# Statistics and Its Interface

## Volume 13 (2020)

### Estimating the mean and variance from the five-number summary of a log-normal distribution

Pages: 519 – 531

DOI: https://dx.doi.org/10.4310/SII.2020.v13.n4.a9

#### Authors

Jiandong Shi (Department of Mathematics, Hong Kong Baptist University, Hong Kong)

Tiejun Tong (Department of Mathematics, Hong Kong Baptist University, Hong Kong)

Yuedong Wang (Department of Statistics and Applied Probability, University of California at Santa Barbara)

Marc G. Genton (Statistics Program, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia)

#### Abstract

In the past several decades, meta-analysis has been widely used to pool multiple studies for evidence-based practice. To conduct a meta-analysis, the mean and variance from each study are often required; whereas in certain studies, the five-number summary may instead be reported that consists of the median, the first and third quartiles, and/or the minimum and maximum values. To transform the fivenumber summary back to the mean and variance, several popular methods have emerged in the literature. However, we note that most existing methods are developed under the normality assumption; and when this assumption is violated, these methods may not be able to provide a reliable transformation. In this paper, we propose to estimate the mean and variance from the five-number summary of a $\log$-normal distribution. Specifically, we first make the log-transformation of the reported quantiles. With the existing mean estimators and newly proposed variance estimators under the normality assumption, we construct the estimators of the log-scale mean and variance. Finally, we transform them back to the original scale for the final estimators.We also propose a biascorrected method to further improve the estimation of the mean and variance. Simulation studies demonstrate that our new estimators have smaller biases and smaller relative risks in most settings. A real data example is used to illustrate the practical usefulness of our new estimators.

#### Keywords

bias correction, five-number summary, $\log$-normal distribution, meta-analysis, variance

Tiejun Tong’s research was supported by the Initiation Grant for Faculty Niche Research Areas (No. RC-IG-FNRA/17-18/13) and the Century Club Sponsorship Scheme of Hong Kong Baptist University, the General Research Fund (No. HKBU12303918), the Health and Medical Research Fund (No. 04150476), and the National Natural Science Foundation of China (No. 11671338).