# Normal {stats}

### Description

Density, distribution function, quantile function and random generation for the normal distribution with mean equal to `mean`

and standard deviation equal to `sd`

.

### 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 ≤ x] otherwise, P[X > x].

### Details

If `mean`

or `sd`

are not specified they assume the default values of ` `

and `1`

, respectively.

The normal distribution has density where μ is the mean of the distribution and σ the standard deviation.

`qnorm`

is based on Wichura's algorithm AS 241 which provides precise results up to about 16 digits.

### Values

`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 parameters for the other functions. The numerical parameters other than `n`

are recycled to the length of the result. Only the first elements of the logical parameters are used.

### 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 *Log*normal distribution.

### Examples

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 R 3.0.2. License: GPL-2.