Normal (Gaussian) Distribution

Probability · Axiom Academy

LESSON Normal (Gaussian) Distribution The most important probability distribution in statistics The normal PDF has its distinctive bell shape: symmetric around the mean μ, with the peak at x = μ. The standard deviation σ controls how spread out the distribution is. About 68% of values fall within one σ of μ, 95% within two σ, and 99.7% within three σ. We write X ~ N(μ, σ²) to denote that X follows a normal distribution with mean μ and variance σ². The parameter μ shifts the center of the distribution left or right, while σ² controls the width. A larger σ² means more spread and a flatter, wider curve. A smaller σ² concentrates values tightly around μ, creating a tall, narrow curve. 3. The Empirical Rule (68-95-99.7) This rule provides quick approximations for probabilities. Approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This makes the normal distribution highly predictable.

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