  In some clinical studies, researchers may report the five number summary
(including the sample median, the first and third quartiles, and the minimum
and maximum values) rather than the sample mean and standard deviation. To
conduct meta-analysis for pooling studies, one needs to first estimate the
sample mean and standard deviation from the five number summary. A number of
studies have been proposed in the recent literature to solve this problem.
However, none of the existing estimators for the standard deviation is
satisfactory for practical use. After a brief review of the existing
literature, we point out that Wan et al.'s method (BMC Med Res Methodol 14:135,
2014) has a serious limitation in estimating the standard deviation from the
five number summary. To improve it, we propose a smoothly weighted estimator by
incorporating the sample size information and derive the optimal weight for the
new estimator. For ease of implementation, we also provide an approximation
formula of the optimal weight and a shortcut formula for estimating the
standard deviation from the five number summary. The performance of the
proposed estimator is evaluated through two simulation studies. In comparison
with Wan et al.'s estimator, our new estimator provides a more accurate
estimate for normal data and performs favorably for non-normal data. In real
data analysis, our new method is also able to provide a more accurate estimate
of the true sample standard deviation than the existing method. In this paper,
we propose an optimal estimator of the standard deviation from the five number
summary. Together with the optimal mean estimator in Luo et al. (Stat Methods
Med Res, in press, 2017), our new methods have improved the existing literature
and will make a solid contribution to meta-analysis and evidence-based
medicine.
