Statistics Learning Center
Explore 31 reference articles on math, probability, statistics, and closely related quantitative methods. These pages are kept public to support the active calculator library with definitions, interpretation, and inspectable reasoning.
Normal Distribution
Explore the bell curve, its properties, and why it appears so often in probability and statistics.
Standard Deviation
Understand the statistical measure that quantifies spread, dispersion, and variance around a mean.
Hypothesis Testing
Learn how null and alternative hypotheses, p-values, confidence intervals, and test selection work together in statistical inference.
Linear Regression
Learn how linear regression models relationships between variables and supports prediction.
Bayesian Statistics
Explore how prior beliefs and observed data combine to update uncertainty with probability theory.
Fibonacci Sequence
See how the Fibonacci pattern grows recursively and why it appears throughout mathematics.
Locally Weighted Scatterplot Smoothing
Understand LOWESS as a flexible curve-fitting method for noisy data without a global model.
Central Limit Theorem
Learn why sample means become normally distributed, how standard error shrinks with sample size, and where the theorem powers everyday inference.
Probability Distributions
Compare the binomial, Poisson, normal, uniform, and exponential distributions and learn how to choose the right model for counts, measurements, and waiting times.
Confidence Intervals
Understand what a confidence interval really claims, how the margin of error is built, when to use z versus t, and how sample size controls precision.
How to Read a Z-Table
Read any z-table in three steps: split the z-score into row and column, read the cumulative area, and convert it into the probability your question needs.
Standard Error vs Standard Deviation
Standard deviation describes the spread of individual values; standard error describes the precision of an estimate. See how one dataset produces both numbers.
Type I vs Type II Errors
False alarms versus missed effects: how α and β trade off, what statistical power means, and a worked example that computes both error rates.
n vs n−1: Bessel's Correction
Why sample variance divides by n−1: where the bias comes from, a tiny population where you can verify the fix by hand, and when plain n is correct.
Binomial vs Poisson vs Hypergeometric
Three questions decide the right counting distribution — fixed trials, independence, replacement — with one inspection scenario computed all three ways.
How to Read a T-Table
Read any t-table in three steps: find the degrees of freedom row, pick the one-tail or two-tail α column, and read the critical value for tests and intervals.
P-Value Explained
What p < 0.05 actually means: the precise definition, a worked coin-flip example, the decision rule against α, and the five misreadings to avoid.
Margin of Error vs Confidence Interval
One is the ± half-width, the other is the full range built from it. The formula anatomy, a worked poll example, and what a ±3% margin really covers.
The Empirical Rule (68-95-99.7)
Why 68% of normal data falls within 1 SD, 95% within 2, and 99.7% within 3 — with a worked IQ example, tail arithmetic, and the Chebyshev fallback.
Correlation vs Causation
The four explanations behind any correlation — causation, reverse causation, confounding, coincidence — with real examples and what supports a causal claim.
How to Interpret R²
R² is the share of variance a regression explains. A worked example from five points, field-dependent benchmarks, and what high or low R² does not mean.
Bayes' Theorem Explained
Why a positive result from a 90%-accurate test is real only 9.2% of the time when the condition is rare — the full base-rate arithmetic, twice.
Expected Value Explained
The probability-weighted average behind every gamble: worked raffle, dice, and insurance examples, the law of large numbers, and where EV alone falls short.
One-Way ANOVA Explained
How ANOVA turns variances into a verdict about means: the full SS/df/MS/F bookkeeping on nine data points, the F-table decision, and what a significant F does not say.
Chi-Square Test Explained
Goodness-of-fit and independence, both worked in full: expected counts, the (O−E)²/E statistic, decisions against the chi-square table, and the validity rules.
Confidence Level vs Confidence Interval
The level is the success rate you choose; the interval is the range your sample produces. One dataset at three levels shows how they relate — and differ.
Percentiles and Quartiles Explained
Position statistics from the ground up: percentile ranks, the quartile split, why computation methods disagree on small samples, and Tukey's outlier fences.
Skewness and Kurtosis Explained
The two shape statistics worked by hand — asymmetry and tail weight — with the interpretation table and the sample-vs-population corrections that surprise on small n.
Covariance vs Correlation
Both measure co-movement; covariance wears the units, correlation standardizes them away. One dataset carried through both computations shows how they relate.
Poisson Distribution Explained
From an average rate to exact count probabilities: worked call-center example, the assumptions checklist, and the binomial approximation shown numerically.
How to Make a Box Plot
From raw data to finished plot: five-number summary, IQR, Tukey fences, the whisker rule most people get wrong, and how to read the result.