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According to the Central Limit Theorem (CLT) in statistics, the distribution of the average of randomly sampled n observations tends to follow normal distribution as the sampling size n becomes larger, even when the parent distribution from which the average is calculated is not normal. The distribution of the average of randomly sampled observations has the same mean as the parent distribution and its variance is equal to the variance of the parent divided by the sampling size.

So, my question is, is there a similar CLT-like statement for the distribution of the median of randomly sampled n observations.

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