A country records: life expectancy at birth = 68.7 years, mean years of schooling = 6.5 years, expected years of schooling = 12.2 years, and GNI per capita (PPP) = 6,829. Using the UNDP method (goalposts: life expectancy 20-85 years; mean years of schooling 0-15 years; expected y

Question

A country records: life expectancy at birth = 68.7 years, mean years of schooling = 6.5 years, expected years of schooling = 12.2 years, and GNI per capita (PPP) = 6,829. Using the UNDP method (goalposts: life expectancy 20-85 years; mean years of schooling 0-15 years; expected years of schooling 0-18 years; GNI per capita 100 to 75,000, PPP-adjusted), compute the country's HDI and state its category.

Accepted Answer

**Step 1: Health Index**
$$I_{	ext{Health}} = \dfrac{68.7 - 20}{85 - 20} = \dfrac{48.7}{65} = 0.749$$

**Step 2: Education Index**

Mean-years-of-schooling index:
$$\dfrac{6.5 - 0}{15 - 0} = \dfrac{6.5}{15} = 0.433$$

Expected-years-of-schooling index:
$$\dfrac{12.2 - 0}{18 - 0} = \dfrac{12.2}{18} = 0.678$$

Education Index (average of the two):
$$I_{	ext{Education}} = \dfrac{0.433 + 0.678}{2} = \dfrac{1.111}{2} = 0.556$$

**Step 3: Income Index** (using natural log of GNI per capita)
$$I_{	ext{Income}} = \dfrac{\ln(6829) - \ln(100)}{\ln(75000) - \ln(100)} = \dfrac{8.829 - 4.605}{11.225 - 4.605} = \dfrac{4.224}{6.620} = 0.638$$

**Step 4: HDI** (geometric mean of the three dimension indices)
$$	ext{HDI} = \sqrt[3]{0.749 	imes 0.556 	imes 0.638} = \sqrt[3]{0.2656} \approx 0.643$$

**Verification (independent re-check):** recomputing 0.749 × 0.556 = 0.4164; 0.4164 × 0.638 = 0.2657; and 0.643³ = 0.6430 × 0.6430 × 0.6430 = 0.2657 — the cube confirms the cube root, so the figures are internally consistent.

**Step 5: Category**

An HDI of 0.643 falls between 0.550 and 0.699, which is the **Medium Human Development** band.

> [!ANSWER]
> HDI ≈ 0.643 — Medium Human Development (0.550-0.699).

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