Normal Distribution
Statistics & ProbabilityThe normal distribution is a symmetric, bell-shaped probability distribution that describes many natural phenomena.
Formula
f(x) = \dfrac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)}
Definition
The normal distribution is a bell-shaped curve where most values cluster near the mean and fewer values appear as you move farther away. It is perfectly symmetric.
Example
Heights of adults, scores on standardized tests, and measurement errors tend to follow a normal distribution: most people cluster around the average, and very tall or very short people are rarer.
Key Insight
The normal distribution is sometimes called the "bell curve" because of its shape. Nature produces it so often because many things result from the sum of many small random effects.
Definition
The normal distribution $N(\mu, \sigma^2)$ is a symmetric, unimodal, continuous probability distribution fully described by its mean $\mu$ and standard deviation $\sigma$. The empirical rule: $68\%$ of data falls within $1\sigma$, $95\%$ within $2\sigma$, $99.7\%$ within $3\sigma$ of the mean.
Example
IQ scores are designed to be $N(100, 15^2)$. About $68\%$ of people score between $85$ and $115$. About $95\%$ score between $70$ and $130$. Only about $0.3\%$ score below $55$ or above $145$.
Key Insight
The normal distribution is important partly because of the central limit theorem: the mean of any large sample is approximately normally distributed regardless of the underlying population shape.
Definition
The normal distribution $N(\mu, \sigma^2)$ has PDF $f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$. It is the maximum entropy distribution for a given mean and variance. The multivariate normal $N(\mu, \Sigma)$ generalizes it with mean vector $\mu$ and covariance matrix $\Sigma$, with PDF $f(x) = (2\pi)^{-k/2}|\Sigma|^{-1/2}\exp\left(-\tfrac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\right)$.
Example
The moment generating function of $N(\mu, \sigma^2)$ is $M(t) = \exp(\mu t + \sigma^2 t^2/2)$, which completely characterizes the distribution. Its log is a quadratic in $t$, a property unique to the normal among all distributions with finite moments.
Key Insight
The central limit theorem (CLT): if $X_1, \ldots, X_n$ are i.i.d. with mean $\mu$ and variance $\sigma^2$, then $\sqrt{n}(\bar{x}-\mu)/\sigma$ converges in distribution to $N(0,1)$. This is the mathematical foundation for normal-based inference and explains the ubiquity of the normal distribution in statistical practice.