= What is the relationship between the median and the mean? =

Hozo et al. (2005) show that for a variable with minimum, min and maximum, max its mean is approximately equal to [min+median+max]/4. They suggest using this for small samples < 25. The standard deviation can be estimated as [Max-Min]/4 (see the pdf [[attachment:Thabane.pdf | here.]]

__A confidence interval for the median__

95% Confidence intervals for the median may also be obtained, for example for assessing the influence of outliers since medians are more robust to outliers. In particular the 95% confidence interval for a median based upon a sample size, are the numbers with ranks
[n/2 - 1.96 sqrt(n/4), n/2 + 1.96 sqrt(n/4) + 1]. So if we have 64 observations the 95% confidence interval for the median has lower bound equal to approximately the 24th (31.5 - 1.96 sqrt(63/4)) highest value and upper bound equal to approximately the 40th (= 63/2 + 1.96 sqrt(63/3) + 1) highest value. Some R code for working out 95% confidence intervals for medians is [[:FAQ/medianCI | is given here.]]

The general formula treats the percentile as binomial proportion, p, with standard error sqrt[np(1-p)]. 


__Reference__

Hozo, S-P, Djulbegovic, B and Hozo, I (2005). [[attachment:meanmedian.pdf|Estimating the mean and variance from the median, range, and the size of a sample.]] ''BMC Medical Research Methodology'' '''5''' 13. doi:10.1186/1471-2288-5-13. See also the link [[http://www.biomedcentral.com/1471-2288/5/13|here.]]