While preparing the data for a meta-analysis, I run into the problem that a few of my sources did not report the outcome of interest as means and standard deviations, but rather as medians and range of values. After looking around, I found this interesting paper which derived (and validated through simple simulations), simple formulas that can be used to convert the median/range into a mean and a variance in a distribution free fashion. With

- a = min of the data
- b = max of the data
- m = median
- n = size of the sample

the formulas are as follows:

**Mean **

**Variance **

The following R function will carry out these calculations

f<-function(a,m,b,n)

{

mn<-(a+2*m+b)/4+(a-2*m+b)/(4*n)

s=sqrt((a*a+m*m+b*b+(n-3)*((a+m)^2+(m+b)^2)/8-n*mn*mn)/(n-1))

c(mn,s)

}

Edit

Surfing around arxiv, I found another paper that handles additional scenarios and proposes alternative formulas

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This entry was posted on December 3, 2015 at 4:24 pm and is filed under Evidence Based Medicine, R. You can follow any responses to this entry through the RSS 2.0 feed.
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December 3, 2015 at 8:15 pm |

You should mention this was tested only for “We drew samples from five different distributions, Normal, Log-normal, Beta, Exponential and Weibull. ” My impression is that these formulas are primarily intended for situations where the full dataset is not available, as the reduction in CPU load is not gigantic.

December 3, 2015 at 8:26 pm |

This is a good point. And yes such formulas are indeed intended when the dataset is not available (e.g. data taken from papers).

December 3, 2015 at 11:41 pm |

An improved version of the Hozo et al method for estimating sample MEAN and SD from median, range and/or IQR was introduced by Wan et al., 2014. This is the link for the paper (http://www.biomedcentral.com/1471-2288/14/135 )

December 3, 2015 at 11:47 pm |

This is the article cited in the post. Note also the arxiv pdf I added later on

December 4, 2015 at 5:23 pm |

[…] Estimating the mean and standard deviation from the median and the range While preparing the data for a meta-analysis, I run into the problem that a few of my sources did not report the outcome of interest as means and standard deviations, but rather as medians and range of values. After looking around, I found this interesting paper which derived (and validated through simple simulations), simple formulas that can be used to convert the median/range into a mean and a variance in a distribution free fashion. […]