So we just started stats at school, and our teacher told us that no matter the data, no matter how distorted or weird it is, there will always be data within 1 standard deviation of the mean. Is this true?? Any way to prove this?
2026-03-31 07:02:53.1774940573
Proof that there will always be data within 1 SD
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The "most extreme" distributions would be ones containing $2N$ data points , half of which have a value of $a$ and the other half a value of $b$
The mean of such a distribution is $\mu = \frac12(a+b)$
the standard deviation is $\sigma = \frac12 |b-a|$
So for this distribution every point is exactly one standard deviation from the mean