# Standardization of a random variable

1. Feb 21, 2016

### K41

I'm learning basic probability and have some understanding of PDF's and CDF's now. (I've not done expected values yet though so I'm not familiar with that notation).

I've come across standardization of a random variable, X, which then gives a new random variable Y with the properties that Y has zero mean and unit variance (and therefore unit standard deviation).

My first question is, does X have to have a normal distribution? I've seen this stated explicitly in some texts but not in others.
My second question is, what is the benefit of defining a new random variable instead of just using our old one? Can anyone give a clear intuitive example of when you may want to standardize data but also when you may not want to standardize data? I'm guessing it involves wanting to compare different data from different sample spaces maybe?

EDIT: Or does it involve comparing data on the same sample space but in vastly different regions of the sample space (i.e one data set involves X looking at temperatures between 0 and 5 degrees whilst another data set looks at temps between 150 and 170 degrees)?

2. Feb 21, 2016

### Staff: Mentor

Yes. [edit: this is wrong, see below]

The primary benefit is that you can look it's p-values up on a table of standard normal p-values. Otherwise you need a separate table for each mean and variance.

Last edited: Feb 24, 2016
3. Feb 21, 2016

### Number Nine

You can standardize any random variable, as long as it has a mean and variance (though the standardized RV will only be standard normal if the original RV was normal). The benefit of standardizing is that the units (standard deviations) become more interpretable for some applications. For example, it's difficult to compare coefficients in regression models when predictors are in different units. Standardizing them solves this problem.

Hypothesis testing (which your learn about soon) is similar, though it standardizes based on a hypothesized mean (usually zero), instead of the sample mean.

4. Feb 23, 2016

### chiro

Standardization - particular in reference to degrees of freedom and 0 mean is used a lot in statistics.

The degrees of freedom is important because of how many statistical tests are built off this basis.

You might want to look at the general theorems in statistical inference and get a feel of the things that stay the same regardless of what is being estimated.

Some things include the Central Limit Theorem and the generalizations there-of - along with the connection these results have with respect to degrees of freedom which is an important standardization regarding that of variance and its use in general statistical inference across general estimators.

5. Feb 24, 2016

### Staff: Mentor

Oh, yes, you are correct. Then you can compare your standardized RV to a standard normal RV, etc. My previous assertion was wrong.