The T-Test: One-Sample, Two-Sample (Pooled & Welch)

In 1908, a statistician named William Sealy Gosset faced a practical problem at the Guinness Brewery in Dublin. He needed to compare the quality of barley batches using small samples, but the statistical methods of his time assumed you knew the population variance, which he didn't. Gosset solved this problem by deriving a new distribution that accounts for the uncertainty of estimating variance from the sample itself. Because Guinness prohibited employees from publishing scientific papers (fearing competitors would realize the advantage of employing statisticians), Gosset published under the pseudonym "Student." The "Student's t-distribution" and the t-test it enables have since become the workhorse of statistical inference.

The t-test is probably the most frequently used statistical test across research and industry, including medicine. Whenever someone asks "Is this average different from that target?" or "Do these two groups differ?", a t-test is typically the answer. This chapter covers the t-test in depth: why it exists, how it works mathematically, and when to use each variant. After mastering t-tests, you'll be ready to explore F-tests for comparing variances, ANOVA for comparing multiple groups, and the broader topics of statistical power and effect sizes.

In the previous chapter on z-tests, we learned to compare means when the population standard deviation $\sigma$ is known. The test statistic was:

This follows a standard normal distribution because the denominator $\sigma / \sqrt{n}$ is a fixed, known quantity. Only the numerator varies from sample to sample.

The problem is that you almost never know $\sigma$. In practice, you estimate the standard deviation from your sample data, giving you the sample standard deviation $s$. Early researchers simply substituted $s$ for $\sigma$ and continued using normal distribution critical values. This seemed reasonable, but it led to systematic errors. They rejected the null hypothesis more often than they should have, especially with small samples.

Gosset discovered why: when you estimate $\sigma$ from the same sample you're testing, you introduce additional variability into the test statistic. The sample standard deviation $s$ is itself a random variable that fluctuates from sample to sample. Sometimes $s$ underestimates $\sigma$, making your test statistic too large. Sometimes $s$ overestimates $\sigma$, making it too small. This extra variability means the test statistic no longer follows a normal distribution. It follows a distribution with heavier tails, one that appropriately penalizes us for not knowing the true population variance.

When we replace $\sigma$ with $s$, the test statistic becomes:

This ratio follows a t-distribution with $n - 1$ degrees of freedom (df). The degrees of freedom represent the amount of independent information available to estimate the variance. When computing $s$ from $n$ observations, we use up one degree of freedom estimating the mean, leaving $n - 1$ degrees of freedom for estimating variance.

The t-distribution has several key properties:

1. Bell-shaped and symmetric: Like the normal distribution, centered at zero under the null hypothesis
2. Heavier tails: More probability in the extreme values than the normal distribution
3. Parameterized by degrees of freedom: The shape depends on df, with smaller df meaning heavier tails
4. Converges to normal: As $df \to \infty$, the t-distribution approaches the standard normal

The heavier tails account for the uncertainty added by estimating variance: with small samples, $s$ can deviate substantially from $\sigma$, so extreme t-values become more probable. The t-distribution therefore requires larger critical values to reject the null hypothesis.

Consider what happens when $s$ happens to underestimate $\sigma$:

• The denominator $s/\sqrt{n}$ is too small
• The t-statistic is inflated
• We might incorrectly reject $H_0$

When $s$ overestimates $\sigma$:

• The denominator is too large
• The t-statistic is deflated
• We might miss a real effect

These errors don't favor either direction on average, but they add variability. The t-distribution captures exactly this additional spread. With only 5 observations (df = 4), the sample standard deviation is quite unreliable, so the t-distribution has very heavy tails. With 100 observations (df = 99), $s$ is a much more reliable estimate of $\sigma$, and the t-distribution is nearly identical to the normal.

The heavier tails translate directly into larger critical values. For a two-sided test at $\alpha = 0.05$:

With df = 5, you need a t-statistic 31% larger than the z-critical value to reject $H_0$. This penalty appropriately reflects the unreliability of variance estimates from small samples. By df = 30, the difference is only about 4%, which is why older textbooks sometimes said "use z for n > 30." Modern practice uses the t-distribution regardless, as computational tools make it trivially easy.

