How to estimate the sample mean and standard deviation from the five number summary?

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.