Empirical Rule: Normally Distributed Variables

For any normal distribution,

$\mu \pm 1\sigma$ encloses about 68.26% of the data.

$\mu \pm 2\sigma$ covers about 95.44% of the data.

$\mu \pm 3\sigma$ covers about 99.73% of the data.

$\mu \pm 6\sigma$ covers about 99.99% of the data.

 

Empirical Rule: Applied to the Inventory Management Example

Remember the example where daily demand $\left(X\right)$ for gasoline at a pump is distributed normally with mean $\left(\mu \right)$ 1,000 gallons and standard deviation $\left(\sigma \right)$ 100 gallons.

• 68.26% of the times demand is between (1,000 - 100) = 900 and (1,000 + 100) = 1,100 gallons
• 95.44% of the times demand is between (1,000 - 2*100) = 800 and (1,000 + 2*100) = 1,200 gallons
• 99.73% of the times demand is between (1,000 - 3*100) = 700 and (1,000 + 3*100) = 1,300 gallons