Normal Distribution
Last Updated: July 29, 2026 | By Mihail Sebastian | AI Dictionary
A symmetric, bell-shaped probability distribution defined entirely by its mean and standard deviation, and a default assumption across statistics and ML.
What is a Normal Distribution?
A normal distribution (also called a Gaussian distribution) is a continuous probability distribution whose values spread symmetrically around the mean in a bell-shaped curve, described completely by two numbers: the mean and the standard deviation.
Its reach comes from the central limit theorem: when an outcome is the sum of many small independent effects, its distribution tends toward this shape. That is why heights, measurement errors, and test scores all trace the same curve.
How a Normal Distribution Works
Two parameters fix the entire curve. The mean \(\mu\) sets the center; the standard deviation \(\sigma\) sets the spread. The density function is:
\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]In practice, the 68-95-99.7 rule does most of the work: about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. A z-score expresses any value in those standard-deviation units, so observations from different scales become directly comparable.
Machine learning depends on this distribution at several points: weight initialization schemes draw starting values from a Gaussian, noise in data is standardly modeled as Gaussian, and many of the tests used in hypothesis testing assume it.
Applications of the Normal Distribution
- Statistical inference: t-tests, ANOVA, and confidence intervals rest on normality assumptions about the data or the sampling distribution.
- Quality control: Manufacturers track whether measurements stay within a set number of standard deviations of a target; excursions signal a process fault.
- Finance: Asset returns are frequently modeled as normal to estimate risk and volatility – an approximation known to understate extreme events.
Example of a Normal Distribution
Suppose adult women’s heights in a population follow a normal distribution with a mean of 160 cm and a standard deviation of 5 cm. The 68-95-99.7 rule then answers concrete questions without any further calculation.
About 68% of women stand between 155 cm and 165 cm, and about 95% between 150 cm and 170 cm. A woman of 175 cm sits three standard deviations above the mean, in territory shared by roughly 0.15% of the population – exactly the reasoning anomaly detectors apply when they flag a data point as improbable.
Related AI terms: Z-Score · Hypothesis Testing · Monte Carlo · Gaussian Process · Weight Initialization
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Mihail Sebastian — Writes about AI governance, regulation, and the technology behind them. Placeholder bio — replace with a real credential line. About