There are many definitions for skewness, such as the one you used Pearson's second skewness. The skewness Xi in fGarch is based on a paper (Fernandez & Steel 2000), and is an inverse scale factor in the positive and the negative directions.
As shown in the figure below, inverting skewness Xi produces the mirror image around zero. When Xi = 1, the distribution is symmetrical. When 0 < Xi < 1, negative skew. When Xi > 1, positive skew.
In your example, the Mean is skewed to the right of the Median. So we're looking for a positive skew (Xi > 1). By trial and error, the Xi could be located around 1.094095 . The median value given by the quantile function qsnorm(0.5, Mean, Sd, xi=1.094095) is indeed 150.
The simulated median is however a bit biased, which results in an unexpected Pearson's second skewness.
Mean <- 155
Sd <- 35
Median <- 150
3 * (Mean - Median) / Sd #Pearson's second skewness
#0.4285714
library(fGarch)
set.seed(1)
x <- rsnorm(100000, Mean, Sd, xi=1.094095)
mean(x)
#154.9353
sd(x)
#35.07564
median(x) #a bit higher than expected
#153.9029
3 * (mean(x) - median(x)) / sd(x) #biased from expected
#0.08830281
Finally, the code for locating Xi using General-purpose Optimization optim method.
fn <- function(log_Xi, Mean, Sd, Median) {
abs(qsnorm(0.5, Mean, Sd, exp(log_Xi)) - Median)
}
tem <- optim(0, fn, gr=NULL, Mean, Sd, Median)
Xi <- exp(tem$par)
#1.094095
And the code for the density plot above.
x_ <- seq(-5,5, by=0.01)
plot(x_, dsnorm(x_, xi=3), type='l', ylab='density', col='red', xlab='x', main='Skewness')
lines(x_, dsnorm(x_, xi=1), col='black', lty=2)
lines(x_, dsnorm(x_, xi=1/3), col='blue')
legend('topright', legend=c('3','1','1/3'), fill=c('red','black','blue'), title='Xi')
