) unknown
- When to use t vs. z: the practical decision rule
⚠ Common Pitfalls
- Using \(df = n\) instead of \(df = n - 1\)
- Applying the z-distribution when \(\sigma\) is unknown and \(n\) is small
- Forgetting that the t-distribution has heavier tails, yielding a wider CI than z at the same confidence level
Standards
L-INF-4
Confidence Intervals for a Proportion
Requires: L-INF-2 Learning Outcome
Construct and interpret confidence intervals for a population proportion, and determine the required sample size.
Prerequisites
L-INF-2
Content
- Point estimate: \(\hat{p} = \frac{x}{n}\); standard error \(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)
- CI formula: \(\hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)
- Conditions: \(n\hat{p} \geq 10\) and \(n(1-\hat{p}) \geq 10\)
- Sample size determination using \(p^* = 0.5\) when no prior estimate is available
- One-sided vs. two-sided CIs for proportions
⚠ Common Pitfalls
- Using unknown \(p\) instead of observed \(\hat{p}\) in the standard error formula
- Failing to check normality conditions before applying the z-interval
- Misinterpreting the margin of error as an absolute cap independent of confidence level
Standards
L-INF-5
Hypothesis Testing for a Population Mean (σ Known)
Requires: L-INF-2 Learning Outcome
Conduct and interpret five-step hypothesis tests for a population mean using the z-distribution when \(\sigma\) is known, explain what a p-value does and does not mean, and distinguish statistical significance from practical importance.
Prerequisites
L-INF-2
Content
- Five steps: state H₀ and Hₐ, check conditions, compute test statistic, find p-value, state conclusion in context
- Test statistic: \(z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}\)
- Type I error (significance level \(\alpha\)) and Type II error (\(\beta\)); the trade-off between them
- p-value: \(P(\text{data this extreme} \mid H_0 \text{ true})\); comparing to \(\alpha\)
- One-tailed vs. two-tailed tests; choosing Hₐ before seeing data
⚠ Common Pitfalls
- "Fail to reject H₀" does not mean "accept H₀" — absence of evidence is not evidence of absence
- Misdefining the p-value as \(P(H_0 \text{ is true})\) — it is a conditional probability given H₀
- Selecting a one-tailed test after inspecting the data (data snooping inflates Type I error)
Standards
L-INF-6
Hypothesis Tests for a Mean (σ Unknown) and for a Proportion
Requires: L-INF-5, L-INF-3 Learning Outcome
Choose and conduct a hypothesis test for a population mean when \(\sigma\) is unknown (the t-test, at any sample size) or for a population proportion, and interpret conclusions in context.
Prerequisites
L-INF-5 L-INF-3
Content
- t-test for a mean with \(\sigma\) unknown: \(t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\), \(df = n-1\), five steps, conditions
- Proportion test: \(z = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}\) — using \(p_0\), not \(\hat{p}\), in the denominator
- One-tailed vs. two-tailed tests for both scenarios
- Approximate p-values from the t-table; exact values with technology
⚠ Common Pitfalls
- Using \(\hat{p}\) instead of \(p_0\) in the denominator of the proportion test statistic
- Applying z instead of t because \(n\) is large, when \(\sigma\) is unknown and estimated by \(s\)
- Confusing the CI approach with the p-value approach when stating conclusions
Standards
Module 5
Regression & Association
Analyzing relationships between two variables — correlation and regression for quantitative data, chi-square for qualitative data.
L-REG-1
Correlation Analysis
Requires: L-DS-5, L-DS-2 Learning Outcome
Calculate and interpret the Pearson correlation coefficient and coefficient of determination.
