← Chapter 12
Statistics · Chapter 12

Paired or Independent?

Two columns of numbers arrive on your desk. Nothing about the numbers themselves tells you how to compare them — that answer lives entirely in how the data were collected. If each value in the first column has a specific partner in the second, belonging to the same person, the same plot of land, the same machine, then the columns are paired, and the honest analysis works on the single column of within‑pair differences. If the two columns are separate sets of subjects with no one‑to‑one link, they are independent, and the machinery of the previous section applies. Get this wrong and nothing crashes; the software returns a p‑value either way, and it is simply the wrong one. Below you can sort real designs into the two bins, and then run a study where the same sixteen numbers are analysed both ways so you can watch, in minutes and in decimals, exactly what the pairing was worth.

One study, and the fact you can choose to see or ignore

Every runner in this study covers the same 5 km loop twice — once in the shoe they already own, once in a new brand, with the order tossed by a coin. That gives two columns of finish times, and each row of the two columns is one runner. Press the toggle: on the left setting the connecting lines are drawn, so the pairing is visible; on the right setting the identical numbers are shown as two anonymous clouds. No value changes between the two views. The only thing that changes is whether you can still tell which time belongs to whom — and that, it turns out, is worth a great deal.
the shoe they own the new shoe runner was faster in the new shoe runner was slower

Collapse each pair to one number

This is the whole paired procedure, and it fits in one line: subtract within each pair first, then forget the two original columns entirely and treat what is left as a single sample. Each dot below is one runner’s personal difference, old shoe minus new shoe, in minutes. A dot to the right of zero means that runner was faster in the new shoe. From here on, everything is one‑sample work: the mean of these differences, their standard deviation, a standard error of sd ÷ √n, and n − 1 degrees of freedom, where n is the number of pairs — not the number of measurements.

The same numbers, analysed both ways

Here is the comparison that makes the section. Both analyses are run on the identical data, and they even agree on the point estimate — the mean of the within‑pair differences and the difference between the two column means are the same number, always, as a matter of arithmetic. What they disagree about is how uncertain that number is. The paired analysis measures the wobble from the spread of the differences; the independent analysis measures it from the spread of the raw times, which is dominated by the fact that some people are simply faster runners than others — a fact that cancelled out completely when you subtracted.
paired — uses the link independent — discards it zero — no difference

Where the gain lives: the correlation inside the pairs

Plot each runner as a single point — old‑shoe time across, new‑shoe time up. A tight diagonal cloud means the pairs track: a runner who is slow in one shoe is slow in the other, because most of what determines a finish time is the runner, not the shoe. That tracking is exactly what subtraction throws away, and the arithmetic is not a metaphor. The variance of the differences is sA² + sB² − 2 r sAsB. The first two terms are what the independent analysis uses; the third term — the one it has no way to see — is the refund the pairing pays you.

Run the whole study a thousand times

One study settles nothing. Each run below recruits fresh runners, times them all twice, and records the mean difference along with both p‑values. The bars are the simulated mean differences; the dark curve is the theoretical sampling distribution, centred on the true effect with spread σ√(2(1 − ρ) / n). Nothing is fitted. Then compare the observed spread of those bars against the two standard errors the analyses reported — whichever standard error matches the real spread is the one telling you the truth.
simulated mean differences theory, not fitted zero — no effect
Studies
run0
Mean of the
differences—
Actual spread
of d̄—
Theory
σ√(2(1−ρ)/n)—
Paired SE
reported—
Independent SE
reported—
Two shoes, one runner at a timeset the study, then look at it both ways

The truth you are sampling from

In a real study you know none of these. Here you set them, so that every number the two analyses report can be checked against an answer already on the table. The fourth dial is the one that matters for this section: it controls how tightly a runner’s two times track each other.
1.0
4.00
0.90
16
Jump to a situation worth seeing

