3.1StatisticsCore

The Normal Distribution

The normal distribution is a continuous, bell-shaped model for data that clusters symmetrically about a mean. It is defined by two parameters: the mean μ and the standard deviation σ.

24 min Video by Zeeshan Zamurred The Normal Distribution

Edexcel A level Maths: 3.1 The Normal DistributionWatch the full walkthrough before the notes below.

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What you'll be able to do

Describe the shape of the normal distribution
Identify μ and σ
Use the notation X ~ N(μ, σ²)
Recall key features and percentages

Shape and notation

A normally distributed variable is written $X \sim N (μ, σ^{2})$ . The curve is symmetric about $μ$ , bell-shaped, and the total area under it is 1.

X \sim N (μ, σ^{2})

Mean μ, variance σ².

Key features

The mean, median and mode all equal $μ$ . About 68% of data lies within $1 σ$ of the mean, 95% within $2 σ$ , and 99.7% within $3 σ$ — the points of inflection are at $μ \pm σ$ .

Tip — In N(μ, σ²) the second parameter is the VARIANCE σ², not σ.

Formula recap

X \sim N (μ, σ^{2})

Standard notation.

mean = median = mode = μ

Symmetry.

\approx 95% within μ \pm 2 σ

Spread.

Common mistakes to avoid

Treating the second parameter as σ.

N(μ, σ²) — the second parameter is the variance.

Forgetting the curve is symmetric.

It is symmetric about μ; use this for probabilities.

Key takeaways

X ~ N(μ, σ²): bell-shaped, symmetric about μ.
Mean = median = mode = μ.
≈68/95/99.7% within 1/2/3 σ of the mean.

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