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Use the original centered quantities
The original five-period cross-product is 21 and benchmark squared sum 10 in percentage points squared. The OLS slope is their direct ratio, 2.1.
Compare consistent and mixed divisors
| Covariance divisor | Variance divisor | Ratio |
|---|---|---|
| n−1: 5.25 | n−1: 2.5 | 2.1 |
| n: 4.2 | n: 2 | 2.1 |
| n: 4.2 | n−1: 2.5 | 1.68 |
| n−1: 5.25 | n: 2 | 2.625 |
Read the estimator labels
Both sample and population moment summaries can reproduce the centered-sum slope when they use the same observations and divisor. A label such as covariance does not specify that choice on its own.
Keep statistical estimation separate
This cancellation explains the slope computation. It does not declare the supplied sample a population or validate any estimator’s assumptions. Retain the paired record and the formula definitions.