To determine the maximum theoretical standard variance from an average value specified for a value-bounded set, the squares must be maximized. If the range of values is from p_a to p_b, and the average value is n, the maximum variance can be calculated fairly simply.

The largest variances possible will be found where the largest possible gaps are. These are from p_a to m = m – p_a, and from m to p_b = p_b – m. The values attained from each of these gaps must contribute to an average value of m. To do this, we require…

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