How to find Pearsonian's Coefficient of Skewness (γ₁) for a distribution?

Question:

Find Pearsonian's Coefficient of Skewness (γ₁) for a distribution when the the first three moments about 2 are 1, 16 and -40 respectively.

Step 1: Note the given data.

Step 2: Convert the moments (u) into central moments (μ) i.e. find μ₂ and μ₃.

μ₂ = u₂ - u₁²

∴ μ₂ = 16 - 1²

∴ μ₂ = 16 - 1

∴ μ₂ = 15

μ₃ = u₃ - 3u₂u₁ + 2u₁²

∴ μ₃ = -40 - 3(16)(1) + 2(1)²

∴ μ₃ = -40 - 48 + 2

∴ μ₃ = -86

Step 3: Find β₁

β₁ = μ₃² / μ₂³

∴ β₁ = (-86)² / (15)³

∴ β₁ = 7396 / 3375

∴ β₁ = 2.19

Step 4: Find γ₁

γ₁ = ±√β₁

∴ γ₁ = ±√2.19

∴ γ₁ = ± 1.48

∵ μ₃ is negative; γ₁ = - 1.48

Pearsonian's Coefficient of Skewness (γ₁) for a distribution is -1.48 when the the first three moments about 2 are 1, 16 and -40 respectively.