# Fitting standard deviation of a normal distribution from one or more quantiles and known mean `get.norm.sd` returns the standard deviation of a normal distribution where the `p`th percentiles match with the quantiles `q`. ```r get.norm.sd(p = c(0.025, 0.5, 0.975), q, show.output = TRUE, plot = TRUE, fit.weights = rep(1, length(p)), scaleX = c(0.1, 0.9), ...) ``` ## Arguments - `p`: numeric, vector of probabilities. - `q`: numeric, vector of quantiles corresponding to p. - `show.output`: logical, if `TRUE` the `optim` result will be printed (default vaule is `TRUE`). - `plot`: logical, if `TRUE` the graphical diagnostics will be plotted (default value is `TRUE`). - `fit.weights`: numerical vector of the same length as a probabilities vector `p` containing positive values for weighting quantiles. By default all quantiles will be weighted by 1. - `scaleX`: numerical vector of the length 2 containing values (from the open interval (0, 1)) for scaling quantile-axis (relevant only if `plot = TRUE`). The smaller the left value, the further the graph is extrapolated within the lower percentile, the greater the right value, the further it goes within the upper percentile. - `...`: further arguments passed to the functions `plot` and `points` (relevant only if `plot = TRUE`). ## Returns Returns an estimated standard deviation or missing value ## Details The number of probabilities and the number of quantiles must be identical and should be at least two. `get.norm.sd` uses the central limit theorem and the linear regression. If `show.output = TRUE` the output of the function `lm` will be shown. The items of the probability vector `p` should lie between 0 and 1. The items of the weighting vector `fit.weights` should be positive values. The function will be meaningful only if the quantile comes from a normal distribution. ## Note It should be noted that the data must be normally distributed, or the central limt theorem must hold for large (enough) samples sizes. ## Examples ```r q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 0, sd = 2) old.par <- graphics::par(mfrow = c(2, 3)) get.norm.sd(q = q) get.norm.sd(q = q, scaleX = c(0.0001, 0.9999)) get.norm.sd(q = q, fit.weights = c(10, 1, 10)) get.norm.sd(q = q, fit.weights = c(1, 10, 1)) get.norm.sd(q = q, fit.weights = c(100, 1, 100)) get.norm.sd(q = q, fit.weights = c(1, 100, 1)) graphics::par(old.par) q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 176, sd = 15) old.par <- graphics::par(mfrow = c(2, 3)) get.norm.sd(q = q) get.norm.sd(q = q, fit.weights = c(10, 1, 10)) get.norm.sd(q = q, fit.weights = c(1, 10, 1)) get.norm.sd(q = q, fit.weights = c(100, 1, 100)) get.norm.sd(q = q, fit.weights = c(1, 100, 1)) graphics::par(old.par) ## The estimation model is not suitable for the following quantiles. ## Because the quantile is unsymmetrical, which could not be from a normally distributed data. q <- c(-2, 30, 31) old.par <- graphics::par(mfrow = c(2, 3)) get.norm.sd(q = q) get.norm.sd(q = q, fit.weights = c(10, 1, 10)) get.norm.sd(q = q, fit.weights = c(1, 10, 1), scaleX = c(0.0001, 0.9999)) get.norm.sd(q = q, fit.weights = c(100, 1, 100)) get.norm.sd(q = q, fit.weights = c(1, 100, 1), scaleX = c(0.0001, 0.9999)) graphics::par(old.par) ## Estimating from actually exponentially distributed data x.exp <- rexp(n = 10, rate = 5) mean(x.exp) stats::sd(x.exp) q <- quantile(x.exp, c(0.025, 0.5, 0.975)) old.par <- graphics::par(mfrow = c(2, 3)) get.norm.sd(q = q) get.norm.sd(q = q, fit.weights = c(1, 10, 1)) get.norm.sd(q = q, fit.weights = c(10, 1, 10)) get.norm.sd(q = q, fit.weights = c(1, 100, 1)) get.norm.sd(q = q, fit.weights = c(100, 1, 100)) graphics::par(old.par) ## other examples q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 1, sd = 1) get.norm.sd(q = q) q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 1, sd = 0.5) get.norm.sd(q = q) q <- stats::qnorm(p = c(0.025, 0.5, 0.975), mean = 0.01, sd = 0.1) get.norm.sd(q = q) ``` ## See Also See `pnorm` for distribution implementation details. ## Author(s) Matthias Greiner matthias.greiner@bfr.bund.de (BfR), Katharina Schueller schueller@stat-up.de (STAT-UP Statistical Consulting), Natalia Belgorodski belgorodski@stat-up.de (STAT-UP Statistical Consulting)

get.norm.sd function