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Visual Labs — Probability & Distributions

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Visual Labs — Probability & Distributions

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1 min read

2026-03-15

Reading compass

Now · 1. Normal Distribution (Gaussian) Curve

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)

Interactive study checkpoint
Distribution reading lab
Change one property at a time. Read the curve as a model of a variable, not as a picture to memorise.
Model result
84.1%
of modelled observations are at or below x = 1.0.
value of xrelative densityμ 0.0x 1.0
The shaded probability is P(X ≤ 1.0). The cut-off sits past the centre into the right side. This is a model probability—not a guarantee about the next individual observation.
Centre (μ)0.00Spread (σ)1.00Read at x1.00Read before calculating
Increase σ: the curve spreads and its peak lowers. Shift μ: the whole model moves, without becoming more or less variable.

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).