# Box Cox Transformation Is it possible to transform every kind of distribution into a normal distribution?

I would like to do a python script for estimating population parameters by means of a sample. The data contained into my sample are measurements. Nevertheless I know that all my measurements are not following a normal distribution... And if the sample does not follow a normal distribution, i can't correctly estimate the parameters ... I would like to know if it is possible to transform every distributions into a normal distribution with a Cox Box transformation ? Is there an other way to do what I want without using the Cox Box transformation ?

I would like to do a script that can be used for many purpose (many distributions) :D

Thank you very much :D

• You can, if $P(X<a)$ is a continuous function in $a$ Commented Aug 24, 2019 at 18:07
• I think this is a better question for stat.stackexchange.com. You have to be careful - just because you can transform to a normal distribution does not mean that your estimates will be good ones when you're done. Commented Aug 24, 2019 at 18:12
• Thank you for your comments :) Ethan Bolker What do you mean ? Do you mean that the Cox box transformation will degrade the estimation of my parameters ? Or are you saying that in any case without or with using a cox box transformation, if i have a normal distribution the estimation of my parameters can not be good. (You re talking about power of the test ?)
– Jean
Commented Aug 24, 2019 at 18:40
• See Probability Integral Transform. However, when you transform your data you change the properties of the estimation procedure, making it difficult to achieve your original estimation objectives. If you would like good advice about succeeding with your original statistical question, why not ask about that directly?
– whuber
Commented Aug 25, 2019 at 13:07

No: a simple example would be the dirac distribution

• Thank for your answer ! Is it possible to encounter the dirac distribution into the real world ? It seems that the dirac distribution is equal to the normal distribution when the standard deviation goes towards 0 (# Wikipedia) ?
– Jean
Commented Aug 24, 2019 at 18:10
• It is if you’re trying to analyse data of a trivial problem... orherwise it’s mostly a theoretical counterexample. In practice though: if your cdf has jumps (which is quite realistic in applications where there are “cut offs”), then I think it’s also not true.
– Applejuicefan
Commented Aug 24, 2019 at 18:14
• Actually my question shoud have been "Is there a way to transform a distribution not known into a normal distribution ? in order to do parametric test ... "
– Jean
Commented Aug 24, 2019 at 18:15
• Ok ... I see thank you for your comment :D
– Jean
Commented Aug 24, 2019 at 18:18