Sampling Distributions (CLT)
Struggle with the Central Limit Theorem? Draw 10,000 samples from highly skewed or bimodal parent populations and watch the perfect normal bell curve emerge.
Power of a Test Curve
Visualize how Statistical Power responds dynamically to Effect Size (mu_a), Sample Size, and Alpha. Plot the complete functional power curve.
ANOVA Variance Ratio
Deconstruct Analysis of Variance geometrically. Shift group means (MSB) and internal scatter (MSW) to see the massive impact on the F-statistic and P-value.
Central Limit Theorem
Visualize how sampling distributions of means approach normality as sample size increases, regardless of population shape. Explore the CLT formula $sigma_{ar{x}} = rac{sigma}{sqrt{n}}$ and see how larger samples produce tighter distributions around the population mean.
t-Distribution vs Normal
Examine the heavy-tailed Student's t-distribution. Increase degrees of freedom (df) to watch it perfectly converge into the standard normal z-distribution.
Normal Distribution Explorer
Adjust mean and standard deviation to see how the normal distribution bell curve shifts and stretches. Shade probability regions to compute areas under the curve and connect z-scores to percentiles.
Chi-Square Goodness of Fit
Calculate and visualize deviations from expected categorical frequencies. Generates the exact right-skewed Chi-Square distribution and P-Value tail.
Type I/II Error Tradeoff & Power
Visualize the fundamental tradeoff between alpha, beta, and statistical power in Hypothesis Testing. Interactive normal distribution sampling curves.
Hypothesis Testing
Conduct hypothesis tests by calculating test statistics and p-values to evaluate null hypotheses. Visualize Type I and Type II errors, significance levels, and the decision-making process for rejecting or failing to reject the null hypothesis based on sample evidence.
Residual Plots
Evaluate the appropriateness of Least Squares Regression Lines (LSRL). Detect curvilinear patterns, heteroscedasticity, and influential outliers.
Scatter Plot & Correlation
Create scatter plots to visualize bivariate relationships and calculate correlation coefficient $r$ to measure linear association strength. Explore how outliers, direction, form, and strength affect correlation, and understand why correlation does not imply causation.
Least Squares Regression Line
Calculate the least squares regression line $hat{y} = a + bx$ that minimizes the sum of squared residuals. Visualize how the slope $b = r rac{s_y}{s_x}$ and intercept relate to correlation, and interpret the line's predictive power for bivariate data.
Confidence Intervals
Construct confidence intervals using $ar{x} pm z^* rac{sigma}{sqrt{n}}$ to estimate population parameters. Visualize how confidence level, sample size, and variability affect interval width, and interpret what it means to be 95% confident about capturing the true parameter.
Binomial Distribution
Model discrete probability distributions for fixed trials with $P(X=k) = inom{n}{k}p^k(1-p)^{n-k}$. Visualize how the number of trials and success probability affect the shape, mean $mu = np$, and standard deviation $sigma = sqrt{np(1-p)}$ of binomial distributions.
Boxplot & Outlier Visualizer
Visualize five-number summaries (minimum, Q1, median, Q3, maximum) in boxplots and identify outliers using the IQR rule. Explore how outliers beyond $Q1 - 1.5 imes IQR$ or $Q3 + 1.5 imes IQR$ affect data distribution and summary statistics.
Chi-Square Goodness of Fit
Calculate and visualize the Chi-Square test statistic. Compare expected vs observed dice rolls across categories and plot how deviations force the P-value into the rejection region.
Type I/II Error & Statistical Power
Interactive hypothesis testing visualizer. Adjust Effect Size, Sample Size, and Alpha to instantly see the tradeoff balance between Type I Error, Type II Error, and Statistical Power.
Central Limit Theorem (Galton)
Drop dynamic particles through a physics-enabled Galton Board. Watch the Binomial Distribution organically construct and flawlessly approximate a continuous Normal Curve.
Simpson's Paradox Visualizer
Examine confounding variables dynamically. Toggle between global and clustered regression data to observe mathematical trends magically reverse polarity (Pearson r) when separated.
