Paired t-Test

Formula

\[ t = \frac{\bar d}{s_d/\sqrt{n}} \]

Plot

fn: 1/(1+exp(-5*(x-0.4)))
xmin: 0
xmax: 1.5
ymin: 0
ymax: 1.05
height: 280
title: Example power curve for paired t-test (illustrative)

Parameters

• \(d_i\): paired differences \(x_i - y_i\)
• \(\bar d\): mean of differences
• \(s_d\): standard deviation of differences
• \(n\): number of pairs

What it means

Tests whether the mean difference between paired measurements is zero.

What it's used for

• Before/after measurements on the same subjects.
• Controlling for subject-level variability.

Key properties

• Under \(H_0\), \(t\) follows Student's t with \(n-1\) degrees of freedom.
• Equivalent to a one-sample t-test on differences.

Common gotchas

• Requires meaningful pairing; otherwise use two-sample t-test.
• Differences should be roughly normal.

Example

If differences are \([2, 0, 1, 3]\), then \(\bar d=1.5\), \(s_d=1.291\), \(n=4\), so \(t=1.5/(1.291/2)=2.324\).

How to Compute (Pseudocode)

Input: data, null hypothesis H0, test statistic T
Output: test statistic and p-value decision summary

compute the observed test statistic T_obs from the data
obtain the null distribution (analytic approximation or exact table, depending on the test)
compute the p-value from the null distribution and tail convention
compare p-value to alpha (if making a decision)
return T_obs and p-value

Complexity

• Time: Depends on the specific test (summary-statistic computation is often linear in sample size; p-value computation may be constant-time with a CDF call or more expensive if resampling is used)
• Space: Depends on whether intermediate summaries or resampled/null distributions are materialized
• Assumptions: Test-specific assumptions (independence, variance structure, distributional assumptions) determine validity and exact computation details