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A z-test tests hypotheses about a population mean when the population standard deviation σ is known (rare in practice). The test statistic is z = (x̄ - μ₀) / (σ/√n). A t-test is used when σ is unknown and replaced by sample standard deviation s; the test statistic is t = (x̄ - μ₀) / (s/√n), which follows a t-distribution with n-1 degrees of freedom. The t-distribution has heavier tails than normal, so t-tests account for additional uncertainty from estimating σ. Both tests compare observed sample means to hypothesized population means.
Compute z and t statistics from data. Use tables or software to find p-values. Compare z-test and t-test on the same data for large and small samples. Understand why t-distribution is appropriate for small samples.
Using z-test when σ is unknown. Confusing degrees of freedom in t-distribution. Thinking larger t values have larger p-values (opposite is true). Misapplying tests without checking normality assumptions for small samples.
You already know the core logic of hypothesis testing: assume the null hypothesis is true, compute how far your sample result is from what the null predicts, and ask how likely such a deviation is by chance. You also know how to convert raw values to z-scores using the standard normal distribution. The z-test and t-test both apply that exact logic to sample means — the only real difference is what you know about the population's variability.
A z-test applies when you know the true population standard deviation σ. The test statistic z = (x̄ − μ₀) / (σ/√n) measures how many standard errors your sample mean sits from the hypothesized population mean μ₀. The denominator σ/√n is the standard error of the mean — the theoretical spread of sample means when repeatedly drawing samples of size n. Because σ is known, this denominator is exact, and the test statistic follows a standard normal distribution precisely. In practice, knowing σ is rare (it would require surveying the entire population), so the z-test is mostly theoretical or used for large samples where the Central Limit Theorem ensures approximate normality.
In real data analysis, you almost always estimate σ from the sample itself using the sample standard deviation s. Plugging s in where σ was gives the t-test statistic: t = (x̄ − μ₀) / (s/√n). But here's the subtle problem: s is itself a random variable — it varies from sample to sample — introducing additional uncertainty into the denominator. This extra randomness makes the test statistic follow a t-distribution rather than a standard normal. The t-distribution looks like a normal distribution but with heavier tails, reflecting that extreme outcomes are somewhat more likely when your estimate of spread is itself imprecise. The exact shape depends on degrees of freedom (df = n − 1): with only a few observations, uncertainty is large and tails are heavy; as n grows, s stabilizes and the t-distribution converges to the standard normal. By n ≈ 30, the two distributions are nearly identical, which is why large-sample z-tests are defensible even when σ is unknown.
The decision rule is the same for both tests: compute the test statistic, find the p-value (the probability of observing a result this extreme under H₀), and compare to your significance level α. A larger absolute value of z or t pushes the p-value *down*, making the result more significant — the opposite of what beginners sometimes assume. When running a two-sided test, you're asking whether your sample mean is far from μ₀ in *either* direction, so you use the total tail probability on both ends. When choosing between z and t, the rule is simple: use t-tests by default. They handle the unknown-σ case correctly, and when n is large enough that z would have been fine, the t-test gives essentially the same answer anyway.