R² After Duplicating the Data

This linear regression question is about how a key goodness-of-fit statistic behaves when you modify the dataset in a very controlled way. In the context of quant prep, it checks whether you really understand what lies behind a popular summary number in regressions, not just how to read it off a stats package. It forces you to connect the formula to the structure of the data used in quant interviews.

It trains your grasp of how ratio-based statistics respond to uniform scaling of the underlying sample and targets intuition about sums of errors, total variation, and relative explanatory power. You practice thinking in terms of how core regression quantities transform when the dataset changes without altering its informational content.

This matters in quant interviews because real-world quant research constantly involves reweighting, resampling, or aggregating data. Interviewers want quants who understand which metrics are stable under these operations and which are not. Being able to reason about such invariances is crucial for interpreting backtests, validating models, and ensuring that your quant prep translates into robust model evaluation on large, messy datasets.

What it tests

The key structure underlying this class of problems is the invariance of ratio-based statistics under uniform duplication or scaling of the dataset. When a statistic is defined as a ratio of two quantities that both scale linearly with the number of data points (such as $SSE$ and $SST$ in $R^2$), duplicating each observation multiplies both the numerator and denominator by the same factor, leaving the ratio unchanged. This is because the relative dispersion or explanatory power measured by such statistics depends on proportions, not absolute sums. The reason this holds is that both the total variation in the data and the unexplained variation (errors) are affected equally by duplication, so their relationship is preserved. This principle applies broadly to any metric where both components are sums over the data and are affected identically by duplication or uniform weighting.