Load your data set into SPSS. In the initial SPSS menu, click on the "file" tab and select "open." A dialogue box will appear. Select the dataset you wish to work with. The data will appear in a column in SPSS's data editor. It will also have a name associated with it, listed at the top of the column. Call this name "n" for later convenience.
Record the mean and standard deviation for your dataset. Click the "analyze" tab at the top of SPSS. Choose "descriptive statistics," "summarize" and then "descriptives." In the resulting dialogue box, click "options." A new dialogue box will appear, showing a list of descriptive statistics. Check at least "mean" and "std. deviation." Select "continue" and then "ok" to get the descriptive statistics. Make note of the numbers for "mean" and "std. deviation." Call the values "m" and "s" for convenience in the next step.
Calculate the Z-scores. Choose the "transform" tab in the SPSS main menu. Click on "compute" in the resulting drop-down box. Under "target variable," type the name of the variable (the name that is listed in the top of the column in the data editor). For "numerical expression" enter the following: (n -- m)/s, where "n," "m" and "s" are the values you recorded earlier. Click "ok." SPSS will return you to the data editor, displaying the Z-variables for your original data.