Estimating Mean and Standard Deviation from the Sample Size, Three Quartiles, Minimum, and Maximum

Abstract

 Background: We sometimes want to include in a meta-analysis data from studies where results are presented as medians and ranges or interquartile ranges rather than as means and standard deviations. In this paper I extend a method of Hozo et al. to estimate mean and standard deviation from median, minimum, and maximum to the case where quartiles are also available.

Methods: Inequalities are developed for each observation using upper and lower limits derived from the minimum, the three quartiles, and the maximum. These are summed to give bounds for the sum and hence the mean of the observations, the average of these bounds in the estimate. A similar estimate is found for the sum of the observations squared and hence for the variance and standard deviation.

Results: For data from a Normal distribution, the extended method using quartiles gives good estimates of sample means but sample standard deviations are overestimated. For data from a Lognormal distribution, both sample mean and standard deviation are overestimated. Overestimation is worse for larger samples and for highly skewed parent distributions. The extended estimates using quartiles are always superior in both bias and precision to those without.

Conclusions: The estimates have the advantage of being extremely simple to carry out. I argue that as, in practice, such methods will be applied to small samples, the overestimation may not be a serious problem.