T-Tests: One-Sample, Two-Sample, and Paired — With Full Worked Examples

A t-test answers one of the most common questions in applied statistics: is this difference between groups real, or could it plausibly be explained by chance alone? Does a new teaching method actually improve test scores, or did this particular class just happen to do a bit better? The t-test is the standard tool for comparing means when your sample size is modest and the true population standard deviation is unknown — which describes the overwhelming majority of real research.

Why the T-Distribution, Not the Normal Distribution?

The t-test extends the general hypothesis testing framework — null and alternative hypotheses, a significance level, a test statistic, a p-value, and a decision — to the specific, extremely common case of comparing means. Everything covered in that framework (Type I and Type II errors, one-tailed vs. two-tailed tests, the meaning of "fail to reject") applies directly here.

A z-test requires knowing the true population standard deviation, σ — something rarely known in practice. When σ must be estimated from the sample (using s), that estimate itself carries uncertainty, especially with small samples. The t-distribution accounts for this extra uncertainty with heavier tails than the normal distribution — meaning more probability sits further from the centre, which is exactly the right adjustment when your estimate of spread is itself imperfect. As sample size grows, the t-distribution converges toward the normal distribution, since a larger sample gives a more reliable estimate of s.

The Three Types of T-Test

• One-sample t-test: compares a single sample's mean against a known or hypothesized value. "Is this batch's average weight different from the target of 500g?"
• Independent (two-sample) t-test: compares the means of two separate, unrelated groups. "Do students taught with Method A score differently than students taught with Method B?"
• Paired t-test: compares two measurements taken on the same subjects, typically before and after some intervention. "Did blood pressure change after patients started a new medication?"

One-Sample T-Test Formula

Here x̄ is the sample mean, μ₀ is the hypothesized population value, s is the sample standard deviation, and n is the sample size. The degrees of freedom for this test is n − 1.

Independent Two-Sample T-Test Formula

This version (Welch's t-test, which doesn't assume the two groups have equal variances — the more robust default choice) compares the difference between two independent sample means, scaled by the combined uncertainty from both samples.

Worked Example: Independent Two-Sample T-Test

Problem: Two teaching methods are compared. Group A (n=25) scores a mean of 78 with SD 8. Group B (n=25) scores a mean of 74 with SD 9. At α = 0.05, is there a significant difference?

Hypotheses: H₀: μ₁ = μ₂ (no difference). H₁: μ₁ ≠ μ₂ (two-tailed).

Test statistic: t = (78 − 74) / √(8²/25 + 9²/25) = 4 / √(2.56 + 3.24) = 4 / √5.8 = 4 / 2.408 ≈ 1.66

Degrees of freedom (Welch-Satterthwaite approximation): approximately 47 for this example.

P-value: for t ≈ 1.66 with df ≈ 47, two-tailed p ≈ 0.103

Decision: p (0.103) > α (0.05) → fail to reject H₀

Conclusion: There isn't statistically significant evidence, at the 5% level, that the two teaching methods produce different mean scores — despite Group A scoring 4 points higher on average, that difference is small enough relative to the natural variability within each group that it's plausibly explained by chance.

Worked Example: Paired T-Test

Problem: Ten patients' blood pressure is measured before and after starting a new medication. The differences (before − after) average 6 mmHg with a standard deviation of 5 mmHg. Is there evidence the medication reduces blood pressure?

Test statistic: t = d̄ / (s_d/√n) = 6 / (5/√10) = 6 / 1.581 ≈ 3.79, with df = n−1 = 9

P-value (one-tailed, since the research question specifically predicts a reduction): p ≈ 0.002

Decision: p (0.002) < α (0.05) → reject H₀. There's strong evidence the medication reduces blood pressure.

Assumptions Behind a T-Test

A t-test's results are only trustworthy when its underlying assumptions roughly hold:

• Independence: observations within each group must be independent of one another (not applicable across paired measurements, which are deliberately dependent by design).
• Approximate normality: the data (or, for larger samples, the sampling distribution of the mean via the Central Limit Theorem) should be roughly normally distributed. T-tests are reasonably robust to mild departures from normality, especially with larger samples, but severe skew or extreme outliers can distort results.
• Roughly similar variances (for the classic "pooled" two-sample t-test — Welch's version, used above, relaxes this requirement, which is why it's the more commonly recommended default today).

When these assumptions are seriously violated — heavily skewed data with a small sample, for instance — a non-parametric alternative like the Mann-Whitney U test may be more appropriate than a t-test.

Effect Size: What a P-Value Alone Doesn't Tell You

A significant t-test tells you a difference likely exists, but not how large or practically meaningful it is. Cohen's d — the standardized difference between two means, in units of pooled standard deviation — fills this gap: d = (x̄₁ − x̄₂) / pooled SD. Conventionally, d ≈ 0.2 is a small effect, 0.5 is medium, 0.8 is large. Reporting effect size alongside a p-value is increasingly standard practice, precisely because a large sample can make a trivially small difference statistically significant, and effect size is what tells you whether that difference actually matters.

Real-World Uses

Clinical trials: comparing a treatment group's outcomes against a control group is one of the most common applications of the independent two-sample t-test in medical research, though large trials increasingly use more sophisticated models that also adjust for other variables.

A/B testing: comparing average conversion value, time-on-page, or other continuous metrics between two website variants often uses a t-test as the underlying statistical engine, deciding whether an observed difference is likely to reflect a genuine difference in user behaviour rather than random noise in who happened to see which variant.

Manufacturing quality control: a one-sample t-test can check whether a batch of products' average measurement (weight, dimension, concentration) differs significantly from a target specification, flagging a process that may be drifting out of tolerance before it produces widespread defects.

Common Mistakes

Running a t-test on clearly non-normal data with a small sample. With a small sample and severely skewed or heavy-tailed data, the t-test's normality assumption can be seriously violated, and a non-parametric alternative should be considered instead.

Treating "not significant" as proof of no difference. A non-significant t-test with a small sample often reflects low statistical power to detect a real but modest effect, not solid evidence that no difference exists. Reporting a confidence interval for the difference alongside the p-value makes this distinction visible.

Ignoring the paired-vs-independent distinction. As covered above, this is a genuinely common error that produces an incorrect result, not just a less powerful one — always check whether your two groups of measurements come from the same subjects before choosing a test.