# Normal

0th

Percentile

##### The Normal Distribution

Density, distribution function, quantile function and random generation for the normal distribution with mean equal to mean and standard deviation equal to sd.

Keywords
distribution
##### Usage
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)
##### Arguments
x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

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 $P[X \le x]$ otherwise, $P[X > x]$.

##### Details

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.

##### Value

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.

##### References

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.

##### See Also

Distributions for other standard distributions, including dlnorm for the Lognormal distribution.

• Normal
• dnorm
• pnorm
• qnorm
• rnorm
##### Examples
library(stats) # NOT RUN { 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) # } 
Documentation reproduced from package stats, version 3.6.0, License: Part of R 3.6.0

### Community examples

lifesonk@gmail.com at Mar 20, 2019 stats v3.5.3

lifesonk@gmail.com at Mar 20, 2019 stats v3.5.3

lifesonk@gmail.com at Mar 20, 2019 stats v3.5.3