independent samples

Two or more samples are independent if there is no systematic relationship between them apart from the fact that they are from the same underlying populations or distribution. This will usually mean that statistical measures of connection are purely random. Often each of the samples are themselves formed by randomly choosing the participants or items in the sample, but there can be more complex sampling methods where each sample has some internal structure (e.g. a stratified sample spread across several subgroups), but the samples are independent from each other. In an experiment choosing the samples of participants for each conditon ensures that there is no statistical bias, but there are also other ways to achieve this.

In an experimental study with people, one would normally ensure each person only participates once, that is a form of sampling without replacement. In principle, this does create a relationship between the samples, for example if one sample of people were (by random chance) unusually tall, there would be less tall people left to choose from and hence the other sample would be expected to be shorter than average; that is a small negative correlation betweeen the average heights of the two samples. However, this could equally well have been the other way, and also the effect is usually very small for sufficiently larger populatuons, so this form of sampling without replacement would usually be still considered as independent.