Probability and Statistics

01Curriculum

Learning Path

Unit 1 · Weeks 1–2

Data and Distributions

3 lessons · Data types · Visualizations · The normal distribution

variable typesdistribution shapez-scores

Types of Data

Categorical vs. quantitative, discrete vs. continuous. Why the type of data determines which methods are valid. Common misclassifications and how to avoid them.

Visualizing Data

Histograms, stem-and-leaf, dotplots, boxplots. Shape, center, spread, and outliers as the four things to describe. Reading and building each type.

The Normal Distribution

Shape, symmetry, and the empirical rule (68-95-99.7). Z-scores as standardized distance from the mean. What a z-score tells you without a table.

Assignment 1

Take one real dataset. Compute summary statistics, create two visualizations, and write a one-paragraph interpretation. Numbers without narrative get sent back.

You describe any dataset completely and correctly before running a single test.

Unit 2 · Weeks 3–4

Descriptive Statistics

3 lessons · Measures of center · Measures of spread · Five-number summary

mean vs. medianstandard deviationIQR

Measures of Center

Mean, median, mode — when each is the right summary. What skewness does to the mean vs. the median. Why the mean is not always the best choice.

Measures of Spread

Range, IQR, variance, and standard deviation. Why variance is squared. How standard deviation relates to the normal distribution.

Five-Number Summary and Outliers

Building and reading a boxplot. Identifying outliers using the IQR rule. What an outlier means and whether to include or exclude it.

Assignment 2

Compute all measures of center and spread for two datasets — one symmetric, one skewed. Write which measure you would report for each and why.

You summarize data with precision and defend your choice of statistic.

Unit 3 · Weeks 5–7

Probability Foundations

4 lessons · Basic probability · Conditional probability · Bayes' theorem

conditional probabilityindependenceBayes

Sample Spaces and Basic Rules

Events, complements, and the addition rule. What probability means as a long-run frequency. Mutually exclusive vs. independent events — these are not the same thing.

Conditional Probability

P(A|B) defined and computed from a table or formula. Independence defined formally: P(A|B) = P(A). The most misunderstood concept in introductory statistics.

Multiplication Rule and Tree Diagrams

Joint probability, with and without replacement. Tree diagrams as the right tool for multi-stage experiments. Counting the branches correctly.

L10

Bayes' Theorem

Worked through from the definition of conditional probability, not memorized as a formula. Applications to medical testing, spam filters, and base rate problems.

Assignment 3

Solve 5 probability problems. Each must include a tree diagram or sample space before any formula.

You calculate any probability and interpret it correctly — including conditional probabilities.

Unit 4 · Weeks 8–9

Discrete Probability Distributions

3 lessons · Expected value · Binomial distribution · Poisson distribution

expected valuebinomial conditionsPoisson parameter

L11

Random Variables and Expected Value

Discrete vs. continuous random variables. Expected value and variance from first principles — not just formulas. What expected value means in context.

L12

The Binomial Distribution

Four conditions for a binomial setting. The formula derived from combinations. Mean and standard deviation. When to use the binomial vs. when not to.

L13

The Poisson Distribution

What Poisson models. The parameter lambda as mean and variance. When Poisson approximates binomial and when it doesn't.

Assignment 4

Classify 8 scenarios as binomial, Poisson, or neither. Solve the ones that fit. Write one sentence justifying each classification.

You model count data correctly and choose the right distribution.

Unit 5 · Weeks 10–11

The Normal Distribution and Sampling

3 lessons · Z-table · Sampling distributions · Central Limit Theorem

z-tablestandard errorCLT

L14

Normal Probability Calculations

Using the z-table to find probabilities and to find values from probabilities. What the area under the curve actually represents. Common errors in table reading.

L15

Sampling Distributions

The sampling distribution of x-bar. Standard error as the standard deviation of a statistic, not of the data. Why larger samples give less variability in the estimate.

L16

The Central Limit Theorem

Why it matters: any sum or average of enough independent observations is approximately normal. What sample size is required. Its limits — what it doesn't say.

Assignment 5

Solve 10 normal probability problems — 5 finding probability, 5 finding the value. Then explain the CLT in three sentences to someone who has never taken statistics.

You use the normal distribution and the CLT as genuine tools, not black boxes.

Unit 6 · Weeks 12–14

Confidence Intervals

3 lessons · One-mean intervals · Proportion intervals · Interpreting correctly

t-distributionmargin of errorcorrect interpretation

L17

Confidence Intervals for One Mean

Z-interval vs. t-interval — when to use each. Conditions for validity. The t-distribution and degrees of freedom.

L18

Confidence Intervals for One Proportion

The np and n(1-p) conditions checked before proceeding. The formula derived from the sampling distribution. Margin of error and sample size determination.

L19

What a Confidence Interval Actually Means

What '95% confident' means and what it doesn't. The most common misinterpretation and why it's wrong. Factors affecting interval width.

Assignment 6

Construct and interpret 3 confidence intervals from real data. Check conditions before proceeding on each. State whether each interval is valid and why.

You construct and correctly interpret any standard confidence interval.

Unit 7 · Weeks 15–18

Hypothesis Testing

4 lessons · Logic of testing · One-sample tests · Two-sample tests · Chi-square

p-valueType I and II errortest selection

L20

The Logic of Hypothesis Testing

Null and alternative hypotheses. Type I and Type II errors and their consequences. Significance level as the tolerance for Type I error. What the p-value actually says.

L21

One-Sample Z-Test and T-Test

Test statistic computed and interpreted. P-value from tables or technology. The conclusion written in context — no statistical shorthand.

L22

Two-Sample Tests

Independent samples t-test, paired t-test — when each applies. The paired design as a strategy to reduce variability. Equal variance assumption checked.

L23

Proportion Tests and Chi-Square

One and two proportion z-tests. Chi-square goodness of fit and test of independence. Expected cell counts and when the test is valid.

Assignment 7

Run a full hypothesis test on a provided scenario. State hypotheses, check conditions, compute, decide, write a conclusion in plain language. No statistical shorthand in the conclusion.

You run any standard hypothesis test and explain the result to someone who doesn't know statistics.

Unit 8 · Weeks 19–26

Regression and Capstone

4 lessons + capstone · Simple linear regression · Model assessment · Full analysis

least squaresr-squaredresidual analysis

L24

Simple Linear Regression

The least squares line derived from minimizing squared residuals — not just handed as a formula. Interpreting slope and intercept in context — always in context.

L25

Correlation, R-squared, and Residuals

The difference between r and r-squared and what each tells you. Residual plots as the diagnostic for model assumptions. When a linear model is not appropriate.

L26

Inference for Regression

The t-test for the slope. Confidence interval for the mean response. Prediction interval for an individual response. What each answers.

L27 (cap)

Capstone: Full Statistical Analysis

A complete analysis of a real dataset chosen with the student. From data description to inference. A written report with interpretation — no bare output.

Assignment 8

A complete statistical analysis of a real dataset: data type identification, descriptive statistics, probability model, confidence interval, hypothesis test, and regression if appropriate. Written report required.

You take a raw dataset and produce a complete, correctly interpreted statistical analysis.

Data, probability,
and the language of
inference.

Interpretation Is the Point

Conditions First, Then Calculations

Data, probability,and the language ofinference.

Interpretation Is the Point

Conditions First, Then Calculations

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Data, probability,
and the language of
inference.