Normal {stats}  R Documentation 
Density, distribution function, quantile function and random
generation for the normal distribution with mean equal to mean
and standard deviation equal to sd
.
dnorm(x, mean = 0, sd = 1, log = FALSE)
pnorm(q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)
qnorm(p, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)
rnorm(n, mean = 0, sd = 1)
x , q 
vector of quantiles. 
p 
vector of probabilities. 
n 
number of observations. If 
mean 
vector of means. 
sd 
vector of standard deviations. 
log , log.p 
logical; if TRUE, probabilities p are given as log(p). 
lower.tail 
logical; if TRUE (default), probabilities are

If mean
or sd
are not specified they assume the default
values of 0
and 1
, respectively.
The normal distribution has density
f(x) =
\frac{1}{\sqrt{2\pi}\sigma} e^{(x\mu)^2/2\sigma^2}
where \mu
is the mean of the distribution and
\sigma
the standard deviation.
dnorm
gives the density,
pnorm
gives the distribution function,
qnorm
gives the quantile function, and
rnorm
generates random deviates.
The length of the result is determined by n
for
rnorm
, and is the maximum of the lengths of the
numerical arguments for the other functions.
The numerical arguments other than n
are recycled to the
length of the result. Only the first elements of the logical
arguments are used.
For sd = 0
this gives the limit as sd
decreases to 0, a
point mass at mu
.
sd < 0
is an error and returns NaN
.
For pnorm
, based on
Cody, W. D. (1993) Algorithm 715: SPECFUN – A portable FORTRAN package of special function routines and test drivers. ACM Transactions on Mathematical Software 19, 22–32.
For qnorm
, the code is based on a C translation of
Wichura, M. J. (1988) Algorithm AS 241: The percentage points of the normal distribution. Applied Statistics, 37, 477–484; doi:10.2307/2347330.
which provides precise results up to about 16 digits for
log.p=FALSE
. For log scale probabilities in the extreme tails,
since R version 4.1.0, extensively since 4.3.0, asymptotic expansions
are used which have been derived and explored in
Maechler, M. (2022) Asymptotic tail formulas for gaussian quantiles; DPQ vignette https://CRAN.Rproject.org/package=DPQ/vignettes/qnormasymp.pdf.
For rnorm
, see RNG for how to select the algorithm and
for references to the supplied methods.
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 13. Wiley, New York.
Distributions for other standard distributions, including
dlnorm
for the Lognormal distribution.
require(graphics)
dnorm(0) == 1/sqrt(2*pi)
dnorm(1) == exp(1/2)/sqrt(2*pi)
dnorm(1) == 1/sqrt(2*pi*exp(1))
## Using "log = TRUE" for an extended range :
par(mfrow = c(2,1))
plot(function(x) dnorm(x, log = TRUE), 60, 50,
main = "log { Normal density }")
curve(log(dnorm(x)), add = TRUE, col = "red", lwd = 2)
mtext("dnorm(x, log=TRUE)", adj = 0)
mtext("log(dnorm(x))", col = "red", adj = 1)
plot(function(x) pnorm(x, log.p = TRUE), 50, 10,
main = "log { Normal Cumulative }")
curve(log(pnorm(x)), add = TRUE, col = "red", lwd = 2)
mtext("pnorm(x, log=TRUE)", adj = 0)
mtext("log(pnorm(x))", col = "red", adj = 1)
## if you want the socalled 'error function'
erf < function(x) 2 * pnorm(x * sqrt(2))  1
## (see Abramowitz and Stegun 29.2.29)
## and the socalled 'complementary error function'
erfc < function(x) 2 * pnorm(x * sqrt(2), lower = FALSE)
## and the inverses
erfinv < function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv < function (x) qnorm(x/2, lower = FALSE)/sqrt(2)