The normal distribution is a continuous, symmetric, bell-shaped family of distributions. It is determined by two parameters: its mean $\mu$ and standard deviation $\sigma>0$.

We write

$$X\sim N(\mu,\sigma^2).$$

The mean fixes the center of the bell, while the standard deviation controls its horizontal spread.

Symmetry and shape

The density is symmetric about $\mu$. Values near the mean have the greatest density, and the tails decrease smoothly in both directions.

Changing $\mu$ shifts the distribution without changing its shape. Increasing $\sigma$ spreads the same total probability over a wider range.

Standardizing

Any normal random variable can be converted to a standard normal variable by

$$Z=\frac{X-\mu}{\sigma}.$$

The transformed variable has mean $0$ and standard deviation $1$:

$$Z\sim N(0,1).$$

For example, $Z=2$ means the original value lies two standard deviations above its mean.

Probabilities and quantiles

Normal probabilities are areas under the normal density. Standardization lets the same standard-normal distribution be used for every choice of $\mu$ and $\sigma$.

Quantiles reverse this process: a probability determines the value below which that proportion of the distribution lies.

A model, not a universal law

Some measured variables are approximately normal, and normal distributions also arise as approximations through the central limit theorem. But skewed, bounded, multimodal or heavy-tailed data may be poorly represented by a normal model.