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 P[X ≤ x] otherwise, P[X > x]. |
If mean
or sd
are not specified they assume the default
values of 0
and 1
, respectively.
The normal distribution has density
f(x) = 1/(√(2 π) σ) e^-((x - μ)^2/(2 σ^2))
where μ is the mean of the distribution and σ 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 a C translation of
Wichura, M. J. (1988) Algorithm AS 241: The percentage points of the normal distribution. Applied Statistics, 37, 477–484.
which provides precise results up to about 16 digits.
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 so-called 'error function' erf <- function(x) 2 * pnorm(x * sqrt(2)) - 1 ## (see Abramowitz and Stegun 29.2.29) ## and the so-called '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)