Recover a Paired Change SD with an Explicit Correlation

Identify the paired observations

A before-and-after measurement on the same participant creates a pair. The variability of the change depends on how those two measurements move together. Subtracting their means is straightforward; subtracting their standard deviations does not recover the standard deviation of the changes.

Start with a record of the outcome, units, baseline and follow-up windows, and number of complete pairs. Confirm that both reported SDs describe the same paired population. If baseline includes everyone enrolled but follow-up includes only completers, combining those summaries may not reconstruct the intended change distribution.

Write the assumption beside the inputs

For paired measurements, the variance of the difference is SD_before² + SD_after² − 2 × r × SD_before × SD_after. Here r is the within-person correlation between the two measurements. Cochrane's effect-measure guidance explains reconstruction using a correlation and recommends sensitivity analysis when that correlation is imputed.

Use the paired-measurement audit worksheet to distinguish a correlation calculated from the relevant paired data from one borrowed or assumed. A correlation between treatment-group averages is not the required participant-level correlation.

Recalculate a fictional example

Suppose the baseline SD is 8 units, the follow-up SD is 9 units, and the assumed within-person correlation is 0.625. The change variance is 64 + 81 − 90 = 55, giving a change SD of approximately 7.4162 units. These values are invented for teaching.

The paired-change SD tool checks this arithmetic. Change r to 0.3 and 0.8: the resulting SDs are approximately 10.0896 and 5.4589. This variation shows why an undocumented correlation can materially alter an analysis. These scenarios are not a confidence interval for the unknown correlation.

Leave a reproducible decision

Seek the reported change SD or relevant paired data before borrowing a correlation. If borrowing is necessary, record the source population, interval between measurements, instrument, and justification for transfer. Do not select a correlation because it produces a smaller standard error or a preferred result.

Keep reported and reconstructed fields separate. Document any sensitivity analysis in the synthesis, not just in the extraction spreadsheet. This calculation does not choose between change-score analysis and baseline-adjusted modeling, repair missing pairs, or establish a treatment effect. Its output is a transparent reconstructed input with stated assumptions.

Source checked on 31 October 2026; fictional arithmetic independently recalculated.