A standardized variable (sometimes called a z-score or a standard score) is a variable that has been rescaled to have a mean of zero and a standard deviation of one. For a standardized variable, each case’s value on the standardized variable indicates it’s difference from the mean of the original variable in number of standard deviations (of the original variable). For example, a value of 0.5 indicates that the value for that case is half a standard deviation above the mean, while a value of -2 indicates that a case has a value two standard deviations lower than the mean. Variables are standardized for a variety of reasons, for example, to make sure all variables contribute evenly to a scale when items are added together, or to make it easier to interpret results of a regression or other analysis.

Standardizing a variable is a relatively straightforward procedure. First, the mean is subtracted
from the value for each case, resulting in a mean of zero. Then, the difference
between the individual’s score and the mean is divided by the standard deviation,
which results in a standard deviation of one.
If we start with a variable **x**, and generate a variable **x***, the process is:

x* = (x-m)/sd

Where **m** is the mean of **x**, and **sd** is the standard deviation of
**x**.

To illustrate the process of standardization, we will use the High School
and Beyond dataset (hsb2). We will create standardized versions of three variables,
**math**, **science**, and **socst**. These variables contain students’
scores on tests of knowledge of mathematics (**math**), science (**science**),
social studies (**socst**). First, we will use the **summarize** command (abbreviated
as **sum** below) to get the mean and standard deviation for each variable.

use https://stats.idre.ucla.edu/stat/stata/notes/hsb2, clear(highschool and beyond (200 cases))sum math science socstVariable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- math | 200 52.645 9.368448 33 75 science | 200 51.85 9.900891 26 74 socst | 200 52.405 10.73579 26 71

The mean of **math** is 52.645, and it’s standard deviation is 9.368448. Based
on this information, we can generate a standardized version of **math** called
**z1math**. The code below does this with the generate command (abbreviated to **gen**), then uses
**summarize** to confirm that the mean of **z1math** is very close to zero
(due to rounding error, the mean of a standardized variable will rarely be exactly
0), and the standard deviation is one.

gen z1math = (math-52.645)/9.368448 sum z1mathVariable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- z1math | 200 -8.51e-09 1 -2.096932 2.386201

Below we do the same for **science** and **socst**, creating
two new variables, **z1science** and **z1socst**, using their
respective means and standard deviations taken from first table of summary
statistics. The table
of summary statistics shown below demonstrates that both variables are indeed
standardized.

gen z1science = (science-51.85)/9.900891 gen z1socst = (socst-52.405)/10.73579 sum z1science z1socstVariable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- z1science | 200 -4.95e-09 1 -2.610876 2.237172 z1socst | 200 4.02e-09 1 -2.45953 1.732057

Standardizing variables is not difficult, but to make this process easier,
and less error prone,
you can use the **egen** command to make standardized variables. The commands
below standardize the values of **math**, **science**, and **socst**,
creating three new variables, **z2math**, **z2science**, and **z2socst**.

egen z2math = std(math) egen z2science = std(science) egen z2socst = std(socst)

Again we can look at a table of summary statistics to confirm that these variables are
standardized. Note that the means are not exactly zero, nor do they match the
means from the set of standardized variables created above using the **generate**
command, in both cases, this is due to very slight rounding error.

sum z2math z2science z2socstVariable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- z2math | 200 3.41e-09 1 -2.096932 2.386201 z2science | 200 -2.94e-09 1 -2.610876 2.237172 z2socst | 200 -7.38e-09 1 -2.459529 1.732056