Statistics: NormalDistribution

Class Definition: `NormalDistribution`

statistics: NormalDistribution

Normal probability distribution object.

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

The normal distribution is a continuous probability distribution that is symmetric about the mean, mu, showing that data near the mean are more frequent in occurrence than data far from the mean. It is defined by location parameter mu and scale parameter sigma.

There are several ways to create a NormalDistribution object.

Fit a distribution to data using the fitdist function.
Create a distribution with fixed parameter values using the makedist function.
Use the constructor NormalDistribution (mu, sigma) to create a normal distribution with fixed parameter values mu and sigma.
Use the static method NormalDistribution.fit (x, censor, freq, options) to fit a distribution to data x.

It is highly recommended to use fitdist and makedist functions to create probability distribution objects, instead of the class constructor or the aforementioned static method.

Further information about the normal distribution can be found at https://en.wikipedia.org/wiki/Normal_distribution

See also: fitdist, makedist, normcdf, norminv, normpdf, normrnd, normfit, normlike, normstat

Source Code: NormalDistribution

Properties

A scalar value characterizing the location of the normal distribution. You can access the mu property using dot name assignment.

Example: 1


 ## Create a Normal distribution with default parameters
 data = randn (10000, 1);
 pd = fitdist (data, "Normal");

 ## Query parameter 'mu' (location parameter, mean)
 pd.mu

 ## Set parameter 'mu'
 pd.mu = 1

 ## Use this to initialize or modify the mean of a Normal distribution.
 ## The mean parameter must be a real scalar, representing the center of symmetry,
 ## useful for shifting the distribution, such as modeling centered data like errors.

ans = -0.011744
pd =
  NormalDistribution

  normal distribution
       mu =        1
    sigma = 0.992727

Example: 2


 ## Create a Normal distribution object by calling its constructor
 pd = NormalDistribution (1.5, 2)

 ## Query parameter 'mu'
 pd.mu

 ## This demonstrates direct construction with a specific mean,
 ## suitable for modeling data with a known central tendency, like IQ scores or heights.

pd =
  NormalDistribution

  normal distribution
       mu = 1.5
    sigma =   2

ans = 1.5000

A positive scalar value characterizing the scale of the normal distribution. You can access the sigma property using dot name assignment.

Example: 1


 ## Create a Normal distribution with default parameters
 data = randn (10000, 1);
 pd = fitdist (data, "Normal");

 ## Query parameter 'sigma' (scale parameter, standard deviation)
 pd.sigma

 ## Set parameter 'sigma'
 pd.sigma = 2

 ## Use this to initialize or modify the standard deviation in a Normal
 ## distribution. The scale parameter must be a positive real scalar, controlling
 ## the spread around the mean.

ans = 0.9891
pd =
  NormalDistribution

  normal distribution
       mu = -0.0142872
    sigma =          2

Example: 2


 ## Create a Normal distribution object by calling its constructor
 pd = NormalDistribution (0, 1.5)

 ## Query parameter 'sigma'
 pd.sigma

 ## This shows how to set the scale parameter directly via the constructor,
 ## ideal for modeling variability in symmetric data, such as measurement precision.

pd =
  NormalDistribution

  normal distribution
       mu =   0
    sigma = 1.5

ans = 1.5000

A character vector specifying the name of the probability distribution object. This property is read-only.

A scalar integer value specifying the number of parameters characterizing the probability distribution. This property is read-only.

A $2×1$ cell array of character vectors with each element containing the name of a distribution parameter. This property is read-only.

A $2×1$ cell array of character vectors with each element containing a short description of a distribution parameter. This property is read-only.

A $2×1$ numeric vector containing the values of the distribution parameters. This property is read-only. You can change the distribution parameters by assigning new values to the mu and sigma properties.

A $2×2$ numeric matrix containing the variance-covariance of the parameter estimates. Diagonal elements contain the variance of each estimated parameter, and non-diagonal elements contain the covariance between the parameter estimates. The covariance matrix is only meaningful when the distribution was fitted to data. If the distribution object was created with fixed parameters, or a parameter of a fitted distribution is modified, then all elements of the variance-covariance are zero. This property is read-only.

