Visual Labs — Probability & Distributions

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2026-03-15

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Statistics I — Interactive Visual Labs

Statistics is the science of uncertainty. These tools help you visualize the "shape" of data and calculate probabilities without getting lost in the calculus.

1. Normal Distribution (Gaussian) Curve

The Bell Curve describes everything from heights to exam scores. Use the sliders to adjust the Mean ( $\mu$ ) and Standard Deviation ( $\sigma$ ), and drag the X-marker to calculate cumulative probability

P(X \dots x)

Probability Lab
Normal Distribution Simulator
84.1%
Area P(X ≤ 1)
Standard Deviations (σ)
Density
Mean (μ)0.0
Std Dev (σ)1.0
Select X1.0
Notice how increasing σ flattens the curve while μ shifts it left/right.

Tip

The Empirical Rule:

$68\%$ of data falls within $1\sigma$ .
$95\%$ of data falls within $2\sigma$ .
$99.7\%$ of data falls within $3\sigma$ . Use the simulator to verify these numbers!

2. Z-Scores and Standardisation

Any normal distribution can be converted to a Standard Normal Distribution (

Z

) where

\mu = 0

and

\sigma = 1

Z = \frac{x - \mu}{\sigma}

Best Practices

Interpreting the Curve:

If the curve is "tall and skinny", $\sigma$ is small (data is consistent).
If the curve is "short and fat", $\sigma$ is large (data is spread out).

Statistics I — Interactive Visual Labs

1. Normal Distribution (Gaussian) Curve

Probability Lab

2. Z-Scores and Standardisation

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