The normal or Gaussian distribution is the classic "bell curve". This is a continuous symmetric distribution defined over all real numbers. A location parameter μ specifies the mean of the distribution, while the variance of the distribution is given by a parameter σ².
| Parameter | Range | Description |
|---|---|---|
| μ | − ∞ < μ < ∞ | Expected value |
| σ² | σ² > 0 | Variance |
Probability Density Function
Support
Mean
Variance
| Example | μ | σ² |
|---|---|---|
| The life of laptop batteries has a normal distribution with mean 4 hours and standard deviation 1 hour. Let X be the battery life of a randomly chosen laptop. | 4.000 | 1.000 |
| Birth weights of babies born in the United States are normally distributed with mean 3.4 kilograms and standard deviation 0.57 kilograms. Let X be the birth weight of a randomly chosen baby. | 3.400 | 0.3249 |
| Heights of male red kangaroos are normally distributed with approximate mean 1.5 meters and standard deviation 0.12 meters. Let X be the height of a randomly chosen male red kangaroo. | 1.500 | 0.0144 |
X ~ Normal(μ, σ²)
E(X) = , Var(X) =
The normal distribution is one of the most commonly used distributions in all of probability theory. This is a consequence of the "Central Limit Theorem", which states that the distribution of the sample means of n independent random variables converges to a normal distribution as n increases. Many quantities which result from the addition of multiple independent factors therefore naturally have a normal distribution.
The illustration above shows a sample of n points chosen independently and at random from a uniform(μ - σ√(3n), μ + σ√(3n)) distribution. The population has mean μ and variance σ2n, while the sample mean X̅ (shown as a light blue line) has mean μ and variance σ2. By the Central Limit Theorem, X̅ converges to a normal(μ, σ2) distribution as n approaches infinity.
Note that the Central Limit Theorem holds for the sample mean of any distribution, as long as the variance is finite.
The simulation above shows a sample of n points (marked red) chosen independently and at random from a uniform(μ - σ√(3n), μ + σ√(3n)) distribution with mean μ and variance σ2n. The light blue line shows the sample mean X̅, which itself has mean μ and variance σ2. By the Central Limit Theorem, X̅ converges to a normal(μ, σ2) distribution as n approaches infinity. The histogram accumulates the results of each simulation.