A $1×2$ logical vector specifying which parameters are fixed and which are estimated. true values correspond to fixed parameters, false values correspond to parameter estimates. This property is read-only.

A $1×2$ numeric vector specifying the truncation interval for the probability distribution. First element contains the lower boundary, second element contains the upper boundary. This property is read-only. You can only truncate a probability distribution with the truncate method.

A logical scalar value specifying whether a probability distribution is truncated or not. This property is read-only.

A scalar structure containing the following fields:

data: a numeric vector containing the data used for distribution fitting.
cens: a numeric vector of logical values indicating censoring information corresponding to the elements of the data used for distribution fitting. If no censoring vector was used for distribution fitting, then this field defaults to an empty array.
freq: a numeric vector of non-negative integer values containing the frequency information corresponding to the elements of the data used for distribution fitting. If no frequency vector was used for distribution fitting, then this field defaults to an empty array.

Methods

NormalDistribution: p = cdf (pd, x)
NormalDistribution: p = cdf (pd, x, "upper")

p = cdf (pd, x) computes the CDF of the probability distribution object, pd, evaluated at the values in x.

p = cdf (…, "upper") returns the complement of the CDF of the probability distribution object, pd, evaluated at the values in x.

Example: 1


 ## Plot various CDFs from the Normal distribution
 x = -5:0.01:5;
 data1 = 0 + 0.5 * randn (10000, 1);
 data2 = 0 + 1.0 * randn (10000, 1);
 data3 = 0 + 2.0 * randn (10000, 1);
 pd1 = fitdist (data1, "Normal");
 pd2 = fitdist (data2, "Normal");
 pd3 = fitdist (data3, "Normal");
 p1 = cdf (pd1, x);
 p2 = cdf (pd2, x);
 p3 = cdf (pd3, x);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r")
 grid on
 legend ({"mu = 0, sigma = 0.5", "mu = 0, sigma = 1", "mu = 0, sigma = 2"}, ...
         "location", "southeast")
 title ("Normal CDF")
 xlabel ("values in x")
 ylabel ("Cumulative probability")

 ## Use this to compute and visualize the cumulative distribution function
 ## for different Normal distributions, showing how probability accumulates,
 ## essential in hypothesis testing or confidence intervals.

NormalDistribution: x = icdf (pd, p)

x = icdf (pd, p) computes the quantile (the inverse of the CDF) of the probability distribution object, pd, evaluated at the values in p.

Example: 1


 ## Plot various iCDFs from the Normal distribution
 p = 0.001:0.001:0.999;
 data1 = 0 + 0.5 * randn (10000, 1);
 data2 = 0 + 1.0 * randn (10000, 1);
 data3 = 0 + 2.0 * randn (10000, 1);
 pd1 = fitdist (data1, "Normal");
 pd2 = fitdist (data2, "Normal");
 pd3 = fitdist (data3, "Normal");
 x1 = icdf (pd1, p);
 x2 = icdf (pd2, p);
 x3 = icdf (pd3, p);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r")
 grid on
 legend ({"mu = 0, sigma = 0.5", "mu = 0, sigma = 1", "mu = 0, sigma = 2"}, ...
         "location", "northwest")
 title ("Normal iCDF")
 xlabel ("Probability")
 ylabel ("values in x")

 ## This demonstrates the inverse CDF (quantiles) for Normal
 ## distributions, useful for finding critical values in statistics,
 ## such as z-scores for confidence levels.

NormalDistribution: r = iqr (pd)

r = iqr (pd) computes the interquartile range of the probability distribution object, pd.

Example: 1


 ## Compute the interquartile range for a Normal distribution
 data = randn (10000, 1);
 pd = fitdist (data, "Normal");
 iqr_value = iqr (pd)

 ## Use this to calculate the interquartile range, which measures the spread
 ## of the middle 50% of the distribution, robust to outliers in symmetric data.

iqr_value = 1.3545

NormalDistribution: m = mean (pd)

m = mean (pd) computes the mean of the probability distribution object, pd.

