I have 10,000 data points referring to data on dogs that follow a positively skewed distribution that I have grouped into quartiles.

I would a second distribution (highly correlated with the first distribution) associated with the same set of dogs whereby the distribution represents a standard normal distribution with mean=0 and variance=1. But if I look at the quartile groups of individuals in the first distribution then these dogs have different variances within the second distribution. Although I should point out the differences are not massive but noticeable.

So my question is:

How can I re-create this using random numbers to essentially simulate the data? I want to create a random normal distribution of 10,000 datapoints whereby I have four equal size groups with different variances that together make up the standard normal distribution.

Any ideas or pointers?

Many thanks.