The normal pdf is

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

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*.

$$erf\left(x\right)=2\Phi \left(x\sqrt{2}\right)-1$$

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.

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 *σ*^{2} 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.

$${s}^{2}=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}}$$ | (B-1) |

$${s}^{2}=\frac{1}{n-1}{\displaystyle \sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}}$$ | (B-2) |

where

$$\overline{x}={\displaystyle \sum _{i=1}^{n}\frac{{x}_{i}}{n}}$$

Equation 1 is the maximum likelihood estimator
for *σ*^{2}, and
equation 2 is the MVUE.

As an example, suppose you want to estimate the mean, *µ*,
and the variance, *σ*^{2},
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 *σ*^{2}, and
confidence intervals for *µ* and *σ*^{2}.
Here is a playful example modeling the heights in inches of a randomly
chosen fourth grade class.

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

mu = 51.1038 s = 2.6001 muci = 50.1329 52.0747 sci = 2.0707 3.4954

Note that `s^2`

is the MVUE of the variance.

s^2

ans = 6.7605

Compute the pdf of a standard normal distribution, with parameters equal to 0 and equal to 1.

x = [-3:.1:3]; norm = normpdf(x,0,1);

Plot the pdf.

figure; plot(x,norm)

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