427. Square-Root Law

For independent demand sources with identical mean and standard deviation , the standard deviation of total demand grows with (not ). This is the square-root law — the central result of risk pooling.

427.1. Statement

Let be independent, identically distributed random variables with . Then for :

427.2. Why variance is additive (proof sketch)

For independent random variables:

Independence forces , so:

Taking square root: . ∎

(The non-i.i.d. case with general covariances is the correlated pooling formula.)

427.3. Safety-stock implication

If safety stock is for each of separate facilities, total:

If pooled into one facility:

Reduction factor:

427.4. Numerical example

A retailer with stores. Each store: weekly demand , .

Each store needs safety stock units for 95% service.

Separate total: units.

Pooled (one warehouse covering all):

Reduction: of original safety stock. 75% savings on safety stock.

427.5. Generalization to non-identical demands

If ‘s differ:

If correlations :

See Correlated Pooling.

427.6. Why this is “square-root”

Demand is sum of random variables. By CLT-like arguments, the sum grows as (in mean) but the fluctuation about the mean grows only as . The fluctuation is what safety stock covers. Hence the savings.

This is the same phenomenon as the standard error of a sample mean shrinking as .

427.7. Limitations

• Independence assumed — positively correlated demands give less benefit
• Identical ‘s assumed — partial concentration in high-variance locations matters
• Service level unchanged — but pooling might allow higher service for same cost
• Ignores transport cost — central stock is farther from some customers

See Risk Pooling for trade-offs.

427.8. See also

• Risk Pooling — overview
• Correlated Pooling — non-independent case
• Location Pooling — application
• Safety Stock