Paired t-test
A paired t-test tests whether the mean difference between matched pairs of observations — such as before-and-after measurements — is zero, by analysing the within-pair differences as a single sample.
FrameworkHypothesis testing
See it move
Ten employees take a time-management course. Their mean daily output rises from 48 units before the course to 52 units after — a mean within-pair difference of +4 units, with a standard deviation of differences of 3.5. Treating those ten differences as a single sample gives t = 4 ÷ (3.5/√10) ≈ 3.61, which exceeds the critical value and signals a statistically significant improvement.
The formula
Variables
- Sample mean of within-pair differences
- Sample standard deviation of the differences
- Number of matched pairs
Degrees of freedom = n − 1. H₀: μ_d = 0 (no mean difference). Pairing removes between-subject variation, increasing test power.
Check yourself
A supermarket chain pilots a new shelf layout in 12 selected stores. Weekly sales are recorded for four weeks before and four weeks after the change in each store. An analyst tests whether mean weekly sales changed. Why is a paired t-test more suitable than an independent-samples t-test?