Example: 1


 ## Compute the mean for different Normal distributions
 data1 = 0 + 0.5 * randn (10000, 1);
 data2 = 0 + 1.0 * randn (10000, 1);
 pd1 = fitdist (data1, "Normal");
 pd2 = fitdist (data2, "Normal");
 mean1 = mean (pd1)
 mean2 = mean (pd2)

 ## This shows how to compute the expected value for Normal
 ## distributions with different scale parameters, representing the average value.

mean1 = -3.8937e-03
mean2 = 5.4939e-05

NormalDistribution: m = median (pd)

m = median (pd) computes the median of the probability distribution object, pd.

Example: 1


 ## Compute the median for different Normal distributions
 data1 = 0 + 0.5 * randn (10000, 1);
 data2 = 0 + 1.0 * randn (10000, 1);
 pd1 = fitdist (data1, "Normal");
 pd2 = fitdist (data2, "Normal");
 median1 = median (pd1)
 median2 = median (pd2)

 ## Use this to find the median value, which equals the mean in symmetric
 ## Normal distributions, useful for central tendency measures.

median1 = -4.2980e-03
median2 = -1.5190e-03

NormalDistribution: nlogL = negloglik (pd)

nlogL = negloglik (pd) computes the negative loglikelihood of the probability distribution object, pd.

Example: 1


 ## Compute the negative loglikelihood for a fitted Normal distribution
 rand ("seed", 21);
 data = randn (100, 1);
 pd_fitted = fitdist (data, "Normal");
 params = [pd_fitted.mu, pd_fitted.sigma];
 nlogL_normlike = normlike (params, data)

 ## This is useful for assessing the fit of a Normal distribution to
 ## data, with lower values indicating a better fit, often used in optimization or AIC/BIC.

nlogL_normlike = 133.39

NormalDistribution: ci = paramci (pd)
NormalDistribution: ci = paramci (pd, Name, Value)

ci = paramci (pd) computes the lower and upper boundaries of the 95% confidence interval for each parameter of the probability distribution object, pd.

ci = paramci (pd, Name, Value) computes the confidence intervals with additional options specified by Name-Value pair arguments listed below.

`Name`		`Value`
`"Alpha"`		A scalar value in the range $(0,1)$ specifying the significance level for the confidence interval. The default value 0.05 corresponds to a 95% confidence interval.
`"Parameter"`		A character vector or a cell array of character vectors specifying the parameter names for which to compute confidence intervals. By default, `paramci` computes confidence intervals for all distribution parameters.

paramci is meaningful only when pd is fitted to data, otherwise an empty array, [], is returned.

Example: 1


 ## Compute confidence intervals for parameters of a fitted Normal
 ## distribution
 rand ("seed", 21);
 data = randn (1000, 1);
 pd_fitted = fitdist (data, "Normal");
 ci = paramci (pd_fitted, "Alpha", 0.05)

 ## Use this to obtain confidence intervals for the estimated parameters (mu
 ## and sigma), providing a range of plausible values given the data.

ci =

  -0.052063   0.952697
   0.071357   1.040062

NormalDistribution: y = pdf (pd, x)

y = pdf (pd, x) computes the PDF of the probability distribution object, pd, evaluated at the values in x.

Example: 1


 ## Plot various PDFs from the Normal distribution
 x = -5:0.01:5;
 data1 = 0 + 0.5 * randn (10000, 1);
 data2 = 0 + 1.0 * randn (10000, 1);
 data3 = 0 + 2.0 * randn (10000, 1);
 pd1 = fitdist (data1, "Normal");
 pd2 = fitdist (data2, "Normal");
 pd3 = fitdist (data3, "Normal");
 y1 = pdf (pd1, x);
 y2 = pdf (pd2, x);
 y3 = pdf (pd3, x);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r")
 grid on
 legend ({"mu = 0, sigma = 0.5", "mu = 0, sigma = 1", "mu = 0, sigma = 2"}, ...
         "location", "northeast")
 title ("Normal PDF")
 xlabel ("values in x")
 ylabel ("Probability density")

 ## This visualizes the probability density function for Normal
 ## distributions, showing the bell-shaped curve for symmetric data.

