# If 100% of data fall within first two standard deviations of the mean, is the distribution Normal?

The empirical rule for a normal distribution suggests that 68% of data will fall within the first standard deviation, 95% within the first two standard deviations, and 99.7% will fall within the first three standard deviations of the mean. If I have a data set where all the elements fall within first two standard deviations of the mean, can I consider this data set as a normally distributed one?

• What is the size of the sample? – Sasha Sep 12 '12 at 4:13
• Isn't there a general rule for this scenario? I have a sample of 100 instances in the current problem that I am dealing with. – LVJung Sep 12 '12 at 4:22

The naive approach would be to expect 5 samples of 100 outside 2 SD, with a standard deviation on that of $\sqrt 5 \approx 2.2$, so getting 0 is only a 2 SD event (and maybe not that bad as the measured SD is probably a bit low).
• Are you calculating with $\sigma$ or with $s_{n-1}$? I guess an unusual sample with all data within $\pm2s_{n-1}$ rather indicates that $s_{n-1}$ is unusually high ... ? – Hagen von Eitzen Sep 12 '12 at 4:56
A normally distributed random variable has a pdf with support over all of $\mathbb R$. If it does not it is some sort of truncated distribution. Check the properties of a truncated normal distribution on Wikipedia to verify if your data satisfies those.