Interactive normal distribution

Drag the sliders to change the mean (μ) and standard deviation (σ). See how the bell curve changes shape and explore the 68-95-99.7 rule.

The normal distribution:

f(x) = (1 / (σ√(2π))) · e^(−(x−μ)² / (2σ²))

μ (mu) is the mean — the centre of the bell. σ (sigma) is the standard deviation — controls how wide or narrow the bell is.

μ = 0.0 — mean

σ = 1.0 — standard deviation

Shade area: P(a < X < b)

a = -1.0

b = 1.0

P(a < X < b)

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Mean (μ)

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Std dev (σ)

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Peak f(μ)

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Variance (σ²)

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68-95-99.7 rule (empirical rule):

μ ± 1σ

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expected: 68.3%

μ ± 2σ

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expected: 95.4%

μ ± 3σ

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expected: 99.7%

Properties of the normal distribution

Symmetric about μ Bell shape

Mean = Median = Mode All equal μ

Area under curve = 1 Total probability

68% within μ ± σ Empirical rule

95% within μ ± 2σ Empirical rule

99.7% within μ ± 3σ Empirical rule

Larger σ → wider bell Spread

Z = (X−μ)/σ Standardisation

Key insight: The shape of the bell only depends on σ — not on μ! Changing μ just slides the curve left or right.

It is the mark of a truly intelligent person to be moved by statistics.

— George Bernard Shaw (1856–1950)