t-test vs z-test

A fast refresher on the two most common parametric hypothesis tests. Use it as a cheat sheet when you’re juggling A/B experiments, manufacturing QC, or any situation where you need to compare means.

Start with the question

Do you know the population standard deviation (σ) or have a very large sample (n ≥ 30)? → lean z-test.
Is σ unknown and your sample is modest (n < 30) or you need extra robustness? → use a t-test.
Matched pairs / before-and-after? → paired t-test.
Two independent samples with unknown, possibly unequal variance? → two-sample t-test (Welch’s variant).

Core formulas

Test	Statistic	Use when
One-sample z-test	`z = (\bar{x} - μ_0) / (σ / √n)`	`σ` known or n ≥ 30 and sampling from (approximately) normal population.
One-sample t-test	`t = (\bar{x} - μ_0) / (s / √n)`, df = n − 1	`σ` unknown, any n (works best for n < 30). Uses sample std dev `s`.
Two-sample t-test (Welch)	`t = (\bar{x}_1 - \bar{x}_2) / √(s_1^2/n_1 + s_2^2/n_2)`	Comparing two independent means, variances not assumed equal. df via Welch–Satterthwaite.
Pooled two-sample t-test	`t = (\bar{x}_1 - \bar{x}_2) / (s_p √(1/n_1 + 1/n_2))`	Only when you can assume equal variance and similar sample sizes.
Paired t-test	`t = (\bar{d}) / (s_d / √n)`	Same subjects measured twice (before/after). Work on the differences `d`.

Decision checklist

z-test

Population variance known, or n ≥ 30 with stable variance estimate.
Sampling distribution of the mean is approximately normal (CLT helps for large n).
Typical use: manufacturing QC, comparing proportions (z-test for proportions), large-scale A/B tests.

t-test

σ unknown, n is small or moderate.
Population roughly normal; t-distribution handles variance uncertainty with df penalty.
Use paired t-tests for “before/after” or matched subjects.
Use Welch’s two-sample t when groups have different variances or sizes.

Interpreting results

Once you compute z or t, compare it with the relevant critical value (or obtain a p-value). For t-tests, degrees of freedom drive the shape of the distribution — smaller df means fatter tails → larger critical values. Always state:

Null hypothesis (H₀) and alternative (H₁)
Chosen alpha (e.g., 0.05)
Test statistic, df, and p-value
Effect size or confidence interval if possible

When the assumptions break

Non-normal data, small sample: consider transformations or nonparametric tests (Wilcoxon, Mann–Whitney).
Severe outliers: winsorize or use robust estimators before testing.
Multiple comparisons: adjust alpha (Bonferroni, Holm) or control false discovery rate.

Need a calculator? Let me know and I’ll add an interactive widget here.