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MATLAB handles the normal distribution through a family of functions in the Statistics and Machine Learning Toolbox: normpdf gives the density, normcdf the cumulative probability, norminv the quantile, normrnd random draws, and normfit or fitdist estimate the parameters from data. This guide walks them as one workflow (simulate, fit, evaluate, read the output) using the examples and outputs published in the MathWorks documentation, and flags the places where the documented behavior is easy to misread.
What it is used for in research
The normal distribution has two parameters: the mean μ (any real number) and the standard deviation σ (which must be greater than zero). Its density is f(x | μ, σ) = 1 / (σ√(2π)) · exp(−(x − μ)² / (2σ²)). In MATLAB you reach for these functions to simulate measurement noise, to compute the probability that a measurement falls below or above a cutoff, to fit a normal model to a sample and report confidence intervals for its parameters, and to supply a reference distribution to a goodness-of-fit test. Simulation-heavy work such as a Monte Carlo simulation usually starts with normrnd. If you are still deciding between tools, see what MATLAB is used for and the MATLAB vs R comparison.
Install or access
The normal distribution functions are documented as part of the Statistics and Machine Learning Toolbox, which includes these functions. The reference pages for each function are under mathworks.com/help/stats. Check the product requirements and your institution’s license on the MathWorks site; this guide does not cover licensing.
Step-by-step: simulate, fit, evaluate
1. Generate normal random numbers with normrnd
The syntax is r = normrnd(mu,sigma) for one value, or r = normrnd(mu,sigma,sz1,...,szN) and r = normrnd(mu,sigma,sz) for arrays. Called as normrnd(mu,sigma,n) it returns an n-by-n matrix. Call rng first if you need repeatable results.
rng('default')
r = normrnd(0,1) % 0.5377 in the MathWorks example
s = rng;
r = normrnd(3,10,[1,5]) % 8.3767 21.3389 -19.5885 11.6217 6.1877
rng(s); % restore the generator state
r = normrnd(3,10,[1,5]) % same five values again2. Fit a normal distribution with normfit or fitdist
[muHat,sigmaHat,muCI,sigmaCI] = normfit(x,alpha) returns the estimates and confidence intervals; alpha defaults to 0.05, which gives 95% intervals. MathWorks’ example draws 1,000 values from a normal with μ = 3 and σ = 5 and requests 99% intervals:
rng('default')
x = normrnd(3,5,[1000,1]);
[muHat,sigmaHat,muCI,sigmaCI] = normfit(x,0.01)
% muHat = 2.8368
% sigmaHat = 4.9948
% muCI = [2.4292; 3.2445]
% sigmaCI = [4.7218; 5.2989]To work with an object instead, pd = fitdist(x,'Normal') returns a NormalDistribution object with mu and sigma properties. pd = makedist('Normal') creates the standard normal (μ = 0, σ = 1) by default.
3. Evaluate the density with normpdf
y = normpdf(x,mu,sigma) evaluates the density at each x. The defaults are μ = 0 and σ = 1, σ must be positive, and array arguments must be the same size (scalars expand).
x = [-2,-1,0,1,2];
y = normpdf(x) % 0.0540 0.2420 0.3989 0.2420 0.0540
y = normpdf(x,2,1) % 0.0001 0.0044 0.0540 0.2420 0.39894. Evaluate probabilities with normcdf
p = normcdf(x,mu,sigma) returns P(X ≤ x). It is equivalent to normcdf((x-mu)/sigma,0,1).
p = normcdf([-1 1]);
p(2) - p(1) % 0.6827, the share of a normal within one σ of the mean
normcdf([-2,-1,0,1,2],2,1) % 0.0000 0.0013 0.0228 0.1587 0.5000Output and how to read it
| Question you are asking | Function | What the number means |
|---|---|---|
| How dense is the distribution at x? | normpdf |
A density, not a probability. Values such as 0.3989 at the mean are heights of the curve and can exceed 1 when σ is small. |
| What is the probability of a value at or below x? | normcdf |
P(X ≤ x), between 0 and 1. |
| What is the probability above x? | normcdf(x,mu,sigma,'upper') |
The complement of the cdf, computed to stay accurate in the extreme upper tail. |
| Which x has cumulative probability p? | norminv |
The quantile; see inverse normal. |
| Draw simulated data | normrnd |
Random values with the given μ and σ. |
| Estimate μ and σ from a sample | normfit, fitdist |
Parameter estimates, with confidence intervals from normfit. |
Tail probabilities: use ‘upper’, not 1 – normcdf
The documentation shows why. 1-normcdf(10) evaluates to 0, because the cdf rounds to 1 in double precision, while normcdf(10,'upper') returns 7.6199e-24. For p-values and extreme-event probabilities, request the upper tail directly.