This study, computed live

The two columns of times, in minutes
Own shoe
mean x̄A— 
New shoe
mean x̄B— 
Mean of the
differences d̄— 
True effect
δ—you set this
Spread — and how tightly the pairs track
sA—own shoe
sB—new shoe
Correlation
r— 
sd of the
differences— 
Analysed as paired — the reading the design earns
Standard
error—sd ÷ √n
Degrees of
freedom—n − 1
t—d̄ ÷ its SE
p‑value
two‑sided— 
Analysed as two independent groups — the same numbers, mis‑read
Standard
error—√((sA²+sB²)/n)
Degrees of
freedom—2n − 2
t—same d̄, bigger SE
p‑value
two‑sided— 
The two 95% intervals for the same quantity
Paired 95% interval for δ— 
Independent 95% interval— 

Sort the designs

Ten studies. For each one, decide from the description alone whether the two columns of numbers are paired or come from independent groups. The test is always the same question: does each number in the first column have one specific partner in the second? If yes — same person, same twin pair, same plot, same specimen — it is paired. If the two columns are different subjects who happen to be in the same study, it is independent.

Say it right

Nine things people say about paired data. Four of them are correct. Tap each one.

What to carry out of this section

The design decides the analysis, and the design was fixed before you saw a number. Pairing is not a setting you choose in software and not a property you can detect by inspecting the two columns. It is a fact about data collection: is there a one‑to‑one link, or is there not? Ask who or what each row is. If the answer is “the same runner, measured twice,” the rows are pairs. If the answer is “a different person in each column,” they are not. Running both analyses and reporting the friendlier p‑value is not a judgement call — it is the thing this section exists to prevent.
Subtract first, then do one‑sample statistics. The paired t‑test is not a new test; it is the one‑sample t‑test wearing a coat. Form di = Ai − Bi within each pair, and everything that follows — d̄, sd, the standard error sd ÷ √n, the interval, the p‑value — is machinery you already own. The one detail people lose is the sample size: n is the number of pairs, so twenty runners timed twice give forty measurements and nineteen degrees of freedom, not thirty‑eight.
What pairing buys is written on one line: SEpaired ÷ SEindependent = √(1 − ρ). When the two spreads are equal, that ratio is the entire story of the section. At ρ = 0.9 the paired standard error is √0.1 ≈ 0.32 of the independent one — to reach that same precision without the pairing you would have to run an independent study about ten times larger, since the sample size has to grow by 1 ÷ (1 − ρ) to compensate. At ρ = 0.5 the factor is 0.71 and the study only doubles. At ρ = 0 it is 1, and the pairing has bought you nothing at all.
Pairing is not free, and it is not always better. Load the Pairing buys nothing preset. With ρ = 0 the two standard errors agree on average, but the paired analysis is judging its t against n − 1 degrees of freedom instead of 2n − 2, so it needs a slightly larger t to clear the same bar — it is now the marginally weaker test. And there is a second cost the arithmetic here cannot show: a crossover design of n runners uses half as many people as an independent study with n per group, and buys that saving by asking each person to do the thing twice, which is not always possible, ethical, or free of carry‑over from the first measurement to the second.
Negative pairing is the dangerous case. Drag ρ below zero — rare in practice, but it happens when the two measurements compete for something fixed, or when pairs were formed by matching on the outcome itself. Now −2 r sAsB is a positive term, the differences are more variable than either column, and the independent analysis reports a standard error that is too small. Set δ to zero as well and run the simulation: the independent analysis’s false‑alarm rate climbs above the 5% you authorised. Positive pairing analysed as independent merely wastes power; negative pairing analysed as independent produces confident conclusions that are not earned.
A small p‑value from a paired test still is not the probability of anything about the truth. It is computed assuming the true mean difference is zero, and it reports how often a mean difference at least this far from zero would appear under that assumption. It is not the probability that the shoes are identical, and it is not the probability that you are wrong. The 95% interval is a statement about the procedure — intervals built this way capture the true δ in 95% of studies — and once the endpoints are printed, the truth is simply inside them or outside them.
Being your own control is a strong design, not a licence for causation. Pairing removes everything stable about a subject — fitness, height, technique, the whole hidden bundle — and that is genuinely powerful. It removes nothing that changed between the two measurements. In a plain before‑and‑after study the second measurement is also later: people practise, illnesses resolve on their own, and extreme first readings drift back toward the middle for purely arithmetic reasons. That is why the shoes here were run in randomised order. A paired comparison with no randomisation and no control group is still an observational one, however small its p‑value gets.