NormalDistribution: plot (pd)
NormalDistribution: plot (pd, Name, Value)
NormalDistribution: h = plot (…)

plot (pd) plots a probability density function (PDF) of the probability distribution object pd. If pd contains data, which have been fitted by fitdist, the PDF is superimposed over a histogram of the data.

plot (pd, Name, Value) specifies additional options with the Name-Value pair arguments listed below.

`Name`		`Value`
`"PlotType"`		A character vector specifying the plot type. `"pdf"` plots the probability density function (PDF). When `pd` is fit to data, the PDF is superimposed on a histogram of the data. `"cdf"` plots the cumulative density function (CDF). When `pd` is fit to data, the CDF is superimposed over an empirical CDF. `"probability"` plots a probability plot using a CDF of the data and a CDF of the fitted probability distribution. This option is available only when `pd` is fitted to data.
`"Discrete"`		A logical scalar to specify whether to plot the PDF or CDF of a discrete distribution object as a line plot or a stem plot, by specifying `false` or `true`, respectively. By default, it is `true` for discrete distributions and `false` for continuous distributions. When `pd` is a continuous distribution object, option is ignored.
`"Parent"`		An axes graphics object for plot. If not specified, the `plot` function plots into the current axes or creates a new axes object if one does not exist.

h = plot (…) returns a graphics handle to the plotted objects.

Example: 1


 ## Create a Normal distribution with fixed parameters mu = 0 and
 ## sigma = 1 and plot its PDF.

 data = randn (10000, 1);
 pd = fitdist (data, "Normal");
 plot (pd)
 title ("Fixed Normal distribution with mu = 0 and sigma = 1")

Example: 2


 ## Generate a data set of 100 random samples from a Normal
 ## distribution with parameters mu = 0 and sigma = 1. Fit a Normal
 ## distribution to this data and plot its CDF superimposed over an empirical
 ## CDF.

 rand ("seed", 21);
 data = randn (100, 1);
 pd_fitted = fitdist (data, "Normal");
 plot (pd_fitted, "PlotType", "cdf")
 txt = "Fitted Normal distribution with mu = %0.2f and sigma = %0.2f";
 title (sprintf (txt, pd_fitted.mu, pd_fitted.sigma))
 legend ({"empirical CDF", "fitted CDF"}, "location", "southeast")

 ## Use this to visualize the fitted CDF compared to the empirical CDF of the
 ## data, useful for assessing model fit in symmetric distributions.

Example: 3


 ## Generate a data set of 200 random samples from a Normal
 ## distribution with parameters mu = 0 and sigma = 1. Display a probability
 ## plot for the Normal distribution fit to the data.

 rand ("seed", 21);
 data = randn (200, 1);
 pd_fitted = fitdist (data, "Normal");
 plot (pd_fitted, "PlotType", "probability")
 txt = strcat ("Probability plot of fitted Normal", ...
               " distribution with mu = %0.2f and sigma = %0.2f");
 title (sprintf (txt, pd_fitted.mu, pd_fitted.sigma))
 legend ({"empirical CDF", "fitted CDF"}, "location", "southeast")

 ## This creates a probability plot to compare the fitted distribution to the
 ## data, useful for checking normality assumptions.

NormalDistribution: [nlogL, param] = proflik (pd, pnum)
NormalDistribution: [nlogL, param] = proflik (pd, pnum, "Display", display)
NormalDistribution: [nlogL, param] = proflik (pd, pnum, setparam)
NormalDistribution: [nlogL, param] = proflik (pd, pnum, setparam, "Display", display)

[nlogL, param] = proflik (pd, pnum) returns a vector nlogL of negative loglikelihood values and a vector param of corresponding parameter values for the parameter in the position indicated by pnum. By default, proflik uses the lower and upper bounds of the 95% confidence interval and computes 100 equispaced values for the selected parameter. pd must be fitted to data.

[nlogL, param] = proflik (pd, pnum, "Display", "on") also plots the profile likelihood against the default range of the selected parameter.

[nlogL, param] = proflik (pd, pnum, setparam) defines a user-defined range of the selected parameter.

