There are situations, e.g. meta analyses, were it might be necessary to use mean and SD when median and IQR are reported in some papers and mean and SD in other papers, AND if there are no hints, that relevant deviations from normal distribution occur for the variable in question.

Then simply use mean=median and SD = IQR/1.35.

However, as noted also in previous answers, one should use these calculations ONLY if there are strong hints, that the overall distribution does not relevantly deviate from normal distribution.

The question remains, why some papers report median and IQR if data are not far from normal distribution. I found two reasons when I performed meta analyses. First, other variables did not follow normal distribution, and authors then used unique estimates (median and IQR) to describe all distrubutions. Second, test for deviation from normal distribution were applied. Given sufficient large sample sizes, these tests show always deviation from normal distribution, even if there are small, irrelevant deviations from it. (Personally, I don't use these tests anymore, because they are not valid to prove hypothesis that data follow normal distribution, they are only valid to show deviation!)

Hozo SP. et al. Estimating the mean and variance from the median, range, and the

size of a sample. BMC Medical Research Methodology 2005, 5:13 described it very well..........but not with IQR.

No, unless you (i) know the distribution and (ii) the IQR is obtained from a large sample. The sample size should be really large (>>100) because the quantiles are quite inefficient estimators.

Example: You know that the distribution is normal. The IQR of a large sample is then a good approximation of the 0.25- and the 0.75-quantile of the normal distribution, and because the distribution is symmetric, the midpoint is an estimation of the 0.5-quantile = median = mean. The variance (or SD) can then be found by seaching the normal distribution for which the 0.25-quantile equals the lower limit of the IQR (or the 0.75-quantile equals the upper limit, or one can use a more sophisticated condition using both limits together).

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as stated by Jochen, you won't "get" them but you can still get lower and upper bounds (and use the mean of such bounds as an estimator) ; this is what Hozo's paper suggested above does, without using the quartiles information

this line of work has been extended by J. Martin Bland to use the quartiles information, work he shared on

https://groups.google.com/forum/#!topic/medstats/hN2NFtuDwUE

specifically, this work does use the full quartile information, not just the iqr but usually, if you know the iqr, you have first computed the quartiles ... (except in meta-analysis !)

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There is a formula to estimate mean and SD from median and range or 95% CI, but not IQR.

https://www.researchgate.net/publication/257302635_Factors_Associated_with_Dengue_Shock_Syndrome_A_Systematic_Review_and_Meta-Analysis?ev=prf_pub

Article Factors Associated with Dengue Shock Syndrome: A Systematic ...

If the work of Hozo's paper has not been published, how do we use it for our research?

There are situations, e.g. meta analyses, were it might be necessary to use mean and SD when median and IQR are reported in some papers and mean and SD in other papers, AND if there are no hints, that relevant deviations from normal distribution occur for the variable in question.

Then simply use mean=median and SD = IQR/1.35.

However, as noted also in previous answers, one should use these calculations ONLY if there are strong hints, that the overall distribution does not relevantly deviate from normal distribution.

The question remains, why some papers report median and IQR if data are not far from normal distribution. I found two reasons when I performed meta analyses. First, other variables did not follow normal distribution, and authors then used unique estimates (median and IQR) to describe all distrubutions. Second, test for deviation from normal distribution were applied. Given sufficient large sample sizes, these tests show always deviation from normal distribution, even if there are small, irrelevant deviations from it. (Personally, I don't use these tests anymore, because they are not valid to prove hypothesis that data follow normal distribution, they are only valid to show deviation!)

Thomas, I see a difference between reporting median/IQR and mean/SD. I understand quantiles as descriptive statistics, so these show properties of the *data*. In contrast, mean and SD are the results of a statistical model behind the data (I know that it is often stated to be "descriptive", but I never saw any explanation why this should be descriptive!). It should be up to the researcher if he/she likes to present the data or some model results (with some implicit assumptions, never really stated, never really thought of - no matter if they would be acceptable or not).

Jochen, thanks, I completely agree. Your comment is very helpful and put my long answer in a nutshell.

Well, there is a way to get mean and SD, but it comes with a whole train of provisos. In some cases, you can generate the pseudo-random data that will, for most practical intents and purposes, match your distribution's Q1, Q2 (median), and Q3. Then you can compute the mean and SD of this pseudo-random distribution. Of course it can't hurt if you can get other percentiles than the quartiles, especially for the tails.

No, you cannot, as it depends on the distribution.

It is another matter that for a unimodal symmetric case, the mean is numerically same as the median. However, for skewed cases, they are obviously different, and to try to infer one from the other in such a case is not quite mathematical!

Thanks Thomas Keller for the formula mean=median and SD = IQR/1.35 to calculate metaanalysis. I would like to ask you how to calculate that example:

median= 3

Iqr= 2-3

so median=mean=3

sd=iqr/1.35 2-3 means 2,5/1.35= sd = 1,85

is that correct?

That formula is only true for normal distribution...

how about this?

Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...

Hozo's formula

Find the calculator here http://vassarstats.net/median_range.html

I know it's been a while but thanks Mihiretu Kebede for the formula!!! I was doing a systematic review and had the exact same problem. In a series of 9 studies they had only medians and ranges reported (don't know why!) and I had (incorrectly) calculated a "mean median age" (that's a statistical 'blasphemy', know that I think); luckily this question/thread save my research!

Kudos to research gate and this team!!

Have a great day,

Thank you. This is a very helpful discussion.

I agree it is not sound to asume normal distribution distribution to declare that " mean=median and SD = IQR/1.35 ". However, if we use maximum and minimum extreme values taken from data, plus media from same dataset, it is possible to find a mathematic distribution that fits the 2 extreme values and media (without using Standar deviation concepts and formulae). For example: if minimum value is Kmin=0.8; maximum Kmax=4; and media U=2, then a plausible solution of cummulative distribution for data ordered from top-to-bottom is:

K>=(p)/U = (Kmax/U) - (Kmax/U- kmin/U).p^((U-Kn)/(Kx-U)) for (0= Kn)

for mentioned values Kx/U=2, Kmin/U=0.4 and exponent e=(1.2)/(2)=0.6 so

K>=(p) = (2 - 1.6* p^0.6) * 2 where U=2 and (2 - 1.6* p^0.6) is K(p) expressed in medias

and inverse function is p(K>=Kj)= ((2-Kj)/3.2)^(1/0.6) for Kj in medias

This is the project I am working these days. Thanks, Emilio Chaves

Please see

http://vassarstats.net/

@Mihiretu M. Kebede thank you so much for sharing the link to the online calculator. It's very helpful!

Please see:

Luo D, Wan X, Liu J, Tong T. Optimally estimating the sample mean from the sample size, median, mid-range, and/or mid-quartile range. Statistical methods in medical research. 2018 Jun;27(6):1785-805. Article Optimally estimating the sample mean from the sample size, m...

It updates the Hozo 2005 method and also the Wan 2014. More data means better estimate of the Mean and SD.

The calculator produced by Luo 2018 for this purpose is very useful and is at this link: http://www.comp.hkbu.edu.hk/~xwan/median2mean.html

Thank you all for useful references and online calculation .