independentsamples

Independent samples refer to samples drawn from two or more populations in such a way that the observations in one sample do not affect those in another. The key property is mutual independence: the measurement in one unit provides no information about measurements in other units across the groups. This contrasts with paired or matched designs, where observations are linked.

In inferential statistics, independence enables comparing central tendencies across groups. The most common setting is the two-sample problem: comparing means of two independent populations. Under normality and equal variances, a two-sample t-test (Student's) tests whether the population means are equal. If variances differ, Welch's t-test adjusts the degrees of freedom.

The standard error of the difference in means for the equal-variance case uses pooled variance; for unequal variances it uses separate variances.

Nonparametric alternatives exist when normality or variance assumptions fail; the Mann-Whitney U test compares distributions based on ranks rather than means.

For more than two independent groups, one-way analysis of variance (ANOVA) assesses whether at least one group mean differs. If ANOVA is significant, follow-up post hoc tests identify specific differences, while keeping independence in mind.

Assumptions generally include random sampling, independence within and between samples, and, for parametric tests, normality of the sampling distributions and homogeneity of variances. Violations of independence can bias results and inflate Type I error rates.

In experimental design, independence is often achieved by random assignment to conditions; observational studies require careful control of confounding factors to approximate independence.