In QuantileTransformer this is available as:

output_distribution : str, optional (default='uniform')
    Marginal distribution for the transformed data.
    The choices are ‘uniform’ (default) or ‘normal’.

I suppose it's not the same. Here it's just a change in scale? Still, it would be nice if the options looked more similar to each other??

I see your point but what would you suggest? This seems a bit clunky?

output_distribution : str, optional (default='non-standard normal')
    Scale distribution for the transformed data to standard normal or keep as non-standard normal.      
    Options 'non-standard normal' or ‘standard normal’

Here it's just a change in scale?

Yes it is just a change in scale. Normally the standard deviation of the output distribution for 0.25,0.75 quartiles is:
std ~= 1/(norm.ppf(.75) - norm.ppf(.25)) = 0.741

The adjusted scale just changes the standard deviation to ~1 by changing the scale:

self.scale_  = (q[1] - q[0]) / self.adjust

where self.adjust is:

self.adjust = norm.ppf(q[1]  / 100.0) - norm.ppf(q[0] / 100.0) 

Maybe:

standard-normal :  boolean, False by default
    If True, scale data to a standard normal distribution.

which is vaguely more similar to QuantileTransformer

Don't know what the right name is... scale_to_unit_variance???

Or just unit_variance? Maybe scale_to_unit_variance is more explicit?
Happy with either name.

Semi-related: should that become the default, i.e. should we deprecate it being False?

Thanks for the reviews. I've used unit_variance but happy to change to scale_to_unit_variance

ping @glemaitre. I also added more explanation of what the parameter does generally as suggested by @adrinjalali, but I'm not sure about the wording...

Could we also expand on this a bit and explain what happens if the feature is not normally distributed? Since this is the only place we're having the explanation of this parameter (no user guide), it'd be nice to elaborate a bit.

Amended!

Thanks @lucyleeow merging