Confidence intervals on a probability
If you want an interval around a cdf value, pass the covariance of the parameter estimates as pCov: [p,pLo,pUp] = normcdf(x,mu,sigma,pCov,alpha). With pCov, x, mu and sigma must be scalars. MathWorks’ example simulates x = normrnd(5,2,1000,1) (after rng('default')), estimates phat = mle(x) as 4.9347 and 1.9969, gets pCov from normlike, and finds normcdf(0,muHat,sigmaHat,pCov) gives p = 0.0067 with bounds 0.0047 and 0.0095.
Checking fit
The normcdf page also shows a chi-square goodness-of-fit test against a fitted normal: chi2gof(x,'cdf',{@normcdf,mean(x),std(x)}) with x = normrnd(50,5,100,1) returns h = 0, meaning the test does not reject normality at the 5% level. For a normality test outside MATLAB, see the Shapiro-Wilk test in R, and for the small-sample counterpart of this distribution see t-distribution vs normal.
Troubleshooting
- An error about σ. σ must be positive in
normpdfand the other functions. Innormrnd, σ = 0 is allowed and every result equals μ. - Size mismatch. If x, mu and sigma are all arrays, they must be the same size; only scalars expand.
- Empty output from normrnd. A size argument of 0 or a negative number returns an empty array.
- The sigma from normfit is not the maximum likelihood estimate. Without censoring,
muHatis the sample mean andsigmaHatis the square root of the unbiased variance estimator. Usemleif you need the MLE of σ. With censoring, both outputs are maximum likelihood estimates. - pCov errors. The
normcdfform withpCovaccepts scalar x, mu and sigma only. - A tail probability of exactly 0. Use the
'upper'option as described above. - Results change between runs. Set the generator with
rng('default'), or save and restore it withs = rng; ... rng(s). - Do not copy the “clone size” one-liner on the normrnd page. It reads
normrnd(1,0,size(A)), which is μ = 1, σ = 0 and returns all ones. For random values of the same size as A, usenormrnd(0,1,size(A)).
Free alternatives
The same four operations exist in free tools. The argument conventions differ, so check the order when translating code.
| Operation | MATLAB | R (stats package) | Python (scipy.stats.norm) |
|---|---|---|---|
| Density | normpdf(x,mu,sigma) |
dnorm(x, mean = 0, sd = 1) |
norm.pdf(x, loc=0, scale=1) |
| Cumulative probability | normcdf(x,mu,sigma) |
pnorm(q, mean = 0, sd = 1, lower.tail = TRUE) |
norm.cdf(x, loc=0, scale=1) |
| Quantile | norminv(p,mu,sigma) |
qnorm(p, mean = 0, sd = 1) |
norm.ppf(q, loc=0, scale=1) |
| Random draws | normrnd(mu,sigma,sz) |
rnorm(n, mean = 0, sd = 1) |
norm.rvs(loc=0, scale=1, size=1) |
In SciPy, loc is the mean and scale is the standard deviation. In R, pnorm(..., lower.tail = FALSE) returns the probability above x, the counterpart of MATLAB’s 'upper' option.
FAQ
What is the difference between normpdf and normcdf in MATLAB?
normpdf returns the height of the density curve at each point; normcdf returns the cumulative probability P(X ≤ x). The pdf is a density, so it is not itself a probability.
How do I generate normally distributed random numbers in MATLAB?
Use normrnd(mu,sigma,sz), for example normrnd(3,10,[1,5]) for five values with mean 3 and standard deviation 10. Call rng beforehand for reproducible output.
How do I fit a normal distribution to data?
Use normfit(x) for estimates and confidence intervals, or fitdist(x,'Normal') for a NormalDistribution object with mu and sigma properties.
What are the default parameters?
normpdf uses μ = 0 and σ = 1 when you omit them, and makedist('Normal') returns the same standard normal.
Is the sigma from normfit the maximum likelihood estimate?
Not without censoring. It uses the unbiased variance estimator; use mle for the maximum likelihood estimate.
Sources
- MathWorks, Normal Distribution, checked 2026-10-29.
- MathWorks, normpdf, normcdf, normrnd and normfit reference pages, checked 2026-10-29.
- MathWorks, Statistics and Machine Learning Toolbox documentation, checked 2026-10-29.
- R Documentation, The Normal Distribution, checked 2026-10-29.
- SciPy documentation, scipy.stats.norm, checked 2026-10-29.
