Normal Distribution :: Distribution Reference (Statistics Toolbox™)

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Normal Distribution

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Definition Background Parameters Example See Also

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Definition

The normal pdf is

Background

The normal distribution is a two-parameter family of curves. The first parameter, µ, is the mean. The second, σ, is the standard deviation. The standard normal distribution (written Φ(x)) sets µ to 0 and σ to 1.

Φ(x) is functionally related to the error function, erf.

The first use of the normal distribution was as a continuous approximation to the binomial.

The usual justification for using the normal distribution for modeling is the Central Limit Theorem, which states (roughly) that the sum of independent samples from any distribution with finite mean and variance converges to the normal distribution as the sample size goes to infinity.

Parameters

To use statistical parameters such as mean and standard deviation reliably, you need to have a good estimator for them. The maximum likelihood estimates (MLEs) provide one such estimator. However, an MLE might be biased, which means that its expected value of the parameter might not equal the parameter being estimated. For example, an MLE is biased for estimating the variance of a normal distribution. An unbiased estimator that is commonly used to estimate the parameters of the normal distribution is the minimum variance unbiased estimator (MVUE). The MVUE has the minimum variance of all unbiased estimators of a parameter.

The MVUEs of parameters µ and σ² for the normal distribution are the sample mean and variance. The sample mean is also the MLE for µ. The following are two common formulas for the variance.

where

Equation 1 is the maximum likelihood estimator for σ², and equation 2 is the MVUE.

As an example, suppose you want to estimate the mean, µ, and the variance, σ², of the heights of all fourth grade children in the United States. The function normfit returns the MVUE for µ, the square root of the MVUE for σ², and confidence intervals for µ and σ². Here is a playful example modeling the heights in inches of a randomly chosen fourth grade class.

height = normrnd(50,2,30,1);             % Simulate heights.
[mu,s,muci,sci] = normfit(height)

mu =
   50.2025

s =
    1.7946

muci =
   49.5210
   50.8841

sci =
    1.4292
    2.4125

Note that s^2 is the MVUE of the variance.

s^2

ans =
    3.2206

Example

The plot shows the bell curve of the standard normal pdf, with µ = 0 and σ = 1.