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

By M. Bland. 2015

132 citations recorded by the source corpus.

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.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/59dd526a4335850fcb364c92bb6f4eb879fb6e59

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Citations in the corpus

• Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range (2014, methodology): “In this setting, Bland’s method [10] extended Hozo et al.”
• Dealing with missing standard deviation and mean values in meta-analysis of continuous outcomes: a systematic review (2018, methodology): “Bland [21] presents a formula for the variance (Table 1) which makes use of the lower quartile and upper quartile in addition to the minimum, median Fig.”
• Optimally estimating the sample standard deviation from the five‐number summary (2020, methodology): “We have the shortcut formula of estimator (10) as”
• Optimally estimating the sample mean from the sample size, median, mid-range, and/or mid-quartile range (2015, background): “Note that the optimal weight woptðnÞ in (5) may not be readily accessible for practitioners as it involves some complicated statistical computation.”