Normal distribution

Normal Distribution is a type of probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, normal distribution will appear as a bell curve.

Definition

The normal distribution is defined by the following probability density function:

$f(x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$

where:

• $\mu$ is the mean or expectation of the distribution (and also its median and mode),
• $\sigma$ is the standard deviation, and
• ${\sigma }^2$ is the variance.

Properties

The normal distribution is the most important probability distribution in statistics because it fits many natural phenomena. For example, heights, blood pressure, measurement error, and IQ scores follow the normal distribution. It is also known as the Gaussian distribution and is often denoted by ${𝒩}(x)$.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution. It is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one.

Central Limit Theorem

The central limit theorem (CLT) states that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the shape of the population distribution.

Applications

The normal distribution is widely used in understanding distributions of factors in the population, and in determining the significance of sampling data in the social sciences, natural sciences and business.

See Also

• Probability distribution
• Standard deviation
• Variance
• Central limit theorem

• Probability Density Function of a Normal Distribution
  Probability Density Function of a Normal Distribution
• Cumulative Distribution Function of a Normal Distribution
  Cumulative Distribution Function of a Normal Distribution
• Standard Deviation Diagram
  Standard Deviation Diagram
• De Moivre-Laplace Theorem
  De Moivre-Laplace Theorem
• Central Limit Theorem with Dice Sums
  Central Limit Theorem with Dice Sums
• Probabilities of Functions of Normal Vectors
  Probabilities of Functions of Normal Vectors
• Quantum Harmonic Oscillator
  Quantum Harmonic Oscillator
• Fisher Iris Versicolor Sepal Width
  Fisher Iris Versicolor Sepal Width
• Fitting a Normal Distribution
  Fitting a Normal Distribution
• Galton Board
  Galton Board
• Carl Friedrich Gauss
  Carl Friedrich Gauss
• Pierre-Simon Laplace
  Pierre-Simon Laplace