Simpson's Paradox Visualizer
Unlock one of statistics' most counter-intuitive phenomenons. Observe two completely separate datasets that both exhibit strong POSITIVE correlations. Merge them together and watch the global line of best fit instantly reverse into a NEGATIVE correlation.
Central Limit Theorem (Galton Board)
A dynamic 2D physics simulation of a Galton Board (Plinko). Drop hundreds of balls that make 50/50 left-right decisions, compounding entirely random events into a mathematically perfect Normal Distribution bell curve.
Least Squares Regression Predictor
Dynamically adjust a line of best fit to actively minimize the sum of squared residuals (SSE) compared to the true OLS regression model.
Margin of Error Simulator
Visualize exactly what the Margin of Error means geometrically. Observe how increasing sample size shrinks it while demanding higher confidence widens it.
Normal Distribution Explorer
Interactive Gaussian curve generator manipulating $\mu$ and $\sigma$ instantly mapped against the 68-95-99.7 Empirical Rule. Highlights discrete $Z$-score thresholds tracking precision standard deviation geometry exactly.
Binomial Probability Model
Interactive Binomial exact probability mass function visualizing discrete histograms. Demonstrates dynamic skew based on fixed $n, p$ ratios and overlays large-sample normal convergence boundaries visually calculating exact vs cumulative density outcomes.
Sampling Distributions (CLT)
Dual-canvas integration rendering Central Limit Theorem convergence. Draws random Monte Carlo datasets mapping sample size variance crushing ($n \geq 30$ rule) converting arbitrary shapes (skewed/bimodal) into stable Gaussian sample-mean distributions.
Confidence Intervals
Visualization generating multiple dynamic Confidence Interval margin 'nets' drawn from random standard errors. Empirically validates the meaning of 95% confidence by revealing the random capture vs failure rate against an unknown stationary population parameter line.
Hypothesis Testing Rules
Inferential hypothesis engine animating null model rejection thresholds vs interactive sample observed P-Value tails. Illuminates the deterministic binary conclusion states driven by fixed Alpha lines mapping Type I geometric error bounds visually.
Linear Regression (LSRL)
Interactive graphical Least Squares regression engine visually proving the $e^2$ square-area minimization property via manual vs automatic fitting while actively mapping high leverage outlier penalties.
Chi-Square Test (Goodness of Fit)
Goodness of Fit interaction engine assessing categorical skew. Graphically maps individual $(O-E)^2/E$ penalty variance blocks onto nominal bar distributions to visually sum into absolute $\chi^2$ scalar outcomes linking directly to cumulative significance rejection limits.
Student's t-Distribution
Interactive dynamic rendering of Student's T probability density logic modeling $df$ expansion from 1 stabilizing towards normal Z convergence. Highlights fat tail density displacement explicitly revealing structural variance uncertainty for small sample designs.
ANOVA Variance Analysis
Interactive ANOVA interaction plotting $MS_{bet}$ vs $MS_{wit}$ signal-to-noise calculations natively. Combines adjustable uniform variance blocks mathematically linking structural layout changes actively to precise numerical F-Distribution test thresholds.
Scatterplots & Correlation (r)
Interactive scatterplot geometry demonstrating Pearson's correlation coefficient mapping bounds ($|r| \to 1$). Employs bivariate normal Box-Muller transformations to visually generate dynamic density clouds calculating $R^2$ determination variances overlaid mathematically on LSRL projection planes natively.
Two-Way Independence
Probability matrices displaying Two-Way intersections visualizing Marginal row sums vs Joint cell probabilities dynamically tracking conditional shrinking denominators highlighting Independence verifications mathematically.
Type I & II Errors & Power
Interactive structural dual-distribution rendering plotting absolute hypothesis decision bounds. Animates the zero-sum mathematical see-saw between Alpha (Type I) restrictions severely punishing statistical Power translating into elevated Type II failures explicitly under sample size constrictions.