# How to make sense of non-linear data transformations? What conclusions drawn can you apply to original data?

In stats class, the professor talked about the interest of transforming skewed data sets to make them more "normal".

From what I've understood so far, the idea is that the normal curve has nice mathematical properties we'd like to work with, so if we have a strongly skewed data set, we can apply non-linear transformations to it to make its distribution closer to a normal distribution.

A few example:

Linear transformations make sense; if we had data in feet and wanted to have it in inches, we could just apply $y=12x$ to the data set. That makes sense.

Even in the case where we have feet but want to deal with square feet; that's a non-linear transformation but the units still make sense (maybe "making sense" is just a question of degree of familiarity)

But now, let's imagine we have a data set for car prices or employee salaries in dollars. What would be the meaning of applying a log tranformation to our data sets? Or an inverse tranformation? What are log dollars or inverse dollars?

Also, even if we can draw conclusions more easily about the new data set, how relevant are those conlusions to our original data set? Can we just assume that our conclusions hold? How relevant is the mean, or SD or variance of a transformed data set to the original data set?

Or for example (I'm seeing this question on the side right now), it seems you can transform a data set to make it more easily linearly separable (makes sense geometrically, I guess).

But does that really work? It feels weird, like "cheating" in a sense. We're messing with the data and then drawing conclusions or coming up with predictive models based on that messed up data. How does that work?

• If you use a log link to model dollars and the coefficient is $b$ then $e^b$ tells you how much dollars are multiplied by. – mdewey Jun 26 '16 at 13:26
• More penetrating thinkers, such as John Tukey, have understood the purpose of independent univariate transformations to be achieving approximate symmetry, rather than normality. Near-symmetry is more generally achievable and leads to more parsimonious descriptions of data. For most purposes, transforming the data to normality has no use or advantage. Here's another way to look at things: if your use of log dollars provides a particularly simple and insightful way to describe and understand salaries, then maybe you should start thinking in terms of log dollars. (Ask any chemist about pH.) – whuber Jun 26 '16 at 16:48
• The result of a log-transform is not in log-dollars; it's unitless (indeed for small fluctuations you can think of differences in logs as essentially percentage changes) – Glen_b -Reinstate Monica Jun 27 '16 at 1:02
• @whuber "then maybe you should start thinking in terms of log dollars". I feel like I almost understand what you're trying to say, but ultimately I think my intuition for stats/science is too weak to really understand. Thankfully I'm pretty stubborn when it comes to trying to understand stuff, so hopefully I'll get there one day :) – jeremy radcliff Jun 27 '16 at 4:18

It should be noted that most transformations quickly become difficult to understand once you leave the basic transformations (i.e. $log(x)$ or $e^x$), and be careful when trying to make conclusions when you have used a transformation. Some effects of transformations on data are mentioned here: http://pareonline.net/getvn.asp?v=8&n=6, which briefly mentions things like changes in the properties of the data, statistical procedure and conclusion issues. It would be wise to consult with a trained mathematician/statistician to determine the effects and implications of a transformation.