Paired data

When two types measurements, X and Y, are made from each individual (or other unit), the data are called bivariate. Sometimes the two measurements are of closely related quantities and may even describe the same quantity at different times.

When the sum or difference of X and Y is a meaningful quantity, the data are called paired data.

Hypotheses of interest

For paired data, We often want to test whether the means of the two variables are equal,

H0 :   μX = μY
HA :   μXμY

Sometimes a one-tailed test is required, such as

H0 :   μX = μY
HA :   μX > μY

Examples

Pre-test, post-test data
This arises when a measurement is made from each individual, then a second measurement of the same type is made after some kind of intervention (e.g. training or medication). Has the intervention "improved" the measurement?
Twin studies
Some experiments or other studies are conducted with identical twins, either human or animal. The members of each pair experience different environments — either two different experimental treatments or two other differences. Are there differences between the two treatments?
Other types of pairing
For example, damaged cars may each be taken to two garages for estimates of the cost of repair. The two estimates for each car are paired data. Does one garage overcharge?