# Quantile transform vs Power transformation to get normal distribution

I was introduced to the concept of quantile-based gaussian transform. To my understanding, it changes the value of the original data by each percentile to the matching percentile of gaussian distribution.

I have a few questions:

1. What is the pros & cons of quantile transform vs power transform (ex: Box-Cox)
2. Let say that you constructed the confidence interval of mean of lognormal distribution after transform. With power transformation, you can back transform the confidence interval by taking the inverse of the power raised. But how does it work in quantile transform? My professor said that you can backtransform the data, but I'm not sure how I can back transform the values of statistics obtained from the transformed data.
3. Is there any case where quantile transform fail to obtain gaussian distribution?
• This paper (ncbi.nlm.nih.gov/pmc/articles/PMC2921808) goes over some of your questions. I'd also add that, when used for prediction, NST can produce results that can be difficult to replicate in small samples -e.g. the log of a given value will be the same transformed value in every sample. But an NST value depends on the quantile in the sample and will differ across samples
– TPM
Commented Sep 18, 2019 at 17:19
• Meant to add that I think this is referring to a transformation often called "normal score transformation" (NST)
– TPM
Commented Sep 18, 2019 at 17:25
• Depending on which of the many possible quantile-based transforms you applied, you can back-transform the data. You can also back-transform some statistics, such as order statistics. But you can't generally back-transform moments (especially the mean or variance). It is impossible for this procedure to produce a true Gaussian distribution: it can only approximate one. It is as arbitrarily complex to compute as there are numbers in your dataset, making it far inferior to almost any other transformation in terms of simplicity and interpretability.
– whuber
Commented May 9, 2022 at 21:06