# Transforming a Random Variable's distribution to Normal using Z-Scores

I have a random variable and many observations of that variable. The random variable is not normally distributed; its distribution is unknown.

However, to analyze this variable and construct a time series model, I need to force this variable to have a normal distribution. I was given this suggestion:

• Take an observation of random variable X=x and compute its CDF value 0 < F(x) < 1
• Take this value F(x) and plug it into the Inverse Normal CDF, and this will return a Z-score corresponding the the observation X=x

Can I use this transformation to convert my data from an unknown distribution to the normal distribution?

• You can do it but you've thrown away all the quirks of your data. That may seem like gain, but it's now unclear precisely what any subsequent analysis implies about your data. I'd back up and tell us which time series method (you think) obliges this kind of force. – Nick Cox Jun 22 '15 at 19:16
• Thanks for the reply! I'm new to time series, so it's very possible I'm misunderstanding the process. I'm trying to model energy output in a generator, and I'm using an ARMA (5,1) model to do so. I have two data series, actual energy output and forecasted output. I want to model the forecasting error (forecast - actual). I was planning on using the ARMA model to predict future errors, but I was going to transform the errors to a Z-score before I modeled them. Is this the wrong way to go? – RPz Jun 22 '15 at 19:25
• Ok, but what tells you that the variable must be normal? – Nick Cox Jun 22 '15 at 19:29
• I assumed that in order to model the series using ARMA, I would need normally distributed data (because the residuals are normally distributed?). It sounds to me like you're saying I can use ARMA on the data as-is, without any transformation. Is that right? – RPz Jun 22 '15 at 19:35
• Arbitrary transformation will alter the very time series structure you hope to model, in ways that may make the very model you wish to apply unsuitable. If you were to transform at all, you wouldn't transform the errors, and you wouldn't use z-scores. In some situations, you might transform the original data (e.g. in some circumstances, you might apply an ARMA model to log data), but it would be something you'd need to do with a good understanding of the way transformation alters not just the distribution, but also the temporal relationships. – Glen_b Jun 23 '15 at 2:38

You can transform a random variable into another one using a wide range of functions. For instance, a normal random variable $\xi$ can be transformed into a lognormal by applying exponentiation as follows $x=e^\xi$. In this case, you can use well known properties of these transformations afterwards.