Statistics and Probability

Probability foundations through hypothesis testing and regression, covering a standard introductory statistics and probability course.

All 26 topics

Every skill the diagnostic can check, grouped the way the map is. Open any topic to start a lesson on it.

Descriptive statistics: Summarizing a dataset with a few numbers, mean, median, variance, or a picture, instead of listing every value.
Probability basics: The formal rules, sample spaces, events, axioms, that make "the chance of something" a well-defined number.

Combinatorics and counting: Counting how many ways something can happen, permutations and combinations, without listing them all.
Conditional probability and independence: How knowing one event happened changes, or doesn't change, the probability of another.
Bayes' theorem: Flipping a conditional probability around, going from P(evidence given cause) to P(cause given evidence).

Discrete random variables: A number assigned to each outcome of a random experiment, when only countably many values are possible.
Continuous random variables: A random variable that can take any value in a range, described by a density curve instead of a list of probabilities.
Expectation and variance: The long-run average value of a random variable, and how spread out it typically is around that average.
Joint distributions: How two random variables behave together, including whether knowing one tells you anything about the other.

Common discrete distributions: The handful of discrete shapes, Bernoulli, Binomial, Poisson, Geometric, that keep showing up because they model real repeated-trial situations.
Common continuous distributions: The handful of continuous shapes, Uniform, Normal, Exponential, that keep showing up in real measurements and waiting times.

Law of large numbers: A sample average gets closer to the true average as you collect more data.
Central limit theorem: Why the average of almost any large sample ends up looking approximately Normal, regardless of the original distribution's shape.

Sampling and sampling distributions: How a statistic, like a sample mean, itself varies from sample to sample if you repeated the study.
Point estimation: Using sample data to produce a single best-guess number for an unknown population value.
Confidence intervals: A range, instead of a single number, built to contain the true value a stated percentage of the time.
Resampling and the bootstrap: Estimating how uncertain a statistic is by repeatedly resampling from the data you already have, instead of relying on a formula.

Hypothesis testing basics: Checking whether observed data is surprising enough to reject a specific default assumption, the null hypothesis.
t-tests: Comparing one or two sample means against a hypothesized value, when the true variance is unknown.
Chi-squared tests: Checking whether observed category counts match an expected pattern, or whether two categorical variables are related.
ANOVA: Comparing the means of three or more groups at once without inflating the false-positive rate.

Correlation: A single number summarizing how strongly, and in what direction, two variables move together.
Linear regression: Fitting a straight line that best predicts one variable from another, and reading its slope as an effect size.
Multiple regression and model fit: Predicting an outcome from several variables at once, and judging how well the resulting model actually fits.

Bayesian inference: Updating a starting belief about a parameter into a revised belief, using Bayes' theorem and observed data.

Markov chains: A sequence of states where the next state depends only on the current one, not the full history.