Independence of observations in regression means

• A. The predictor is uncorrelated with the outcome.
• B. The residuals are uncorrelated across observations.
• C. Residuals are uncorrelated across observations.
• D. The data are normally distributed.

Answer: (C)

Independence of observations in regression is about the errors (residuals) you get after fitting the model not being related from one observation to the next. The statement that best captures this is that the residuals are uncorrelated across observations. If residuals show correlation, it means the model isn’t accounting for some pattern or structure in the data, which can lead to biased or unreliable standard errors and hypothesis tests.

Think of it this way: each data point should provide new information, and the error for one case shouldn’t inform the error for another. When you see patterns in residuals over time or order—like clusters of positive residuals followed by clusters of negative ones—that signals a violation of independence and suggests you may need to address autocorrelation, perhaps by modeling time structure or using robust standard errors.

The other statements miss the core point. Saying the predictor is uncorrelated with the outcome describes a relationship between X and Y, not the independence of the observations themselves. And assuming the data are normally distributed speaks to the distributional form of the errors, not whether those errors are independent from one observation to the next.