# normcdf

Normal cumulative distribution function

## Syntax

``p = normcdf(x)``
``p = normcdf(x,mu)``
``p = normcdf(x,mu,sigma)``
``[p,pLo,pUp] = normcdf(x,mu,sigma,pCov)``
``[p,pLo,pUp] = normcdf(x,mu,sigma,pCov,alpha)``
``___ = normcdf(___,'upper')``

## Description

example

````p = normcdf(x)` returns the cumulative distribution function (cdf) of the standard normal distribution, evaluated at the values in `x`.```
````p = normcdf(x,mu)` returns the cdf of the normal distribution with mean `mu` and unit standard deviation, evaluated at the values in `x`.```

example

````p = normcdf(x,mu,sigma)` returns the cdf of the normal distribution with mean `mu` and standard deviation `sigma`, evaluated at the values in `x`.```

example

````[p,pLo,pUp] = normcdf(x,mu,sigma,pCov)` also returns the 95% confidence bounds [`pLo`,`pUp`] of `p` when `mu` and `sigma` are estimates. `pCov` is the covariance matrix of the estimated parameters.```
````[p,pLo,pUp] = normcdf(x,mu,sigma,pCov,alpha)` specifies the confidence level for the confidence interval `[pLo,pUp]` to be `100(1–alpha)`%.```

example

````___ = normcdf(___,'upper')` returns the complement of the cdf, evaluated at the values in `x`, using an algorithm that more accurately computes the extreme upper-tail probabilities. `'upper'` can follow any of the input arguments in the previous syntaxes.```

## Examples

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Compute the probability that an observation from a standard normal distribution falls on the interval `[–1 1]`.

```p = normcdf([-1 1]); p(2)-p(1)```
```ans = 0.6827 ```

About 68% of the observations from a normal distribution fall within one standard deviation of the mean 0.

Compute the cdf values evaluated at the values in `x` for the normal distribution with mean `mu` and standard deviation `sigma`.

```x = [-2,-1,0,1,2]; mu = 2; sigma = 1; p = normcdf(x,mu,sigma)```
```p = 1×5 0.0000 0.0013 0.0228 0.1587 0.5000 ```

Compute the cdf values evaluated at zero for various normal distributions with different mean parameters.

```mu = [-2,-1,0,1,2]; sigma = 1; p = normcdf(0,mu,sigma)```
```p = 1×5 0.9772 0.8413 0.5000 0.1587 0.0228 ```

Find the maximum likelihood estimates (MLEs) of the normal distribution parameters, and then find the confidence interval of the corresponding cdf value.

Generate 1000 normal random numbers from the normal distribution with mean 5 and standard deviation 2.

```rng('default') % For reproducibility n = 1000; % Number of samples x = normrnd(5,2,n,1);```

Find the MLEs for the distribution parameters (mean and standard deviation) by using `mle`.

`phat = mle(x)`
```phat = 1×2 4.9347 1.9969 ```
```muHat = phat(1); sigmaHat = phat(2);```

Estimate the covariance of the distribution parameters by using `normlike`. The function `normlike` returns an approximation to the asymptotic covariance matrix if you pass the MLEs and the samples used to estimate the MLEs.

`[~,pCov] = normlike([muHat,sigmaHat],x)`
```pCov = 2×2 0.0040 -0.0000 -0.0000 0.0020 ```

Find the cdf value at zero and its 95% confidence interval.

`[p,pLo,pUp] = normcdf(0,muHat,sigmaHat,pCov)`
```p = 0.0067 ```
```pLo = 0.0047 ```
```pUp = 0.0095 ```

`p` is the cdf value using the normal distribution with the parameters `muHat` and `sigmaHat`. The interval `[pLo,pUp]` is the 95% confidence interval of the cdf evaluated at 0, considering the uncertainty of `muHat` and `sigmaHat` using `pCov`. The 95% confidence interval means the probability that `[pLo,pUp]` contains the true cdf value is 0.95.

Determine the probability that an observation from a standard normal distribution will fall on the interval `[10,Inf]`.

`p1 = 1 - normcdf(10)`
```p1 = 0 ```

`normcdf(10)` is nearly 1, so `p1` becomes 0. Specify `'upper'` so that `normcdf` computes the extreme upper-tail probabilities more accurately.

`p2 = normcdf(10,'upper')`
```p2 = 7.6199e-24 ```

You can also use `'upper'` to compute a right-tailed p-value.

Use the probability distribution function `normcdf` as a function handle in the chi-square goodness-of-fit test (`chi2gof`).

Test the null hypothesis that the sample data in the input vector `x` comes from a normal distribution with parameters µ and σ equal to the mean (`mean`) and standard deviation (`std`) of the sample data, respectively.

