🎲 Normal Distribution

Normal Distribution#

The bell-shaped Gaussian distribution defined by its mean and variance.

Important

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What it is#

The normal (Gaussian) distribution is the continuous, bell-shaped, symmetric distribution defined by two parameters — the mean \(\mu\) (its center) and the standard deviation \(\sigma\) (its spread; variance \(\sigma^2\)):

\[X \sim \mathcal{N}(\mu, \sigma^2).\]

Its mean, median and mode coincide, and it extends from \(-\infty\) to \(+\infty\).

The density#

\[f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right).\]

About 68% of values lie within \(1\sigma\) of the mean, 95% within \(2\sigma\), and 99.7% within \(3\sigma\) (the 68–95–99.7 rule). Standardizing with \(z = (x-\mu)/\sigma\) maps any normal onto the standard normal \(\mathcal{N}(0, 1)\), so a single table serves all.

Why it’s everywhere#

The Central Limit Theorem — averages of many independent, finite-variance quantities tend toward a normal — makes it the default model for measurement errors and aggregates. But it has light tails: with heavy-tailed data or frequent outliers (e.g. Cauchy, Pareto) it fits poorly and least-squares methods grow unreliable.

Theme: Probability & Statistics Foundations · All terminology

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Mind map — connected ideas

Probability Distribution · Probability Density · Cumulative Distribution Function (CDF) · Z-Score · Standard Error (SE) · Bootstrap Confidence Intervals (CIs)

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More in Probability & Statistics Foundations

Beta Distribution · Confidence Level · Correlation · Critical Value · Cumulative Distribution Function (CDF) · Frequentist · IID (Independent and Identically Distributed) · Likelihood · Margin of Error (MoE) · Mean · Median · Outlier · Population Proportion · Probability