The one-sample t-test compares a sample mean to a hypothesized population value when the population variance is unknown. This is the simplest and most direct application of the t-distribution.

Hypotheses:

• $H_0: \mu = \mu_0$ (population mean equals the hypothesized value)
• $H_a: \mu \neq \mu_0$ (two-sided), or $\mu > \mu_0$ / $\mu < \mu_0$ (one-sided)

Test statistic:

where:

• $\bar{x}$ = sample mean
• $\mu_0$ = hypothesized population mean
• $s$ = sample standard deviation (with Bessel's correction: divide by $n-1$)
• $n$ = sample size
• $s / \sqrt{n}$ = standard error of the mean

Distribution under $H_0$: $t \sim t_{n-1}$ (t-distribution with $n-1$ degrees of freedom)

P-value calculation:

• Two-sided: $p = 2 \times P(T > |t|)$ where $T \sim t_{n-1}$
• One-sided (greater): $p = P(T > t)$
• One-sided (less): $p = P(T < t)$

Decision rule: Reject $H_0$ if $p < \alpha$, or equivalently, if $|t| > t_{\alpha/2, n-1}$ for two-sided tests.

A coffee shop claims their large drinks contain 16 ounces. A quality inspector measures 15 randomly selected drinks to verify this claim.

The two-sample t-test compares means from two independent groups. This is perhaps the most common application: comparing treatment vs. control, method A vs. method B, or any two distinct populations.

The choice of t-test depends on one question: do the two groups have equal variance?

When comparing two means, we need to estimate the standard error of the difference $\bar{x}_1 - \bar{x}_2$. If both groups share the same variance $\sigma^2$, we can pool their data to get a more precise estimate. But if variances differ, pooling gives incorrect results, biasing our inference.

This leads to two variants:

• Pooled (Student's) t-test: Assumes equal variances, pools data for efficiency
• Welch's t-test: Makes no variance assumption, handles unequal variances correctly

When we assume $\sigma_1^2 = \sigma_2^2 = \sigma^2$, we can estimate this common variance using data from both groups.

Pooled variance estimate:

This is a weighted average of the two sample variances, with weights proportional to their degrees of freedom. Larger samples contribute more to the pooled estimate because they provide more reliable information.

Standard error of the difference:

The term $\sqrt{1/n_1 + 1/n_2}$ accounts for the fact that uncertainty in the difference comes from uncertainty in both means.

Test statistic:

Degrees of freedom: $df = n_1 + n_2 - 2$

Each group contributes $n - 1$ degrees of freedom for variance estimation, and we combine them.

When variances may differ, we cannot pool them. Instead, we estimate standard errors separately and combine them using a different formula.

Standard error of the difference:

This follows from the variance of a difference: $\text{Var}(\bar{x}_1 - \bar{x}_2) = \text{Var}(\bar{x}_1) + \text{Var}(\bar{x}_2)$ for independent samples.

Test statistic:

Degrees of freedom (Welch-Satterthwaite approximation):

This complex formula estimates the effective degrees of freedom when variances are unequal. The result is typically non-integer, which modern software handles through interpolation.

Why Welch's is the recommended default:

• When variances are equal, Welch's test loses only slightly in power compared to the pooled test
• When variances are unequal, the pooled test can give seriously incorrect results
• The asymmetric risk/reward favors Welch's as the default choice

A researcher compares test scores from two teaching methods: Method A (traditional lectures) and Method B (active learning).

Before choosing between pooled and Welch's tests, you might want to check whether variances are equal. Two common approaches:

1. Visual inspection: Compare the spread in boxplots or calculate the ratio of sample variances. A ratio greater than 2 suggests meaningful inequality.

2. Levene's test: A formal hypothesis test for equality of variances. Unlike older alternatives (Bartlett's test), Levene's test is less sensitive to departures from normality.

