How do I determine a Standard deviation with the mean and range known?
2026-03-31 15:50:31.1774972231
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How do I determine a standard deviation with the mean and range known?
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You cannot; below are a few examples demonstrating the variance can be pretty much arbitrary.
Mean: $\frac{1}{2}$. Range: $[0,1]$.
Example #1: $\operatorname{Bernoulli}(\frac12)$. Variance: $\frac14$.
Example #2: $\operatorname{Uniform}([0,1])$. Variance: $\frac{1}{12}$.
Example #3: $\operatorname{Dirac}(\frac12)$. Variance: $0$.
Example #4: $\frac12(\operatorname{Dirac}(\epsilon)+\operatorname{Dirac}(1-\epsilon))$. Variance: $\epsilon(1-\epsilon)$.
Example #5: $\operatorname{Triangle}(0,1,\frac12)$. Variance: $\frac{1}{24}$.
Example #6: $\operatorname{Beta}(\alpha,\alpha)$. Variance: $\frac{1}{4\sqrt{2\alpha+1}}$.
If you want the effective range (ie, only the set of values with probability non zero), you still get #2, #5 and #6.
In particular, #6 by itself shows that any variance in $(0,\frac{1}{4})$ is achievable.
You can't. Without knowing what the distribution is, you can't really know what the standard deviation.