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**HallsofIvy** The average of 35 samples, from a normal distribution of mean $\displaystyle /mu$ and standard deviation $\displaystyle \sigma$ is normally distributed with mean $\displaystyle \mu$ and standard deviation $\displaystyle \sigma \sqrt{35}$. The average of 50 such samples is normally distributed with mean $\displaystyle \mu$ and standard deviation $\displaystyle \sqrt{50}(\sqrt{35}\sigma)= \sqrt{(50)(35)}\sigma$. That is the same as the distribution of (35)(50) separate samples.