The paired t-test is used when observations come in matched pairs, where each pair shares some characteristic that makes them non-independent. Common scenarios include:

• Before-after measurements: Same subjects measured twice
• Matched case-control studies: Cases matched to controls on key characteristics
• Twin studies: Comparing outcomes between twins
• Left-right comparisons: Same subject, different sides

Consider testing whether a training program improves performance. You could:

1. Independent design: Randomly assign people to training or control groups, compare group means
2. Paired design: Measure each person before and after training, analyze the changes

The paired design is often more powerful because it eliminates between-subject variability. People differ enormously in baseline ability, but by looking at changes within each person, we focus only on the training effect.

The paired t-test is simply a one-sample t-test on the differences:

Test statistic:

where:

• $\bar{d}$ = mean of the paired differences
• $s_d$ = standard deviation of the differences
• $n$ = number of pairs
• $df = n - 1$

Null hypothesis: $H_0: \mu_d = 0$ (no average change)

A study tests whether a new medication reduces blood pressure. Ten patients have their systolic blood pressure measured before starting medication and again after 4 weeks of treatment.

You might wonder: what if we ignored the pairing and used an independent two-sample t-test? Let's compare:

The paired test detects the effect with much higher confidence because it accounts for the correlation between before and after measurements within each subject.

All t-tests rely on assumptions. Proper application requires understanding these assumptions, how to check them, and how violations affect the tests.

1. Independence

• One-sample: Observations must be independent of each other
• Two-sample (independent): Observations between groups must be independent; observations within groups must be independent
• Paired: The pairs must be independent of each other (though observations within a pair are dependent by design)

Independence is often the most critical and hardest to verify assumption. It depends on study design rather than data characteristics.

2. Normality

The sampling distribution of the mean should be approximately normal. This is satisfied when:

• The population is normally distributed, OR
• The sample size is large enough for the Central Limit Theorem to apply

Tolerance of non-normality: Departures from normality usually have little effect on the t-test for:

• Two-sided tests
• Larger sample sizes (n > 30 per group)
• Symmetric distributions

Departures matter more for:

• Small samples from highly skewed distributions
• One-sided tests
• Comparing variances or making precise probability statements

3. Homogeneity of Variance (pooled t-test only)

The pooled two-sample t-test assumes equal population variances. Violations affect:

• Type I error rate (can be inflated or deflated)
• The bias is worse with unequal sample sizes

Solution: Use Welch's t-test, which doesn't assume equal variances.

Selecting the appropriate t-test depends on your research design and data characteristics:

Quick Reference:

The t-test is the fundamental tool for comparing means when population variance is unknown, which is nearly always. Key takeaways:

The t-distribution:

• Accounts for uncertainty from estimating variance
• Has heavier tails than the normal distribution
• Converges to normal as degrees of freedom increase
• Requires larger critical values for small samples

One-sample t-test:

• Compares sample mean to hypothesized value
• Test statistic: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$
• Degrees of freedom: $n - 1$

Two-sample t-test:

• Compares means from two independent groups
• Pooled variant assumes equal variances
• Welch's variant (recommended default) handles unequal variances
• Degrees of freedom differ between variants

Paired t-test:

• For matched pairs or repeated measures
• Analyzes within-pair differences
• Often more powerful than independent tests by eliminating between-subject variability

Assumptions:

• Independence (critical, design-based)
• Normality (less restrictive for large samples)
• Equal variances (pooled test only; avoid by using Welch's)

When in doubt: Use Welch's t-test for two-sample comparisons. You sacrifice minimal power when variances are equal but avoid the equal-variance assumption when they differ.

This chapter covered t-tests for comparing one or two means. But what if you want to compare variances rather than means? Or compare more than two groups? The next chapters extend these ideas:

• The F-Test introduces the F-distribution for comparing variances. This is important both as a standalone test (are two groups equally variable?) and as foundation for ANOVA.

• ANOVA extends the t-test to compare three or more groups simultaneously, avoiding the multiple comparison problems that arise from running many t-tests.

• Error types, statistical power, effect sizes, and multiple comparison corrections are all essential for interpreting and designing studies that use t-tests and related methods.

The t-test you've learned here is the building block for all these extensions. Master it, and the more advanced methods will follow naturally.

Ready to test your understanding? Take this quick quiz to reinforce what you've learned about the t-test.