Documentation

# NormalDistribution

Normal probability distribution object

## Description

A NormalDistribution object consists of parameters, a model description, and sample data for a normal probability distribution.

The normal distribution, sometimes called the Gaussian distribution, is a two-parameter family of curves. 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.

The normal distribution uses the following parameters.

ParameterDescriptionSupport
mu (μ)Mean$-\infty <\mu <\infty$
sigma (σ)Standard deviation$\sigma \ge 0$

## Creation

There are several ways to create a NormalDistribution probability distribution object.

• Create a distribution with specified parameter values using makedist.

• Fit a distribution to data using fitdist.

• Interactively fit a distribution to data using the Distribution Fitter app.

## Properties

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### Distribution Parameters

Mean of the normal distribution, specified as a scalar value.

Data Types: single | double

Standard deviation of the normal distribution, specified as a nonnegative scalar value.

You can specify sigma to be zero when you create an object by using makedist. Some object functions support an object pd with zero standard deviation. For example, random(pd) always returns mu, and cdf(pd,x) returns either 0 or 1. The output is 0 if x is smaller than mu, and 1 otherwise. mean, std, and var return the mean, standard deviation, and variance of pd, respectively.

Data Types: single | double

### Distribution Characteristics

Logical flag for truncated distribution, specified as a logical value. If IsTruncated equals 0, the distribution is not truncated. If IsTruncated equals 1, the distribution is truncated.

Data Types: logical

Number of parameters for the probability distribution, specified as a positive integer value.

Data Types: double

Covariance matrix of the parameter estimates, specified as a p-by-p matrix, where p is the number of parameters in the distribution. The (i,j) element is the covariance between the estimates of the ith parameter and the jth parameter. The (i,i) element is the estimated variance of the ith parameter. If parameter i is fixed rather than estimated by fitting the distribution to data, then the (i,i) elements of the covariance matrix are 0.

Data Types: double

Logical flag for fixed parameters, specified as an array of logical values. If 0, the corresponding parameter in the ParameterNames array is not fixed. If 1, the corresponding parameter in the ParameterNames array is fixed.

Data Types: logical

Distribution parameter values, specified as a vector.

Data Types: single | double

Truncation interval for the probability distribution, specified as a vector containing the lower and upper truncation boundaries.

Data Types: single | double

### Other Object Properties

Probability distribution name, specified as a character vector.

Data Types: char

Data used for distribution fitting, specified as a structure containing the following:

• data: Data vector used for distribution fitting.

• cens: Censoring vector, or empty if none.

• freq: Frequency vector, or empty if none.

Data Types: struct

Distribution parameter descriptions, specified as a cell array of character vectors. Each cell contains a short description of one distribution parameter.

Data Types: char

Distribution parameter names, specified as a cell array of character vectors.

Data Types: char

## Object Functions

 cdf Cumulative distribution function icdf Inverse cumulative distribution function iqr Interquartile range mean Mean of probability distribution median Median of probability distribution negloglik Negative loglikelihood of probability distribution paramci Confidence intervals for probability distribution parameters pdf Probability density function proflik Profile likelihood function for probability distribution random Random numbers std Standard deviation of probability distribution truncate Truncate probability distribution object var Variance of probability distribution

## Examples

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Create a normal distribution object using the default parameter values.

pd = makedist('Normal')
pd =
NormalDistribution

Normal distribution
mu = 0
sigma = 1

Create a normal distribution object by specifying the parameter values.

pd = makedist('Normal','mu',75,'sigma',10)
pd =
NormalDistribution

Normal distribution
mu = 75
sigma = 10

Compute the interquartile range of the distribution.

r = iqr(pd)
r = 13.4898

Load the sample data and create a vector containing the first column of student exam grade data.

Create a normal distribution object by fitting it to the data.

pd = fitdist(x,'Normal')
pd =
NormalDistribution

Normal distribution
mu = 75.0083   [73.4321, 76.5846]
sigma =  8.7202   [7.7391, 9.98843]

The intervals next to the parameter estimates are the 95% confidence intervals for the distribution parameters.