The Square-Root Law says that the standard error (SE) of the sample sum of n random draws with replacement from a box of tickets with numbers on them is

SE(sample sum) = n^½×SD(box),

and the standard error of the sample mean of n random draws with replacement from a box of tickets is

SE(sample mean) = n^−½×SD(box),

where SD(box) is the standard deviation of the list of the numbers on all the tickets in the box (including repeated values).

---

The Square Root Law of Inventory Management

he effect on maintaining the same item in more than one location on the overall levels of inventory escapes many practitioners. The principal problem is that an item stocked in multiple location increases the overall inventory levels. The key to the problem is the term location. Whilst we all understand physical warehouses being a location where inventory is stored, location can also be virtual locations or contracts where inventory is ring-fenced. Let us explore what this all means; we start with the relationship of inventory levels and the number of locations described in the square root law.

The square root law was first published by D.H. Maister in 1976 and describes the relationship levels of inventory levels and the number of locations an item in stored in.

The square root law of inventory management gives you an estimate of how the number of warehouse locations affect the size of your inventory. Its formula is: X2 = (X1) * v(n2/n1) where v is the square root. Therefore, a single warehouse reduces the inventory by one half.

---

Penrose square root law

In the mathematical theory of games, the Penrose square root law, originally formulated by Lionel Penrose, concerns the distribution of the voting power in a voting body consisting of N members. It states that the a priori voting power of any voter, measured by the Penrose–Banzhaf index ψ scales like 1/sqrt(N).

This result was used to design the Penrose method for allocating the voting weights of representatives in a decision-making bodies proportional to the square root of the population represented.

To estimate the voting index of any player one needs to estimate the number of the possible winning coalitions in which his vote is decisive. Assume for simplicity that the number of voters is odd, N = 2j + 1, and the body votes according to the standard majority rule. Following Penrose one concludes that a given voter will be able to effectively influence the outcome of the voting only if the votes split half and half: if j players say 'Yes' and the remaining j players vote 'No', the last vote is decisive.

Assuming that all members of the body vote independently (the votes are uncorrelated) and that the probability of each vote 'Yes' is equal to p = 1/2 one can estimate likelihood of such an event using the Bernoulli trial. The probability to obtain j votes 'Yes' out of 2j votes reads:

Pj = (1/2)^2j(2j)!/(j!)².