# 50% interval of observations in a Gaussian distribution

In the one sigma interval around the mean (i.e., one standard distribution) fall 68.2% of the observations.

In which interval would exactly 50% of the observations fall?

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Use a table of the normal distribution function $\Phi(x)$ to figure out the solution to $\Phi(x) = 0.75$. The (shortest of many possible) interval you seek is $(-x,x)$. –  Dilip Sarwate Dec 24 '12 at 23:36
... to which the OP will have to add the mean. –  Stephan Kolassa Dec 25 '12 at 19:55
@StephanKolassa Actually, following the sense of the OP's statement: taking "one sigma interval around the mean" to mean that $68.2$% of the probability mass lies within one standard deviation from the mean, the value of $x$ that is the solution to $\Phi(x)=0.75$ should be taken to mean that $50$% of the probability mass lies within $x$ standard deviation from the mean. –  Dilip Sarwate Dec 26 '12 at 1:12