if I have data that satisfy 68–95–99.7 rule, does it mean the data is normally distributed? Thanks
2026-04-04 10:36:56.1775299016
68–95–99.7 rule mean normal distribution
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No, the distribution is not necessarily normal if it satisfies the $68-95-99.7$ rule. Take a normal distribution, then move all the events that are within $(\mu+0.25 \sigma, \mu + 0.75 \sigma)$, some of them to the mean and some of them to $\mu + 0.99 \sigma$ Do the same for the ones within the range $(\mu - 0.75 \sigma, \mu - 0.25 \sigma)$ in the other direction. It should be clear that the mean has not changed, and you can choose the fraction to move farther to keep $\sigma$ the same. You now have a distribution that is not normal but satisfies $68-95-99.7$ just fine. In the other direction, yes if the distribution is normal it will satisfy $68-95-99.7$ as those values are calculated from the normal distribution.