Prerequisites
L-DS-5 L-DS-2
Content
- Scatter plots: direction, form (linear vs. non-linear), strength, outliers
- Pearson \(r\): range \(-1 \leq r \leq 1\), sign (direction), magnitude (strength)
- Coefficient of determination \(r^2\): proportion of variance in \(y\) explained by the linear relationship with \(x\)
- Spearman's rank correlation (overview for non-normal data and ordinal variables)
- Correlation vs. causation: confounding variables and spurious correlations
- What \(r\) ignores: units, which variable is \(x\), and the slope (\(r\) is not a slope)
- Restricted range and single influential points
⚠ Common Pitfalls
- Interpreting \(r = 0\) as "no relationship" — there could be a strong non-linear relationship
- Confusing \(r\) with \(r^2\) (e.g., \(r = 0.7\) explains only \(r^2 = 49\%\) of variance, not 70%)
- Inferring causation from a strong correlation
Standards
L-REG-2
Linear Regression
Requires: L-REG-1 Learning Outcome
Determine the equation of the least-squares regression line and interpret its slope and intercept.
Prerequisites
L-REG-1
Content
- Least-squares regression line: \(\hat{y} = a + bx\); minimizing the sum of squared residuals
- Slope: \(b = r \cdot \frac{s_y}{s_x}\); intercept: \(a = \bar{y} - b\bar{x}\)
- Residuals: \(e_i = y_i - \hat{y}_i\); sum of squared errors (SSE); \(\sum e_i = 0\)
- The regression line always passes through \((\bar{x}, \bar{y})\)
- Regression to the mean: in standard units \(\hat{z}_y = r\,z_x\)
- Predicting \(x\) from \(y\) needs its own line, with slope \(r \cdot s_x/s_y\)
⚠ Common Pitfalls
- Swapping dependent variable \(y\) and independent variable \(x\)
- Using the slope formula with \(s_x\) and \(s_y\) reversed
- Forgetting that the regression line is guaranteed to pass through \((\bar{x}, \bar{y})\)
- Judging a line by how many points it passes through instead of by its SSE
- Explaining regression to the mean with a cause
Standards
L-REG-3
Regression Interpretation and Prediction
Requires: L-REG-2, L-INF-5 Learning Outcome
Use the regression equation for prediction and critically evaluate model quality and limitations.
Prerequisites
L-REG-2 L-INF-5
Content
- Interpreting slope: a one-unit increase in \(x\) is associated with \(b\) units of change in \(\hat{y}\), on average
- Interpreting intercept: predicted \(y\) when \(x = 0\) (only meaningful when \(x = 0\) is in context)
- Making predictions; interpolation (safe) vs. extrapolation (risky)
- Residual plots for checking linearity and homoscedasticity assumptions
- Outliers and influential points; their effect on slope and intercept
- Significance test for \(r\): \(t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\), \(df = n-2\); H₀: \(\rho = 0\) (no linear relationship in the population)
⚠ Common Pitfalls
- Extrapolating far beyond the observed range of \(x\)
- Interpreting the intercept as meaningful when \(x = 0\) falls outside the observed data
- Ignoring influential points when evaluating the quality of a model
- Treating a statistically significant \(r\) as practically meaningful — a small \(r\) can reach significance with large \(n\) yet explain very little variance (\(r^2\) is what matters for prediction quality)
Standards
L-REG-4
Chi-Square Test of Independence
Requires: L-INF-5 Learning Outcome
Perform and interpret a chi-square test of independence for two qualitative variables using a contingency table.
Prerequisites
L-INF-5
Content
- Contingency tables (two-way tables): observed frequencies and marginal totals
- Expected frequencies: \(E = \frac{\text{row total} \times \text{column total}}{\text{grand total}}\)
- Chi-square test statistic: \(\chi^2 = \sum \frac{(O - E)^2}{E}\)
- Degrees of freedom: \(df = (r-1)(c-1)\); p-values from the \(\chi^2\) distribution
- Five steps of the chi-square test; condition: all expected frequencies \(\geq 5\)
- Cramér's V as an effect size measure
- Reading a two-way table by row percentages: no link means every row splits alike, not equal counts
- Simpson's paradox (Berkeley, 1973): an association that vanishes or reverses within groups
- Choosing a chi-square test or correlation and regression from the types of the two variables
⚠ Common Pitfalls
- Applying chi-square when expected cell frequencies are less than 5
- Confusing observed \(O\) and expected \(E\) frequencies in the formula
- Concluding that statistical dependence implies causation
- Reading a tiny p-value as a strong association; report Cramér's V
- Treating coded categories (postal codes, jersey numbers, rating labels) as quantitative