Erthiox Labs - Statistics & Probability Sandbox

Central Limit Theorem Monte Carlo Simulation

Experiment Type

Batch Size

Rolls per batch100

Z-Score Shading

Show ±σ Region1.0 σ

Overlay Curve

Descriptive Statistics

Total Samples (n)0

Sample Mean (x̄)—

Std Deviation (σ)—

Std Error (SEM)—

Min—

Max—

Z-Score (±σ Area) 68.27%

Theory: Central Limit Theorem

The Central Limit Theorem (CLT) states that the distribution of sample means approximates a normal distribution (bell curve) as the sample size becomes large, regardless of the population's distribution shape. The standard deviation of the sample means is the **Standard Error of the Mean (SEM)**, representing the spread.

Interactive Lab Tasks

Task 1: Generate Batches
Select 'Roll Dice' preset, set Rolls per batch to 100, and click 'Run Batch' multiple times. Notice how the histogram gathers around the mean (3.5). Task 2: Watch the Bell Shape Form
Click 'Auto-Simulate (10k/s)' and watch the sample counts climb. Verify that the random samples converge into a smooth normal distribution curve. Task 3: Normal Standard Shading
In the Distribution tab, adjust the 'Show ±σ Region' slider. Observe the corresponding Z-score area percentage in the Readouts tab.