probability-and-statistics

Data often depends on when it happened. We explore temporal structure, autocorrelation (correlation with past values), and trends, moving beyond independent samples to model history and forecast future values.

A shift in philosophy. Instead of fixed parameters, we treat them as random variables. We combine prior beliefs with observed data (likelihood) to calculate a "Posterior" probability, mathematically updating our view of the world.

When data isn't Normal or sample sizes are tiny, standard tests fail. We use methods based on ranks (like Mann-Whitney or Kruskal-Wallis) to draw valid conclusions without assuming a specific distribution shape (Normality).

Are two categorical variables (like "Gender" and "Voting Preference") related? This test checks if knowing one variable helps predict the other. If the variables are independent, the observed pattern will match random chance.

Does our data match expectations? This test compares observed counts to theoretical expected counts (e.g., "is this die fair?"). A large discrepancy suggests the data does not follow the hypothesized distribution.

ANOVA tells us "there is a difference," but not where. Post-hoc tests (like Tukey’s HSD) act as detectives after a significant ANOVA result, comparing specific pairs of groups while correcting for false positives to identify the exact outlier.

The engine of ANOVA. It compares the "signal" (variance between group means) to the "noise" (variance within groups). A high F-ratio suggests the differences between groups are real, rather than just random fluctuations.

Comparing two means is easy, but what about three or more? ANOVA tests if any group mean differs from the others by analyzing variance. It avoids the statistical errors that occur when running multiple separate t-tests.

How well does the line fit? R square measures the percentage of variance in the dependent variable that is explained by our model. An R square of 1.0 indicates a perfect fit, while 0.0 means the model explains no variability at all.

We diagnose models by examining residuals—the difference between observed and predicted values. A random cloud of points in a residual plot indicates a healthy model; specific patterns suggest we are missing a key variable or relationship.