```rng('default') % For reproducibility x = normrnd(50,5,100,1); h = chi2gof(x,'cdf',{@normcdf,mean(x),std(x)})```
```h = 0 ```

The returned result `h = 0` indicates that `chi2gof` does not reject the null hypothesis at the default 5% significance level.

## Input Arguments

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Values at which to evaluate the cdf, specified as a scalar value or an array of scalar values.

If you specify `pCov` to compute the confidence interval `[pLo,pUp]`, then `x` must be a scalar value.

To evaluate the cdf at multiple values, specify `x` using an array. To evaluate the cdfs of multiple distributions, specify `mu` and `sigma` using arrays. If one or more of the input arguments `x`, `mu`, and `sigma` are arrays, then the array sizes must be the same. In this case, `normcdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `p` is the cdf value of the distribution specified by the corresponding elements in `mu` and `sigma`, evaluated at the corresponding element in `x`.

Example: `[-1,0,3,4]`

Data Types: `single` | `double`

Mean of the normal distribution, specified as a scalar value or an array of scalar values.

If you specify `pCov` to compute the confidence interval `[pLo,pUp]`, then `mu` must be a scalar value.

To evaluate the cdf at multiple values, specify `x` using an array. To evaluate the cdfs of multiple distributions, specify `mu` and `sigma` using arrays. If one or more of the input arguments `x`, `mu`, and `sigma` are arrays, then the array sizes must be the same. In this case, `normcdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `p` is the cdf value of the distribution specified by the corresponding elements in `mu` and `sigma`, evaluated at the corresponding element in `x`.

Example: `[0 1 2; 0 1 2]`

Data Types: `single` | `double`

Standard deviation of the normal distribution, specified as a nonnegative scalar value or an array of nonnegative scalar values.

If `sigma` is zero, then the output `p` is either 0 or 1. `p` is 0 if `x` is smaller than `mu`, or 1 otherwise.

If you specify `pCov` to compute the confidence interval `[pLo,pUp]`, then `sigma` must be a scalar value.

To evaluate the cdf at multiple values, specify `x` using an array. To evaluate the cdfs of multiple distributions, specify `mu` and `sigma` using arrays. If one or more of the input arguments `x`, `mu`, and `sigma` are arrays, then the array sizes must be the same. In this case, `normcdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `p` is the cdf value of the distribution specified by the corresponding elements in `mu` and `sigma`, evaluated at the corresponding element in `x`.

Example: `[1 1 1; 2 2 2]`

Data Types: `single` | `double`

Covariance of the estimates `mu` and `sigma`, specified as a 2-by-2 matrix.

If you specify `pCov` to compute the confidence interval `[pLo,pUp]`, then `x`, `mu`, and `sigma` must be scalar values.

You can estimate `mu` and `sigma` by using `mle`, and estimate the covariance of `mu` and `sigma` by using `normlike`. For an example, see Confidence Interval of Normal cdf Value.

Data Types: `single` | `double`

Significance level for the confidence interval, specified as a scalar in the range (0,1). The confidence level is `100(1–alpha)`%, where `alpha` is the probability that the confidence interval does not contain the true value.

Example: `0.01`

Data Types: `single` | `double`

## Output Arguments

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cdf values, evaluated at the values in `x`, returned as a scalar value or an array of scalar values. `p` is the same size as `x`, `mu`, and `sigma` after any necessary scalar expansion. Each element in `p` is the cdf value of the distribution specified by the corresponding elements in `mu` and `sigma`, evaluated at the corresponding element in `x`.

Lower confidence bound for `p`, returned as a scalar value or an array of scalar values. `pLo` has the same size as `p`.

Upper confidence bound for `p`, returned as a scalar value or an array of scalar values. `pUp` has the same size as `p`.

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### Normal Distribution

The normal distribution is a two-parameter family of curves. The first parameter, µ, is the mean. The second parameter, σ, is the standard deviation.

The standard normal distribution has zero mean and unit standard deviation.

The normal cumulative distribution function (cdf) is

`$p=F\left(x|\mu ,\sigma \right)=\frac{1}{\sigma \sqrt{2\pi }}{\int }_{-\infty }^{x}{e}^{\frac{-{\left(t-\mu \right)}^{2}}{2{\sigma }^{2}}}dt,\text{ }\text{for}\text{\hspace{0.17em}}x\in ℝ.$`

p is the probability that a single observation from a normal distribution with parameters μ and σ falls in the interval (-∞,x].

## Algorithms

• The `normcdf` function uses the complementary error function `erfc`. The relationship between `normcdf` and `erfc` is

`$\text{normcdf}\left(x\right)=\frac{1}{2}\text{erfc}\left(-\frac{x}{\sqrt{2}}\right).$`

The complementary error function `erfc(x)` is defined as

`$\text{erfc}\left(x\right)=1-\text{erf}\left(x\right)=\frac{2}{\sqrt{\pi }}{\int }_{x}^{\infty }{e}^{-{t}^{2}}dt.$`

• The `normcdf` function computes confidence bounds for `p` by using the delta method. `normcdf(x,mu,sigma)` is equivalent to `normcdf((x–mu)/sigma,0,1)`. Therefore, the `normcdf` function estimates the variance of `(x–mu)/sigma` using the covariance matrix of `mu` and `sigma` by the delta method, and finds the confidence bounds of `(x–mu)/sigma` using the estimates of this variance. Then, the function transforms the bounds to the scale of `p`. The computed bounds give approximately the desired confidence level when you estimate `mu`, `sigma`, and `pCov` from large samples.

## Alternative Functionality

• `normcdf` is a function specific to normal distribution. Statistics and Machine Learning Toolbox™ also offers the generic function `cdf`, which supports various probability distributions. To use `cdf`, create a `NormalDistribution` probability distribution object and pass the object as an input argument or specify the probability distribution name and its parameters. Note that the distribution-specific function `normcdf` is faster than the generic function `cdf`.

• Use the Probability Distribution Function app to create an interactive plot of the cumulative distribution function (cdf) or probability density function (pdf) for a probability distribution.

 Abramowitz, M., and I. A. Stegun. Handbook of Mathematical Functions. New York: Dover, 1964.

 Evans, M., N. Hastings, and B. Peacock. Statistical Distributions. 2nd ed., Hoboken, NJ: John Wiley & Sons, Inc., 1993.