The unit information standard deviation (UISD) reflects the
“within-study” variability, which, depending on the effect
measure considered, sometimes is a somewhat
heuristic notion (Roever et al., 2020).
For a single study, presuming that standard errors result as
$$\sigma_i=\frac{\sigma_\mathrm{u}}{\sqrt{n_i}},$$
where \(\sigma_\mathrm{u}\) is the within-study (population) standard
deviation, the UISD simply results as
$$\sigma_\mathrm{u} = \sqrt{n_i \, \sigma_i^2}.$$
This is often appropriate when assuming an (approximately) normal likelihood.
Assuming a constant \(\sigma_\mathrm{u}\) value across studies, this
figure then may be estimated by
$$s_\mathrm{u} \;=\; \sqrt{\bar{n} \, \bar{s}^2_\mathrm{h}} \;=\; \sqrt{\frac{\sum_{i=1}^k n_i}{\sum_{i=1}^k \sigma_i^{-2}}},$$
where \(\bar{n}\) is the average (arithmetic mean) of the
studies' sample sizes, and \(\bar{s}^2_\mathrm{h}\) is the
harmonic mean of the squared standard errors (variances).
The estimator \(s_\mathrm{u}\) is motivated via meta-analysis
using the normal-normal hierarchical model (NNHM). In the special case
of homogeneity (zero heterogeneity, \(\tau=0\)), the overall
mean estimate has standard error
$$\left(\sum_{i=1}^k\sigma_i^{-2}\right)^{-1/2}.$$
Since this estimate corresponds to complete pooling, the
standard error may also be expressed via the UISD as
$$\frac{\sigma_\mathrm{u}}{\sqrt{\sum_{i=1}^k n_i}}.$$
Equating both above standard error expressions yields
\(s_\mathrm{u}\) as an estimator
of the UISD \(\sigma_\mathrm{u}\) (Roever et al, 2020).