StatLab Interactive Probability & Statistics Learning Hub

Understand Statistics Through Data, Change, and Inference

Explore CLT, Law of Large Numbers, Skewness, Normal vs Lognormal, and Hypothesis Testing (z, t, Chi-Square, F) with real-time visual simulations, bilingual explanations, and IPA.

6 Core Modules
6 Live Simulators
15+ Phonetic Terms
Module 1

Mean, Median, Mode, Standard Deviation & Skewness

Understand how measures of central tendency behave under different distribution shapes and extreme outliers.

Central Tendency Measures

  • Mean (均值): /miːn/ The arithmetic average. Highly sensitive to outliers.
    $$\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i$$
  • Median (中位数): /ˈmiːdiən/ The 50th percentile value. Robust to extreme outliers.
  • Mode (众数): /moʊd/ The most frequently occurring value in the distribution dataset.

Dispersion & Skewness

  • Standard Deviation (标准差): /ˈstændərd ˌdiːviˈeɪʃn/ Measures average distance of observations from the mean.
    $$s = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2}$$
  • Skewness (偏度): /ˈskjuːnəs/ Measures asymmetry of probability distribution.
    $$\gamma_1 = \text{E}\left[\left(\frac{X-\mu}{\sigma}\right)^3\right]$$
  • Kurtosis (峰度): /kɜːrˈtoʊsɪs/ Measures tail-heaviness and peak sharpness relative to normal distribution.

Interactive Skewness & Central Tendency Simulator

Drag the skewness slider to see how Mean, Median, and Mode shift relative to each other!

Symmetric (正态/对称)
Left (-1.5) Right (+1.5)
0.0
+0 Outlier
Mean: 50.00
Median: 50.00
Mode: 50.00
Symmetric distribution: Mean = Median = Mode
Module 2

Limit Theorems & Core Distributions

Explore Law of Large Numbers (LLN), Central Limit Theorem (CLT), and Normal vs. Lognormal transformations.

Law of Large Numbers (LLN) 大数定律 /lɔː əv lɑːrdʒ ˈnʌmbərz/

As the sample size $n$ increases, the sample mean $\bar{X}_n$ converges towards the true population mean $\mu$ with probability 1 (Strong LLN).

$$\lim_{n \to \infty} P\left(|\bar{X}_n - \mu| < \epsilon\right)=1$$

LLN Sample Convergence Simulator

Simulate tossing coins / rolling dice to see sample mean converge to theoretical expectation.

300 Trials
Theoretical Mean ($\mu$): 0.500
Final Sample Mean ($\bar{x}_n$): -
Absolute Error ($|\bar{x}_n - \mu|$): -

Central Limit Theorem (CLT) 中心极限定理 /ˈsɛntrəl ˈlɪmɪt ˈθɪərəm/

Regardless of the underlying population distribution shape (Uniform, Exponential, Bimodal), the distribution of sample means $\bar{X}$ approaches a Normal distribution with Asymptotic Normality /ˌæsɪmˈptɑːtɪk/ as sample size $n \ge 30$.

$$\bar{X}_n \xrightarrow{d} \mathcal{N}\left(\mu, \frac{\sigma^2}{n}\right)$$

CLT Interactive Sampling Laboratory

Select a non-normal parent population, adjust sample size $n$, and watch the sampling distribution become Gaussian!

n = 30
K = 800
Std Error $SE = \frac{\sigma}{\sqrt{n}}$: 0.053

1. Parent Population Shape

2. Distribution of Sample Means ($\bar{X}$)

Normal vs. Lognormal Distributions 正态与对数正态 /ˈlɔːɡnɔːrməl/

If random variable $X \sim \mathcal{N}(\mu, \sigma^2)$, then $Y = e^X$ follows a Lognormal distribution. Lognormal variables are non-negative and right-skewed, common in financial stock prices, income distributions, and biological sizes.

$$X = \ln(Y) \sim \mathcal{N}(\mu, \sigma^2) \iff Y = e^X \sim \text{Lognormal}(\mu, \sigma^2)$$

Normal & Lognormal Comparative Explorer

Adjust $\mu$ and $\sigma$ to observe how exponential transformation stretches the right tail.

