T-Test — Types, Formula & When to Use

Reach for a t-test when you need to decide whether a sample mean, or the difference between two sample means, is bigger than random variation alone would produce. The method fits when the outcome is numeric and the population standard deviation is unknown — the usual real-world case.

When this method applies

A t-test is a good fit when all of these are reasonably true:

1. The outcome variable is numeric.
2. Observations are independent within the chosen design, unless you intentionally use a paired setup.
3. The question is about a mean or a mean difference.
4. The sample is not so small and distorted by outliers or strong skew that the mean and standard deviation become misleading.

If the population standard deviation were known exactly, a textbook $z$-test would be the direct alternative. In practice $\sigma$ is usually unknown, which is why t-tests are so common. The test must match the design of the data: it is for mean-based questions, not categorical counts.

The procedure, step by step

Step 1 — Match the design. Decide whether you have one sample, two independent groups, or paired observations from the same units. This choice selects the formula:

• One-sample — compare one sample mean to a benchmark $\mu_0$:

• Two-sample (Welch) — compare two independent group means without assuming equal variances:

Welch's degrees of freedom are not simply $n_1 + n_2 - 2$, so software usually handles that part.

• Paired — for before-and-after or matched pairs, run the test on the pairwise differences, not the two raw columns:

Often the null value is $\mu_{d,0} = 0$, meaning the average change is zero.

Step 2 — State the hypotheses. Write the null and alternative mean statements and decide one-sided or two-sided.

Step 3 — Compute the t statistic. Every version has the same shape,

so $t$ grows when the mean difference is large and shrinks when the data are noisy or the sample is small.

Step 4 — Read the result. Under the null hypothesis the statistic follows a $t$ distribution, not a normal $z$ distribution. The $t$ distribution has heavier tails, especially for small samples, so it is more cautious about significance. Use the right degrees of freedom to get a p-value or compare with a critical value.

Step 5 — Interpret carefully. Check the assumptions and explain the conclusion in context, not just the arithmetic.

A full worked example: one-sample t-test

A packaging process should average $100$ grams. A random sample of $25$ packages gives

and you want to know whether the true mean differs from $100$ grams. One sample against a target value means a one-sample t-test.

State the hypotheses:

Standard error:

Test statistic:

Degrees of freedom:

For a two-sided test with $df = 24$, $t = 2.5$ gives a p-value below $0.05$. The result is significant at the $5\%$ level, so you reject $H_0$: the sample gives evidence the process mean differs from $100$ grams — provided the sample is reasonably independent and not badly distorted by outliers.

Where each step trips people up, and how to self-check

• Design step: Picking the wrong version. If the same people, machines, or units are measured twice, the data are paired — an independent two-sample t-test is wrong.
• Read step: Treating the $t$ value as if it followed a normal distribution; small samples need the heavier-tailed $t$.
• Interpret step: Reading "not statistically significant" as "there is no difference." It usually means the sample lacked strong enough evidence against the null.
• Data check: Skipping it. A tiny sample with one extreme outlier still produces a number, but the conclusion may not be trustworthy.

Your turn

Change the example so the sample mean is $101$ instead of $102$, keeping $n = 25$ and $s = 4$. Recompute $t$ and decide whether the evidence still clears the $5\%$ level — a clean way to see how the conclusion shifts as the mean moves toward the null. T-tests like this run throughout experiments, quality control, medicine, psychology, education, and A/B-style comparisons whenever the outcome is numeric.