Normal distribution (Gaussian bell curve)

In short: The normal distribution is a bell-shaped distribution in which values in the middle are most frequent and become evenly rarer on both sides. Many natural and technical quantities follow it approximately.

In more detail: It is fully determined by the mean μ and the standard deviation σ. The curve is symmetric around μ.

In Depth

• 68-95-99.7 rule: About 68 % of values lie within μ ± σ, 95 % within μ ± 2σ, 99.7 % within μ ± 3σ.
• Z-score: How many standard deviations a value lies from the mean: z = (x − μ) / σ. This lets you compare different scales.
• Central limit theorem: Sums or means of many independent random quantities are approximately normally distributed, even if the individual ones are not. That explains why the bell appears so often.
• Examples: Body heights, measurement errors, dice sums of many dice, noise in signals.
• Counter-examples: Incomes, wealth, page views, downtimes are skewed (power laws, lognormal). Do not assume normality blindly.
• Applications: Hypothesis testing, quality control, initialising weights in neural networks (initialisation).
• Tools: scipy.stats.norm, spreadsheets (NORM.DIST).

See also: standard deviation, probability