[nlogL, param] = proflik (pd, pnum, setparam, "Display", "on") also plots the profile likelihood against the user-defined range of the selected parameter.

For the normal distribution, pnum = 1 selects the parameter mu and pnum = 2 selects the parameter sigma.

When opted to display the profile likelihood plot, proflik also plots the baseline loglikelihood computed at the lower bound of the 95% confidence interval and estimated maximum likelihood. The latter might not be observable if it is outside of the used-defined range of parameter values.

Example: 1


 ## Compute and plot the profile likelihood for the scale parameter of a fitted
 ## Normal distribution
 rand ("seed", 21);
 data = randn (1000, 1);
 pd_fitted = fitdist (data, "Normal");
 [nlogL, param] = proflik (pd_fitted, 2, "Display", "on");

 ## Use this to analyze the profile likelihood of the scale parameter (sigma),
 ## helping to understand the uncertainty in parameter estimates for symmetric data.

NormalDistribution: r = random (pd)
NormalDistribution: r = random (pd, rows)
NormalDistribution: r = random (pd, rows, cols, …)
NormalDistribution: r = random (pd, [sz])

r = random (pd) returns a random number from the distribution object pd.

When called with a single size argument, normrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Example: 1


 ## Generate random samples from a Normal distribution
 rand ("seed", 21);
 samples = randn (500, 1);
 hist (samples, 50)
 title ("Histogram of 500 random samples from Normal(mu=0, sigma=1)")
 xlabel ("values in x")
 ylabel ("Frequency")

 ## This generates random samples from a Normal distribution, useful
 ## for simulating symmetric data like noise or natural variations.

NormalDistribution: s = std (pd)

s = std (pd) computes the standard deviation of the probability distribution object, pd.

Example: 1


 ## Compute the standard deviation for a Normal distribution
 data = randn (10000, 1);
 pd = fitdist (data, "Normal");
 std_value = std (pd)

 ## Use this to calculate the standard deviation, which equals sigma
 ## and measures the spread around the mean in symmetric distributions.

std_value = 0.9928

NormalDistribution: t = truncate (pd, lower, upper)

t = truncate (pd, lower, upper) returns a probability distribution t, which is the probability distribution pd truncated to the specified interval with lower limit, lower, and upper limit, upper. If pd is fitted to data with fitdist, the returned probability distribution t is not fitted, does not contain any data or estimated values, and it is as it has been created with the makedist function, but it includes the truncation interval.

Example: 1


 ## Plot the PDF of a Normal distribution, with parameters mu = 0
 ## and sigma = 1, truncated at [-2, 2] intervals. Generate 10000 random
 ## samples from this truncated distribution and superimpose a histogram scaled
 ## accordingly

 rand ("seed", 21);
 data_all = randn (20000, 1);
 data = data_all(data_all >= -2 & data_all <= 2);
 data = data(1:10000);
 pd = fitdist (data, "Normal");
 t = truncate (pd, -2, 2);
 [counts, centers] = hist (data, 50);
 bin_width = centers(2) - centers(1);
 bar (centers, counts / (sum (counts) * bin_width), 1);
 hold on;
 
 ## Plot histogram and truncated PDF
 x = linspace (-2, 2, 500);
 y = pdf (t, x);
 plot (x, y, "r", "linewidth", 2);
 hold off
 title ("Normal distribution (mu = 0, sigma = 1) truncated at [-2, 2]")
 legend ("Truncated PDF", "Histogram")

 ## This demonstrates truncating a Normal distribution to a specific
 ## range and visualizing the resulting distribution with random samples,
 ## useful for bounded symmetric data.

NormalDistribution: v = var (pd)

v = var (pd) computes the variance of the probability distribution object, pd.

