# Limits on median given mean and sd

Given N, a mean, a standard deviation, and the range, what does that tell us about the median? Obviously and trivially it must be within the range, but can we get tighter limits? To give context, I'm trying to find a median by repeated search. I may not rearrange the data and I get charged for each pass through it. So the tighter the limits on the initial estimate the better, and I can obtain range, mean and standard deviation from one pass.

• For continuous distributions, the mean and the median cannot differ by more than one standard deviation. One way to prove that involves Jensen's inequality. I'm not sure what might need to be changed for discrete distributions. $\qquad$ – Michael Hardy Aug 21 '16 at 17:45
• Stat 100 answ: Depends on whether you want an absolute rule, or something that works most of the time. The Empirical Rule, applicable to many kinds of datasets (not just ones very close to 'normal') says about 68% of data fall within $\bar X \pm S$, and that clearly also contains the median. // Interesting question. But with modern computers and efficient search/sort algorithms, it is not 'expensive' to find the exact median unless $N$ is huuuge, so I have trouble viewing this as a practical problem. – BruceET Aug 21 '16 at 18:01
• Bruce - you don't just need medians of human-gathered datasets. You might need to take several hundred every video frame refresh with the data coming at you from sensors, other other algorithms, e.g shape recognition problems. – Malcolm McLean Aug 21 '16 at 18:07