How to calculate Standard deviation with mean 0 and Min and max value on x-axis is -1 and 1 respectively?

Question

How to calculate Standard deviation with mean 0 and Min and max value on x-axis is -1 and 1 respectively? It is of-course a normalize distribution.

I apologize in advance for stupid question.

Answer 1

You're looking for the z-value $z(x)$ of $x=1$; the z-value measures precisely ( by definition) , the number of deviations of any value from the mean. In a standard normal, the mean is $0$, and the standard deviation is $1$. We then have : $$z(x)=\frac{x-\mu}{\sigma} $$ , which, in this case is equal to $$z(1)=\frac{1-0}{1}=1 $$ , so, in a standard normal, i.e., when $\mu=0$ and $\sigma=1$ , the value $1$ is exactly one deviation from the mean. By symmetry of the normal, there is also -1 deviations from the mean between $0$ and $-1$. So there is a total of 2 deviations from the mean between $-1$ and $1$ in a standard normal.

Notice that in any normal distribution; whether normal or not, the $68-95-99.7$ rule will apply, so that the intervals $(-\sigma, \sigma), (-2\sigma, 2\sigma), (-3\sigma, 3\sigma)$ will contain respectively $68%-95%-99.7%$ of all data.