One-Sample & Paired t-Tests
The one-sample t-test asks a simple question: is my sample's mean far enough from some claimed value to be more than chance? It's the hypothesis-testing logic applied to a single mean — with one important complication that gives the test its name.
The test statistic
You compare your sample mean x̄ to a hypothesized value μ₀, scaled by the standard error:
t = (x̄ − μ₀) / (s / √n)
All three members of the family (one-sample, paired, independent) sit next to each other on the formula sheet, which is the quickest way to see how little actually changes between them.
This looks just like a z-score — except for one thing. We almost never know the population standard deviation, so we estimate it with the sample's s. That estimate is itself uncertain, especially with small samples, and that extra wobble means the statistic no longer follows the normal curve. It follows the t-distribution instead.
The t-distribution
The t-distribution looks like the normal's slightly nervous cousin: same bell shape, centered at zero, but with heavier tails. Those fat tails say "with a small sample, extreme values are more common than the normal would predict." Its shape is governed by the degrees of freedom (df = n − 1). The smaller the sample, the fatter the tails — and the further out you must go to call a result significant.
🎮 The t-Distribution vs. the Normal
Shrink the degrees of freedom and watch the t-curve's tails fatten and its 5% critical value drift out past the normal's 1.96.
Notice how at df = 5 you need a t beyond about ±2.57 to reach significance, but by df = 60 the critical value has shrunk back toward the familiar 1.96. As samples grow, the t-distribution becomes the normal distribution. That's why, with large n, the t-test and a z-test give nearly identical answers. For the exact critical value at any df and α, and for the p-value behind a t you have already computed, use the statistical tables calculator.
The paired t-test: a clever reframe
What about before-and-after measurements on the same people — a weight-loss program, a training intervention, a drug trial? Those two columns aren't independent; each person is measured twice. The trick is to collapse them: compute each person's difference (after − before), and run a one-sample t-test on those differences against μ₀ = 0.
Why pairing is powerful: by looking at within-person change, you cancel out the huge person-to-person variability ("Ana is just heavier than Ben") and isolate the effect of the treatment. A paired design often detects an effect that an independent-groups design would miss, which is why choosing between the two is a design decision, not an analysis one.
What you're assuming
- The data (or the differences, for a paired test) are roughly normal — or the sample is large enough for the CLT to rescue you.
- Observations are independent (for the paired test, the pairs are independent).
Why it matters: the one-sample and paired t-tests are the simplest members of a family that runs through the entire course. Understand the t-distribution here and the independent-samples t-test next is a short step.
Both tests are worked end to end in the practice problems: Problem 10 from summary statistics, Problem 11 from a column of difference scores.
Common questions
When should I use a paired t-test instead of an independent t-test?
Use paired when the two sets of scores belong to the same (or matched) units — before/after measurements, twins, left/right comparisons. The pairing lets you analyze within-unit differences, canceling stable individual differences and usually gaining a lot of power. Use the independent-samples test when the groups contain different, unrelated people.
What are degrees of freedom in a t-test?
Roughly, the number of values free to vary once you've estimated the necessary quantities — for a one-sample or paired test, df = n − 1 (estimating the mean uses up one). Degrees of freedom set the t-distribution's shape: small df means fatter tails and a stricter critical value, reflecting the extra uncertainty of estimating the SD from little data.
Is a paired t-test the same as a repeated-measures ANOVA with two conditions?
They give the same verdict. Run both on the same two-condition data and you get F = t² with an identical p-value, because a repeated-measures ANOVA on two levels is doing what the paired test does: analyzing each person's own difference score. The ANOVA earns its keep once there are three or more conditions, which is what repeated-measures ANOVA covers.