# data tranformation

#### Mada

##### New Member
How do I interpret this notation?

If QN(p) is the normal quantile function, then F(Y)= QN[G(Y)] follows a normal for any G ∼ G(Y)

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#### Dason

##### Ambassador to the humans
I don't understand your notation for describing the notation you don't understand. Do you have a link to what you're talking about? Or possibly an image?

#### hlsmith

##### Less is more. Stay pure. Stay poor.
In your face @Dason , in your face!

#### GretaGarbo

##### Human
I am sorry that the OP deleted this. It was an interesting question.

#### Dason

##### Ambassador to the humans
It was. I even had a response I was going to post once I had time.

#### Mada

##### New Member
I am sorry that the OP deleted this. It was an interesting question.
It was. I even had a response I was going to post once I had time.
Sorry for removing the question. I have posted it again.

#### GretaGarbo

##### Human
It was. I even had a response I was going to post once I had time.
Well, what is your answer?

If you have a random (let’s say a continuous) variable Y with the distribution function F() and you take p = F(Y), then p will be uniformly distributed.

And then if you take the inverse standard normal distribution function, i.e. the standard normal quantile function Q(p) you will get a normal random variable. That is a method to do a normalising transformation. Is that correct?

#### Dason

##### Ambassador to the humans
Yup. That's basically it although I think their question was mainly on the notation but you covered it. This is a case where we do need to care if Y is continuous or not though (which you mentioned but it deserves an extra special call out here)

#### GretaGarbo

##### Human
Yup. That's basically it although I think their question was mainly on the notation but you covered it. This is a case where we do need to care if Y is continuous or not though (which you mentioned but it deserves an extra special call out here)
Thanks!

Lets talk about an empirical issue. Suppose if we identify the correct distribution F() but that we estimate some parameters with some errors. How sensitive would that be for the resulting normal distribution and the inference from a normal model (say an anova).

Suppose now that we identify an incorrect distribution F() but a similar one. How sensitive would that be for our normal distribution based inference?

I would not be surprised if there has been a lot written in this area. Has anybody seen anything about this?

#### Dason

##### Ambassador to the humans
I'm not sure it would make much sense to do an empirical version of this transformation so that you could do further analysis on it if that's what you're suggesting.

#### GretaGarbo

##### Human
Yes, that is what I suggested.

But also for multivariate analysis. If the marginal distribution is not normal then it can not be a multivariate normal distribution. If the marginal distribution is normal it does not have to multivariate normal, but the possibility exist. Maybe the transformation can make it come closer to a multivariate normal.

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