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 socst
Variable | 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 z1math
Variable | 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 z1socst
Variable | 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 z2socst
Variable | 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