Median may be [more robust][1] than mean, but this does not make it robust for such extreme cases like sample consisting of single datapoint. Let's get more general for a moment, $p$ [quantile][2] (median is $p=0.5$ quantile) of distribution is the value $x$ such that $$ \Pr(X \leq x) = p $$ Sample quantile for ordered sample $$ x_1 \le x_2 \le \dots \le x_n $$ then median is middle value of such sample, where we have multiple definitions on how to define it precisely (Hyndman and Fan, 1996). If you think of it other way around, we want $np$ out of $n$ values of $X$ be smaller or equal than $x$ where $p=0.5$, so $X$ follows binomial distribution with parameters $n$ and $p$. [Wald confidence interval][3] for $p$ can be easily calculated as $p \pm z_{\alpha/2} \sqrt{p(1-p)/n}$ (but see [here][4], [here][5], and Brown and DasGupta, 2001). Relating to your second question, the interval [can be used][6] to find such value of $n$ that makes the interval acceptably narrow. (Notice that for $n=1$ the confidence interval yields improper values that fall beyond the $[0,1]$ interval for $p$ and Wald's method should be used for samples of at least five.) It is convenient to define sample median for sample of size one as the data point that you have, but if you think of it, single data point at the same time is also the minimal $x_1$, or maximal value $x_n$ of this sample. "Intuitively", it may seem that if you have single datapoint, then it is unlikely the extreme value, but this does not have to be true. For example, if your variable is uniformly distributed, than single datapoint can be *any* value with equal probability (this is consistent with very wide confidence intervals for $n=1$). Hopefully, with single datapoint you can compute [confidence intervals for mean][9], e.g. $95\%$ interval would be $$ x \pm 9.68 |x| $$ As you can see, the intervals are pretty wide, but also the situation is extreme, so our uncertainty about mean is also greater. If you can make any distributional assumptions you could get more precise. If you have *a priori* knowledge about your problem you could try Bayesian approach with informative priors. --- Hyndman, R. J., & Fan, Y. (1996). [Sample quantiles in statistical packages.][10] *The American Statistician, 50*(4), 361-365. Brown, L. D., Cai, T. T., & DasGupta, A. (2001). [Interval estimation for a binomial proportion.][11] *Statistical science*, 101-117. [1]: https://stats.stackexchange.com/questions/2547/why-is-median-age-a-better-statistic-than-mean-age [2]: https://en.wikipedia.org/wiki/Quantile [3]: https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval#Normal_approximation_interval [4]: https://stats.stackexchange.com/questions/105972/averaging-binomial-confidence-intervals [5]: https://stats.stackexchange.com/questions/82720/confidence-interval-around-binomial-estimate-of-0-or-1/82724#82724 [6]: https://stats.stackexchange.com/questions/165035/number-of-samples-needed-in-monte-carlo-simulation-how-good-is-this-approximati [7]: https://stats.stackexchange.com/questions/21103/confidence-interval-for-median [8]: https://stats.stackexchange.com/questions/122001/confidence-intervals-for-median [9]: https://stats.stackexchange.com/questions/157582/what-can-we-say-about-population-mean-from-a-sample-size-of-1 [10]: https://www.amherst.edu/media/view/129116/original/Sample+Quantiles.pdf [11]: http://projecteuclid.org/download/pdf_1/euclid.ss/1009213286