The normal distribution, also known as the Gaussian distribution, is the most ubiquitous probability model in quantitative science.
Its probability density function produces the familiar symmetric, bell-shaped curve that extends infinitely in both directions.
The curve is governed entirely by two parameters: the mean mu, which sets location, and the standard deviation sigma, which sets scale.
The density reaches its peak at the mean, where the distribution is perfectly symmetric and inflection points occur at one standard deviation.
The standard normal distribution is a special case with a mean of zero and a standard deviation of one, denoted by the letter Z.
Any normal random variable can be converted to standard normal form by subtracting its mean and dividing by its standard deviation.
This transformation produces a z-score, which measures an observation distance from the mean in standard deviation units.
The empirical rule establishes that 68, 95 and 99.7 percent of values fall within 1, 2 and 3 standard deviations.
Because probabilities outside three standard deviations are tiny, normal models predict that extreme outliers are exceptionally rare.
In financial markets, assuming asset returns follow normal distributions has caused catastrophic underestimation of tail risk.
Real market returns have heavier tails than a normal curve, generating crashes far more often than Gaussian theory predicts.
Despite this limitation, normal distributions describe measurement noise, biological dimensions, and physical traits with remarkable fidelity.
Normal distributions are mathematically convenient: linear combinations of independent normal variables remain strictly normal.
In linear regression, model errors are traditionally assumed to be normally distributed to justify standard hypothesis tests.
A frequent mistake is assuming that all symmetric bell-shaped data must be Gaussian, ignoring heavy-tailed alternatives like Student t distributions.