Statistics Learning Center
Explore 57 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.
Core Concepts
The reference articles behind the calculators: distributions, inference, regression, probability and how to read the results.
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.
TI-84 Statistics Functions
Every TI-84 statistics function: DISTR (normalcdf, invNorm, tcdf, binompdf…) and STAT TESTS (T-Test, TInterval, 2-SampFTest, LinRegTTest…), with syntax, a decision guide, and an online calculator for each.
How-To Guides
Task-oriented walk-throughs: which test to use, spreadsheet formulas, R and Python commands, and how to read statistical tables.
Which Statistical Test Should I Use?
Choose the right test in three questions (what you measure, how many groups you compare and whether the data are paired) with a decision table and worked decisions.
Statistics Formulas Cheat Sheet
The core formulas in one place (descriptive statistics, probability, distributions, confidence intervals, hypothesis tests and regression) with a checked worked example and a calculator for each.
Standard Deviation in Excel and Google Sheets
STDEV.S or STDEV.P? The Excel and Google Sheets formulas for standard deviation, variance, standard error and coefficient of variation, checked on one worked example.
P-Value in Excel: t, z, Chi-Square and F Formulas
Excel and Google Sheets formulas for p-values (T.DIST.2T, NORM.S.DIST, CHISQ.DIST.RT, F.DIST.RT and the T.TEST, CHISQ.TEST, Z.TEST and F.TEST functions) checked on worked examples.
T-Test in Excel: T.TEST and the Data Analysis ToolPak
Run a t-test in Excel with T.TEST or the Analysis ToolPak: the type and tails arguments, paired and unequal-variance tests, a checked example and how to read the output.
Confidence Interval in Excel: CONFIDENCE.T and CONFIDENCE.NORM
Build a confidence interval for a mean or a proportion in Excel with CONFIDENCE.T, CONFIDENCE.NORM and the Descriptive Statistics ToolPak, and compare Wald with Wilson intervals.
Chi-Square Test in Excel: CHISQ.TEST Step by Step
Run a chi-square test of independence or goodness of fit in Excel: expected counts, CHISQ.TEST, cell contributions, Cramér's V and a checked 2×3 example.
Correlation and Regression in Excel: CORREL, SLOPE and LINEST
Excel functions for correlation and linear regression (CORREL, RSQ, SLOPE, INTERCEPT, LINEST), the Regression ToolPak output explained and Spearman correlation via RANK.AVG.
How to Read a Chi-Square Table
Find the chi-square critical value from the degrees of freedom and significance level, with a table of common values, three worked examples and software equivalents.
How to Read an F Table
Look up the F critical value from the numerator and denominator degrees of freedom, with worked ANOVA examples, why the order matters and software equivalents.
T-Test in R and Python: t.test() and scipy.stats
Run one-sample, two-sample, Welch and paired t-tests in R and Python side by side, see why the default variance option differs and how one outlier can flip the result: checked against SciPy.
Concept Explainers
The ideas that decide which method to use: hypotheses, tails, degrees of freedom, power, effect size, sample size and multiple comparisons.
Null and Alternative Hypothesis
How to write H0 and H1 for means, proportions, correlations and tables, with two worked z-test examples and the correct wording of a conclusion.
One-Tailed vs Two-Tailed Tests
How the p-value and critical value change with the direction of the test, a worked example where the choice flips the conclusion, and when a one-sided test is legitimate.
Degrees of Freedom Explained
What degrees of freedom are, why the sample variance divides by n − 1, and the df formula for the t-test, ANOVA, chi-square and regression, with worked examples.
Sample Size Explained: How Many Responses Do You Need?
Sample size formulas for a proportion and a mean, the finite population correction, an adjustment for dropout and a lookup table for margins of error, with worked examples.
Statistical Power Explained
What 80% power means, how effect size, sample size and alpha change it, and the sample size per group needed for small, medium and large effects.
Effect Size Explained: Cohen's d, r and Eta Squared
What Cohen's d, Hedges' g, r, eta squared and Cramér's V measure, how to calculate d by hand, and why a small p-value is not enough.
Multiple Comparisons: Bonferroni, Holm and FDR
Why many tests inflate false positives and how Bonferroni, Holm, Šidák, Tukey and Benjamini-Hochberg corrections fix it, with a worked eight-test example.
Normality Tests Explained: Shapiro-Wilk and Q-Q Plots
How to check whether data are normal with Shapiro-Wilk, Q-Q plots and skewness, the small- and large-sample traps, and what to do when the data are not normal.
Parametric vs Nonparametric Tests
What each family assumes, the rank-based alternative to every common test, a worked comparison of the t-test with Mann-Whitney U, and when to switch to ranks.
Cronbach's Alpha Explained: Formula and Meaning
What Cronbach's alpha measures, its formula worked by hand for five respondents and three items, how the number of items drives it, and what it cannot tell you.
Relative Risk vs Odds Ratio: How They Differ
Relative risk compares risks, the odds ratio compares odds. Both worked on one 2×2 table, why they differ, when they agree and which one to report.
Sensitivity, Specificity, PPV and NPV Explained
What sensitivity, specificity, PPV and NPV mean, how to calculate them from a 2×2 table, and why the predictive values fall when a condition is rare.
Independent vs Mutually Exclusive Events Explained
Independent events do not affect each other; mutually exclusive events cannot happen together. Formulas, worked examples and a test to tell them apart.
Permutations vs Combinations: The Difference
Permutations count arrangements where order matters, combinations count selections where it does not. Formulas, worked examples and how to choose between them.