Example: 1


 ## Compute the variance for a Normal distribution
 data = randn (10000, 1);
 pd = fitdist (data, "Normal");
 var_value = var (pd)

 ## Use this to calculate the variance, which is sigma squared
 ## and quantifies the spread in symmetric distributions.

var_value = 0.9870

Examples

 pd_fixed = makedist ("Normal", "mu", 0, "sigma", 1)
 randn ("seed", 2);
 data = random (pd_fixed, 5000, 1);
 pd_fitted = fitdist (data, "Normal")
 plot (pd_fitted)
 msg = "Fitted Normal distribution with mu = %0.2f and sigma = %0.2f";
 title (sprintf (msg, pd_fitted.mu, pd_fitted.sigma))

pd_fixed =
  NormalDistribution

  normal distribution
       mu = 0
    sigma = 1

pd_fitted =
  NormalDistribution

  normal distribution
       mu = 0.00166669   [-0.0251795, 0.0285129]
    sigma =   0.968309   [0.949696, 0.987671]

Categories &

Functions List

Clustering

Clustering

Classification Classes

Classification Classes

Clustering Classes

Clustering Classes

Regression Classes

Regression Classes

Data Manipulation

Data Manipulation

Descriptive Statistics

Descriptive Statistics

Distribution Classes

Distribution Classes

Distribution Fitting

Distribution Fitting

Distribution Functions

Distribution Functions

Distribution Statistics

Distribution Statistics

Distribution Wrappers

Distribution Wrappers

Experimental Design

Experimental Design

Machine Learning

Machine Learning

Model Fitting

Model Fitting

Hypothesis Testing

Hypothesis Testing

I/O

I/O

Plotting

Plotting

Regression

Regression

Transforms

Transforms

Class Definition: NormalDistribution

statistics: NormalDistribution

Properties

mu

Example: 1

Example: 2

sigma

Example: 1

Example: 2

DistributionName

NumParameters

ParameterNames

ParameterDescription

ParameterValues

ParameterCovariance

ParameterIsFixed

Truncation

IsTruncated

InputData

Methods

cdf

NormalDistribution: p = cdf (pd, x)

NormalDistribution: p = cdf (pd, x, "upper")

Example: 1

icdf

NormalDistribution: x = icdf (pd, p)

Example: 1

iqr

NormalDistribution: r = iqr (pd)

Example: 1

mean

NormalDistribution: m = mean (pd)

Example: 1

median

NormalDistribution: m = median (pd)

Example: 1

negloglik

NormalDistribution: nlogL = negloglik (pd)

Example: 1

paramci

Class Definition: `NormalDistribution`

`mu`

`sigma`

`DistributionName`

`NumParameters`

`ParameterNames`

`ParameterDescription`

`ParameterValues`

`ParameterCovariance`

`ParameterIsFixed`

`Truncation`

`IsTruncated`

`InputData`

`cdf`

`NormalDistribution: p = cdf (pd, x)`

`NormalDistribution: p = cdf (pd, x, "upper")`

`icdf`

`NormalDistribution: x = icdf (pd, p)`

`iqr`

`NormalDistribution: r = iqr (pd)`

`mean`

`NormalDistribution: m = mean (pd)`

`median`

`NormalDistribution: m = median (pd)`

`negloglik`

`NormalDistribution: nlogL = negloglik (pd)`

`paramci`

`NormalDistribution: ci = paramci (pd)`

`NormalDistribution: ci = paramci (pd, Name, Value)`

`pdf`

`NormalDistribution: y = pdf (pd, x)`

`plot`

`NormalDistribution: plot (pd)`

`NormalDistribution: plot (pd, Name, Value)`

`NormalDistribution: h = plot (…)`

`proflik`

`NormalDistribution: [nlogL, param] = proflik (pd, pnum)`

`NormalDistribution: [nlogL, param] = proflik (pd, pnum, "Display", display)`

`NormalDistribution: [nlogL, param] = proflik (pd, pnum, setparam)`

`NormalDistribution: [nlogL, param] = proflik (pd, pnum, setparam, "Display", display)`

`random`

`NormalDistribution: r = random (pd)`

`NormalDistribution: r = random (pd, rows)`

`NormalDistribution: r = random (pd, rows, cols, …)`

`NormalDistribution: r = random (pd, [sz])`

`std`

`NormalDistribution: s = std (pd)`

`truncate`

`NormalDistribution: t = truncate (pd, lower, upper)`

`var`

`NormalDistribution: v = var (pd)`