1. Sampling distribution of the mean & the CLT
Draw repeated samples of size $n$ from a chosen population and pile up each sample mean $\bar X$. The histogram of means is the sampling distribution; it becomes ever more Normal and narrows with standard error $\mathrm{SE}=\sigma/\sqrt{n}$, regardless of the population shape (the CLT).
2. Maximum likelihood estimation
For a fixed sample, slide the parameter and watch the log-likelihood $\ell(\theta)=\sum_i \log f(x_i;\theta)$. The maximiser is the MLE $\hat\theta$. For the exponential, $\hat\lambda=1/\bar x$; for the Normal mean, $\hat\mu=\bar x$.
3. Method of moments vs. MLE
Two estimators on the same sample. The method of moments equates sample moments to population moments; the MLE maximises the likelihood. Repeatedly resample to compare their sampling spread — for the Gamma shape they differ, for the exponential rate they coincide ($1/\bar x$).
4. Confidence-interval coverage
Each horizontal bar is a 95% CI $\bar x \pm t_{0.025,n-1}\,s/\sqrt n$ from one fresh sample. Bars that miss the true mean $\mu$ are drawn in red. Over many samples the covered fraction approaches the stated confidence level — the meaning of “95% confident”.
5. CI width vs. sample size & confidence
The half-width of a mean CI is $t_{\alpha/2,\,n-1}\,s/\sqrt n$. It shrinks like $1/\sqrt n$ (quadrupling $n$ halves the width) and grows as you demand more confidence. The curve plots half-width against $n$ for the chosen level.
6. One-sample z / t test
Test $H_0:\mu=\mu_0$. The statistic is $z=\dfrac{\bar x-\mu_0}{\sigma/\sqrt n}$ (known $\sigma$) or $t=\dfrac{\bar x-\mu_0}{s/\sqrt n}$ (estimated). The shaded rejection region holds the most extreme $\alpha$ of the null distribution; reject when the statistic lands inside it.
7. The p-value as a tail area
Given an observed test statistic on the standard Normal null, the p-value is the probability of a result at least as extreme. Drag the statistic and the alternative; the shaded area is the p-value. Small tail area ⟹ the data are surprising under $H_0$.
8. Type I / II error & statistical power
Two distributions of $\bar X$: under $H_0$ (centre $\mu_0$) and under $H_1$ (centre $\mu_0+\delta$). The critical line splits them. Right of it under $H_0$ is $\alpha$ (Type I); left of it under $H_1$ is $\beta$ (Type II); power $=1-\beta$ is the rest. Move effect size, $n$ and $\alpha$ to grow the power.
9. Two-sample t-test (pooled vs. Welch)
Compare two group means. The pooled t assumes equal variances; Welch's t does not and adjusts the degrees of freedom (Satterthwaite). When variances or sizes differ, the two disagree — Welch is the safer default. Edit each group's mean, SD and size.
10. Paired vs. unpaired analysis
The same before/after data, two ways. Pairing analyses the within-subject differences $d_i$, removing between-subject variation; the unpaired test ignores the pairing. When subjects vary a lot but the change is consistent, pairing shrinks the standard error and lifts the t statistic.
11. One-way ANOVA — F = between / within
Several groups. ANOVA splits total variation into between-group (MSB) and within-group (MSE) parts; the ratio $F=\mathrm{MSB}/\mathrm{MSE}$ is large when group means are spread far relative to the noise. Move the group means apart or raise the noise and watch $F$ and its p-value react.
12. Chi-square test of independence
A contingency table of observed counts $O_{ij}$. Under independence the expected count is $E_{ij}=\dfrac{R_i\,C_j}{N}$, and $\chi^2=\sum\dfrac{(O-E)^2}{E}$ with $(r-1)(c-1)$ degrees of freedom. Edit the cells; the expected table, contributions and p-value update live.