Continuous probability distribution
Normal (Gaussian) distribution
X ~ Normal(μ, σ²)
The bell curve. By the Central Limit Theorem, sums and averages of many independent influences tend toward a Normal distribution — which is why it appears throughout statistics and nature. About 68% of values fall within one standard deviation of the mean, and 95% within two.
Explore it interactivelyKey facts
- Notation
- X ~ Normal(μ, σ²)
- Type
- continuous
- Parameters
- μ — mean (centre); σ — standard deviation (spread, σ > 0)
- Support
- (−∞, ∞)
- f(x) = 1 / (σ√(2π)) · e^(−(x − μ)² / (2σ²))
- Mean
- μ
- Variance
- σ²
When to use it
- Measurement error
- Heights, weights, and test scores
- The sampling distribution of a mean
- Modelling noise
Related distributions
Frequently asked questions
What is the Normal (Gaussian) distribution?
The bell curve. By the Central Limit Theorem, sums and averages of many independent influences tend toward a Normal distribution — which is why it appears throughout statistics and nature. About 68% of values fall within one standard deviation of the mean, and 95% within two.
What are the mean and variance of the Normal (Gaussian) distribution?
For X ~ Normal(μ, σ²), the mean is μ and the variance is σ².
When is the Normal (Gaussian) distribution used?
It is commonly used for: Measurement error; Heights, weights, and test scores; The sampling distribution of a mean; Modelling noise.