Explanation

To compute the standard deviation of compensation, we apply linear transformation principles to sales data:

Step 1: Define the Compensation Formula

Compensation (Y) is given by:

Y = α + βX
Y = 30,000 + 0.05X

Y = α + βX
Y = 30,000 + 0.05X

where:

• Y = compensation (transformed variable)
• X = sales variable
• β = scale constant ($0.05/$1)
• α = shift constant ($30,000)

Step 2: Calculate Variance of Compensation

For linear transformations, the variance of Y is:

Var(Y) = Var(α + βX) = β²Var(X) = β²σ²

Var(Y) = Var(α + βX) = β²Var(X) = β²σ²

Given:

• Var(X) = σ² = 5,000,000
• β = 0.05

Var(Y) = (0.05)² × 5,000,000 = 0.0025 × 5,000,000 = 12,500

Var(Y) = (0.05)² × 5,000,000 = 0.0025 × 5,000,000 = 12,500

Step 3: Calculate Standard Deviation

Standard deviation is the square root of variance:

SD(Y) = √Var(Y) = √12,500 = `$111.8` ≈ `$112`

SD(Y) = √Var(Y) = √12,500 = `$111.8` ≈ `$112`

Key Insight: The basic salary ($30,000) doesn't affect the variance or standard deviation since it's a constant shift that gets canceled out in variance calculations. Only the variable component (bonus) contributes to the dispersion of compensation.