Standardized testing (statistics)
A Z-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. Because of the central limit theorem, many test statistics are approximately normally distributed for large samples. For each significance level, the Z-test has a single critical value (for example, 1.96 for 5% two tailed) which makes it more convenient than the Student's t-test which has separate critical values for each sample size. Therefore, many statistical tests can be conveniently performed as approximate Z-tests if the sample size is large or the population variance known. If the population variance is unknown (and therefore has to be estimated from the sample itself) and the sample size is not large (n < 30), the Student's t-test may be more appropriate.
If T is a statistic that is approximately normally distributed under the null hypothesis, the next step in performing a Z-test is to estimate the expected value θ of T under the null hypothesis, and then obtain an estimate s of the standard deviation of T. After that the standard score Z = (T − θ) / s is calculated, from which one-tailed and two-tailed p-values can be calculated as Φ(−Z) (for upper-tailed tests), Φ(Z) (for lower-tailed tests) and 2Φ(−|Z|) (for two-tailed tests) where Φ is the standard normal cumulative distribution function.
The term "Z-test" is often used to refer specifically to the one-sample location test comparing the mean of a set of measurements to a given constant. If the observed data X1, ..., Xn are (i) uncorrelated, (ii) have a common mean μ, and (iii) have a common variance σ2, then the sample average X has mean μ and variance σ2 / n. If our null hypothesis is that the mean value of the population is a given number μ0, we can use X −μ0 as a test-statistic, rejecting the null hypothesis if X − μ0 is large.
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