Square-Root Law

431. Square-Root Law

For $𝑁$ independent demand sources with identical mean $𝜇$ and standard deviation $𝜎$ , the standard deviation of total demand grows with $\sqrt{𝑁}$ (not $𝑁$ ). This is the square-root law — the central result of risk pooling.

431.1. Statement

Let $𝐷_{1}, \dots, 𝐷_{𝑁}$ be independent, identically distributed random variables with $𝐸 [𝐷_{𝑖}] = 𝜇, Var (𝐷_{𝑖}) = 𝜎^{2}$ . Then for $𝑆 = \sum_{𝑖 = 1}^{𝑁} 𝐷_{𝑖}$ :

𝐸 [𝑆] = 𝑁 𝜇

Var (𝑆) = 𝑁 𝜎^{2} (variance is additive)

𝜎_{𝑆} = 𝜎 \sqrt{𝑁} (standard deviation grows as \sqrt{𝑁})

431.2. Why variance is additive (proof sketch)

For independent random variables:

Var (𝐷_{1} + 𝐷_{2} + \dots + 𝐷_{𝑁}) = \sum_{𝑖 = 1}^{𝑁} Var (𝐷_{𝑖}) + 2 \sum_{𝑖 < 𝑗} Cov (𝐷_{𝑖}, 𝐷_{𝑗})

Independence forces $Cov (𝐷_{𝑖}, 𝐷_{𝑗}) = 0$ , so:

Var (𝑆) = \sum_{𝑖 = 1}^{𝑁} 𝜎^{2} = 𝑁 𝜎^{2}

Taking square root: $𝜎_{𝑆} = 𝜎 \sqrt{𝑁}$ . ∎

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

431.3. Safety-stock implication

If safety stock is $𝑧 𝜎$ for each of $𝑁$ separate facilities, total:

{SS}_{separate} = 𝑁 𝑧 𝜎

If pooled into one facility:

{SS}_{pool} = 𝑧 𝜎_{𝑆} = 𝑧 𝜎 \sqrt{𝑁}

Reduction factor:

\frac{{SS}_{pool}}{{SS}_{separate}} = \frac{\sqrt{𝑁}}{𝑁} = \frac{1}{\sqrt{𝑁}}

$𝑁$	SS separate	SS pooled	reduction
4	$4 𝑧 𝜎$	$2 𝑧 𝜎$	50%
9	$9 𝑧 𝜎$	$3 𝑧 𝜎$	67%
25	$25 𝑧 𝜎$	$5 𝑧 𝜎$	80%
100	$100 𝑧 𝜎$	$10 𝑧 𝜎$	90%

431.4. Numerical example

A retailer with $𝑁 = 16$ stores. Each store: weekly demand $𝜇 = 100$ , $𝜎 = 30$ .

Each store needs safety stock $𝑧 𝜎 = 1.65 \cdot 30 = 49.5$ units for 95% service.

Separate total: $16 \cdot 49.5 = 792$ units.

Pooled (one warehouse covering all):

\begin{matrix} 𝜎_{pool} = 30 \sqrt{16} = 120 \\ {SS}_{pool} = 1.65 \cdot 120 = 198 units \end{matrix}

Reduction: $\frac{198}{792} = 25 %$ of original safety stock. 75% savings on safety stock.

431.5. Generalization to non-identical demands

If $𝜎_{𝑖}$ ’s differ:

𝜎_{pool} = \sqrt{\sum_{𝑖} 𝜎_{𝑖}^{2}}

If correlations $𝜌_{𝑖 𝑗} \neq 0$ :

𝜎_{pool}^{2} = \sum_{𝑖} 𝜎_{𝑖}^{2} + 2 \sum_{𝑖 < 𝑗} 𝜌_{𝑖 𝑗} 𝜎_{𝑖} 𝜎_{𝑗}

See Correlated Pooling.

431.6. Why this is “square-root”

Demand is sum of $𝑁$ random variables. By CLT-like arguments, the sum grows as $𝑁$ (in mean) but the fluctuation about the mean grows only as $\sqrt{𝑁}$ . 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 $\frac{1}{\sqrt{𝑛}}$ .

431.7. Limitations

Independence assumed — positively correlated demands give less benefit
Identical $𝜎$ ’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.

431.8. See also

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