A Z-score, or Zx [Z of x], allows you to make inferences about your sample and how much of your sample deviates from the larger population's mean. You need a lot of data, however, to use it. A Z-score requires that you know your population mean ∪, the population standard deviation, or ∂, your sample size, or "n," and your sample's standard deviation, or "S."
If you know ∂, then use the Z-test; if you don't know ∂, then estimate (find S) and use the T-test:
s = √'(X - X line over)2/n - 1 = 'SS/n - 1
To calculate the T-test, first calculate the standard error of the estimate S from ∂ using the formula sx (s of x) = s/√n and now calculate t = X line over - ∪/sx (s of x).
For both tests, Z-scores and T-scores, you must compare your result against a critical value found in charts at the appendixes of statistics textbooks or online. For Z-scores, however, you can assume a normal distribution and find your critical values in the normal distribution table. You cannot assume a normal distribution for T-scores, as they are based on an estimation and -- usually -- a smaller study population, or "n."