🔁 Paired-Samples t-Test

1. The big idea

Last unit, the independent t-test compared two separate groups of people — coffee drinkers versus non-drinkers, one mean against another. This unit, the paired (also called dependent or within-subjects) t-test compares two scores from the SAME people — a before and an after, or a matched pair. The two columns aren't strangers; they're tied together, row by row, because the same person produced both.

The rule of thumb: same people measured twice = paired. Two separate, unconnected groups = independent. That single distinction decides which test you reach for — and it's set by your design, long before you touch a number.

2. Recognizing a paired (within-subjects) design

The paired test fits whenever your two scores are linked in pairs. There are two flavors, and both count:

• Same people, twice (repeated measures). You measure each person before and after something — a treatment, a lesson, an exercise. In our dataset, mood_pre and mood_post are exactly this: every student gave a mood rating before a gratitude exercise and again after.
• Matched pairs. Different people, but deliberately paired off so each pair is as alike as possible — twins, married couples, or two participants matched on age and baseline score, one assigned to each condition. The pairing makes their scores belong together just like a before/after pair.

The tell is always the same: can you line the two columns up row by row, with a meaningful reason each row's two values belong together? If yes — same person, or a matched partner — it's paired. If the two columns are just two piles of unrelated people, it's independent.

3. The trick — it works on the differences

Here's the clever part, and it's the single most important idea in the unit. The paired t-test doesn't really juggle two columns at all. First it does subtraction. For each person it collapses the two scores into one difference score, usually written d:

That subtraction is the whole magic move. Once you have a column of d's, the paired test is nothing more than a one-sample t-test on those differences, with a test value of zero. You already learned that test in Unit 10 — so you already know how to run a paired t-test. It just asks one simple question:

Is the average difference (d̄) different from zero?

• If nothing really changed, the ups and downs cancel and the differences average out to about 0.
• If something genuinely happened, the differences pile up on one side — and their average sits clearly above (or below) zero, far enough that chance can't comfortably explain it.

That's the entire test. Turn two scores into one difference, then check whether the mean difference is far enough from zero to believe. Two columns in; one column of d's; a one-sample t against 0 comes out.

4. The formula — a one-sample t in disguise

Because the paired test is just a one-sample t on the difference scores, the formula is the one you already know, dressed in d's:

Read it in plain English: the top is "how big is the average change?" and the bottom is "how much do the changes wobble around from person to person?" A big, consistent change over a small wobble makes t large — large enough to clear the critical value and reject the null. Watch it work on five clean numbers:

5. Why pairing is so powerful

Everyone starts at a different baseline. One student is a naturally cheerful 8; another is a quiet 3. In the independent t-test, those huge person-to-person gaps become noise in the denominator that swamps the real effect — the spread between people drowns out the signal. Pairing cancels that noise. Because each person is compared to themselves, their personal baseline subtracts away in the very first step (the difference score). The cheerful student and the quiet student can both show a clean +1 jump, and pairing sees both jumps with no baseline static in the way.

Statistically, this is the payoff: subtracting each person's own baseline removes the between-person variability, which shrinks the standard error in the denominator. A smaller denominator makes t larger for the same average change — so a real effect is easier to detect. That is what we mean by more statistical power.

6. The headline study — a gratitude lift

We ran a quick gratitude exercise in class: students listed a few things they were grateful for. We measured mood before (column mood_pre) and mood after (column mood_post) for all 200 students. Reduce each student to a difference score, average them, and run a one-sample t against zero — exactly the recipe above, just with 200 pairs instead of 5.

t(199) = 11.9, p < .001.

Because p is far below .05, we reject the null hypothesis of no change: the lift is real, not a fluke. The gratitude exercise significantly raised mood — about a full point on the scale, and not by chance. Our result echoes a famous study: Emmons and McCullough (2003) had people keep weekly lists of things they were grateful for and found the gratitude group reported higher well-being than a comparison group. A tiny exercise, a real and reliable lift — exactly what we saw in our own data.

🧮 Try it yourself

Here's the live Paired-Samples t-Test calculator. Hit Load class data to drop in the mood pre/post numbers, then Run and read the plain-English result. Behind the scenes it does precisely the six steps above — difference scores, their mean and SD, then t = d̄ ÷ (s_d/√n).

📏 How big? Cohen's d for paired designs

Same lesson as the last two units: t(199) = 11.9 says the lift is real, not how big it is. For a paired design, Cohen's d takes the cleanest possible form — the average change divided by the SD of the change scores, both of which you already computed on the way to t:

d = d̄ ÷ s_d = 1.12 ÷ 1.33 = 0.84

✍️ Report it in APA style

The APA (7th ed.) sentence for a paired t gives both time-points' means, then the statistics — same house rules as always (italicized letters, two decimals, no leading zero on p, "p < .001" never "= .000", effect size included):

Direction in plain words, both means, then the full inferential sentence. Note the df quietly telling the reader this was 200 pairs — in a paired design the write-up describes the same people at two moments, not two separate groups.

💻 In Google Sheets

Both scores live in their own columns — one for before, one for after. There are two equally good ways to do this, and they give the same answer.

The one-formula way. In any empty cell:

=T.TEST(V2:V201, U2:U201, 2, 1)

Reading the arguments left to right:

• V2:V201 = mood after
• U2:U201 = mood before
• 2 = two-tailed (we'd accept a change in either direction)
• 1 = PAIRED — this last digit (the type argument) is the switch that makes it a paired test

That single 1 is the whole difference from last unit. A 2 there would run an independent test instead. One digit changes everything. T.TEST hands you the p-value directly.

The build-the-difference-column way (and the one that shows you what's really happening). In a new column, subtract row by row, then run a one-sample test on that single column against 0:

W2:  =V2 - U2          → drag down to W201: a difference score per person
=AVERAGE(W2:W201)      → d̄  (the mean difference)
=STDEV(W2:W201)        → s_d (SD of the differences)
=AVERAGE(W2:W201) / (STDEV(W2:W201) / SQRT(200))   → t
df = 200 - 1 = 199

The two recipes agree because they are the same test. The first hides the difference column; the second shows it. Either way you're testing whether d̄ is reliably different from zero.

🐮 Before and after, on the farm question

Before-and-after designs are everywhere, and they're a natural fit for studying how attitudes shift. Imagine showing students a short documentary about modern animal agriculture, then measuring their animal-compassion score both before and after. Same people, two scores, one difference apiece — a textbook paired design, scored with exactly the formula above.

Pairing is precisely the right tool here, because people walk in at wildly different starting points: some already deeply concerned, some never having thought about it. An independent comparison would drown the real shift in all that baseline variation — that between-person spread would inflate the standard error and bury the signal. Pairing strips it out — each person is compared to their own earlier score — so even a modest, genuine change in compassion becomes easy to detect. The same logic that caught a one-point mood lift would catch an attitude shift after a single film. Matter-of-fact, and entirely testable.

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