0.0
0.50
Lognormal Mean $E[Y] = e^{\mu + \sigma^2/2}$: 1.13
Lognormal Median $= e^\mu$: 1.00
Lognormal Mode $= e^{\mu - \sigma^2}$: 0.78

Normal Distribution $X \sim \mathcal{N}(\mu, \sigma^2)$

Lognormal Distribution $Y = e^X$

Module 3

Hypothesis Testing: Z-test, T-test, Chi-Square & F-test

Visualizing Null Hypotheses $H_0$, Significance Levels $\alpha$, Critical Values, and P-values.

Z-Test Parameter Workbench & Density Plot

Adjust sample stats and significance level to visualize rejection region & P-value.

Fail to Reject H₀

z = 1.96
Observed statistic z = 1.96
Critical Value(s): ±1.960
Calculated P-value: p = 0.0500

Fail to Reject H₀

Accept Region ($1-\alpha$) Rejection Region ($\alpha$) P-value Area
1
HypothesesH₀: no effect
2
Test ruleTwo-tailed, α = 0.05
3
Evidencez = 1.96 · p = 0.0500
4
DecisionFail to reject H₀

Hypothesis Test Selector Matrix

Test Type Primary Purpose Assumptions & Sample Size Test Statistic Formula
Z-Test /ziː test/ Compare sample mean to population mean when variance is known. Known $\sigma$, Normal distribution or large sample size ($n \ge 30$). $$Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}$$
t-Test /tiː test/ Compare mean(s) when population variance is unknown. Unknown $\sigma$, normally distributed population, small sample ($n < 30$). $$t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}, \quad df = n-1$$
Chi-Square ($\chi^2$) /ˈkaɪ skwer/ Goodness of fit test or test of independence between categorical variables. Categorical data, expected frequency per cell $\ge 5$. Right-skewed. $$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}, \quad df = k-1$$
F-Test (ANOVA) /əˈnoʊvə/ Compare two population variances or test overall equality of 3+ means (ANOVA). Independent normally distributed samples. Ratio of sample variances. $$F = \frac{s_1^2}{s_2^2}, \quad df_1 = n_1-1, \; df_2 = n_2-1$$
Module 4

Interactive Knowledge Check & Quiz

Test your understanding of CLT, LLN, Distributions, Descriptive Stats, and Hypothesis Tests.

Question 1 of 6 Score: 0
Central Limit Theorem

Loading question...

Module 5

Probability & Statistics Formula Sheet

Essential mathematical formulas, properties, and key statistical concepts.

1. Central Tendency & Dispersion

Sample Mean
$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$$
Sample Variance
$$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$
Standard Error (SE)
$$SE = \frac{\sigma}{\sqrt{n}}$$

2. Common Probability Distributions

Normal Distribution PDF
$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$
Lognormal Distribution PDF
$$f(y) = \frac{1}{y\sigma\sqrt{2\pi}} e^{-\frac{(\ln y - \mu)^2}{2\sigma^2}}, \quad y > 0$$

3. Hypothesis Test Statistics

One-Sample Z Statistic
$$Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}$$
One-Sample t Statistic
$$t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}, \quad df = n-1$$
Chi-Square ($\chi^2$) Goodness-of-Fit
$$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}, \quad df = k-1$$
Two-Sample F Test
$$F = \frac{s_1^2}{s_2^2}, \quad df_1 = n_1-1, \, df_2 = n_2-1$$

⚡ Key Rules & Concepts

  • P-value Definition: P-value is the probability of obtaining test results at least as extreme as observed, assuming $H_0$ is TRUE. It is NOT the probability that $H_0$ is true!
  • Type I vs Type II Error:
    • Type I ($\alpha$): Reject $H_0$ when $H_0$ is actually true (False Positive).
    • Type II ($\beta$): Fail to reject $H_0$ when $H_0$ is false (False Negative).
  • Skewness Order:
    • Right-Skewed (正偏态): $\text{Mean} > \text{Median} > \text{Mode}$
    • Left-Skewed (负偏态): $\text{Mean} < \text{Median} < \text{Mode}$