How to convert to gaussian distribution with given mean and standard deviation I was just curious. I was watching this movie phd comics where one of the professor says grade the papers so that it has a gaussian distribution with mean 81 and standard deviation 12.
I am a bit confused, to get the data to follow such distribution, I will have to change the data or the grades that I have already given. Is there a standard procedure to do this? I mean lets say I have already graded. So how can I change it to the given distribution
 A: Dimitriy’s answer is ok if the grades are Gaussian already. In the general case, just perform quantile renormalization : modify your grades to map their quantiles on the Gaussian quantiles.
The following R code generates normal grades. Pay attention to the use of rank to deal with ties.
# generate uniform grades
grades <- sample(0:100, 50, replace = TRUE)

# map on quantiles
L <- length(grades)
normal.grades <- qnorm( rank(grades)/(L+1), mean = 81, sd = 12)

I have to warn you that many teacher tell this (with varying values for mean and standard deviation), but this is just a joke. Don’t do this. 
I once made a normal qq-plot for around 150 grades. It was almost perfect. This is not very nice, because it’s what you expect if answers are random and independent... I never tried that again, since then.
A: Standardize the original score by subtracting the sample mean and dividing by the standard deviation. Call that the $z$-score. It will have a mean of zero and standard deviation of one. Then create a rescaled score by multiplying the $z$-score by 12 and adding 81.
A: In fact, you need to use a copula-like transformation. You can use the empirical cdf of the data to transform them into uniformly distributed data and use the inverse CDF of Gaussian to transform them into Gaussian distributed data.
A: As you see there are many ways to get this! :)
Here's my two cents in this:
Box-Cox transform [1,2] your data first so you set the higher order moments (skewness and kurtosis) to the desire values (0 and 3 for the case of a Gaussian). That can be easily done by trying different parameters for the power transformation and testing to see the respective skewness and kurtosis of the transformed sample. Afterwards you follow the idea by Dimitriy; you subtract the sample mean and divide by the standard deviation to make your sample $N(0,1)$ (that won't affect the higher order moments) and then set your desire scale by multiplying the sample by 12 and adding 81.
The power transformation in the first step actually takes care of Douglas' comment on Dimitriy's solution for the "non-Gaussianity" of the original data.
And there you have it, grades $ \sim N(81,144)$ (almost). Truth be told, with such StdDev for a large class of students you 'll be expecting to have some people scoring 100+ grades...  (0.0557 = 1-pnorm( 100.1, mean=81, sd=12))
