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

## Authors

• Martin Bland Prof. of Health Statistics, University of York, York YO10 5DD, England

## Keywords:

Quartile, minimum, maximum, mean, standard deviation, systematic review.

## 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.

## Author Biography

### Martin Bland, Prof. of Health Statistics, University of York, York YO10 5DD, England

Prof. of Health Statistics

## References

Hozo SP, Djulbegovic B, Hozo I. Estimating the mean and variance from the median, range, and the size of a sample. BMC Med Res Methodol 2005; 5: 13. http://dx.doi.org/10.1186/1471-2288-5-13

Bland M. An Introduction to Medical Statistics, 3rd ed. Oxford, University Press 2000.

2014-01-27

## How to Cite

Bland, M. (2014). Estimating Mean and Standard Deviation from the Sample Size, Three Quartiles, Minimum, and Maximum. International Journal of Statistics in Medical Research, 4(1), 57–64. https://doi.org/10.6000/1929-6029.2015.04.01.6

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