Theory: The Gaussian (Normal) distribution is defined by its mean (μ) and variance (σ²). The mean sets the center, and the variance (or its square root, the standard deviation σ) sets the spread. The 3σ rule states that for any Gaussian distribution, about 95.45% of data falls within two standard deviations of the mean, i.e., in the range [μ - 2σ, μ + 2σ].

Procedure:

1. Use the "Mean" and "Variance" sliders to shape the Probability Density Function (PDF). Observe how the curve changes.
2. Calculate the peak height of the current PDF using the formula: \(height = \frac{1}{\sigma\sqrt{2\pi}}\). Enter it and click "Check Answer". Your answer must be within +/- 0.01 of the correct value.
3. Next, use the "Number of Samples" slider to choose how many random points to generate from your configured distribution.
4. Click "Generate Samples". The samples will be drawn on the chart. Samples in green fall within the [μ - 2σ, μ + 2σ] range; those in red fall outside.
5. Compare the measured percentage of samples inside the range to the theoretical 95.45%. Notice how it gets closer as the number of samples increases (Law of Large Numbers).