# Human Development Index (HDI) & Its Formula

## Human Development Index (HDI)

The Human Development Index (HDI) is a composite statistic used to rank countries based on their level of human development. It was introduced by the United Nations Development Programme (UNDP) in its first Human Development Report in 1990. The HDI aims to provide a broader picture of a country’s development level beyond just economic indicators like GDP per capita. It focuses on three basic dimensions of human development:

### 1. Life Expectancy at Birth

This component measures the average expected lifespan of a population, reflecting the country’s health status and longevity. It indicates the ability of people to live long and healthy lives.

### 2. Education

This dimension is assessed through two indicators:

**Mean Years of Schooling**for adults aged 25 years and older: This reflects the average number of years of education received by people in this age group, showing the level of education among the adult population.**Expected Years of Schooling**for children of school-entry age: This measures the total number of years of schooling a child of school-entry age can expect to receive if prevailing patterns of age-specific enrollment rates persist throughout the child’s life. It indicates the commitment to education.

### 3. Gross National Income (GNI) per Capita

Adjusted to purchasing power parity (PPP), this component reflects the average income of a country’s citizens, indicating the standard of living. It is adjusted for the cost of living and inflation rates to make fair comparisons between countries.

## Calculation of HDI

The HDI is calculated by geometrically averaging the normalized indices for each of the three dimensions. The normalization is done to ensure that each indicator falls between 0 and 1, allowing them to be averaged. The formula for calculating the HDI value is:

**HDI = ∛(I _{Health}) × (I_{Education}) × (I_{Income})**

Where:

*I*is the index for Life Expectancy,_{Health}*I*is the average of the indices for Mean Years of Schooling and Expected Years of Schooling,_{Education}*I*is the index for GNI per capita._{Income}

### 1. *I*_{Health}: Index for Life Expectancy

_{Health}

The index for Life Expectancy (*I _{Health}*) is calculated using the formula:

**I _{Health} = (LE – 20) / (85 – 20)**

Where:

**LE**is the Life Expectancy at birth.- 20 years is considered the minimum life expectancy.
- 85 years is considered the maximum life expectancy.

This formula normalizes the life expectancy at birth within a scale of 0 to 1, where 20 years is the minimum expected value (set to 0) and 85 years is the maximum (set to 1).

### 2. *I*_{Education}: Index for Education

_{Education}

The Education Index (*I _{Education}*) is the average of two indices: the Mean Years of Schooling Index and the Expected Years of Schooling Index. It is calculated as follows:

**I _{Education} = (MYSI + EYSI) / 2**

Where:

**MYSI**(Mean Years of Schooling Index) = MYS / 15**MYS**is the Mean Years of Schooling for the adult population (ages 25 and older).- 15 years is considered the maximum of mean years of schooling.

**EYSI**(Expected Years of Schooling Index) = EYS / 18**EYS**is the Expected Years of Schooling for children of school-entering age.- 18 years is considered the maximum expected years of schooling.

This formula averages the normalized values of mean years of schooling and expected years of schooling, each scaled from 0 to 1.

### 3. *I*_{Income}: Index for Gross National Income (GNI) per Capita

_{Income}

The index for GNI per capita (*I _{Income}*) is calculated using the formula:

**I _{Income} = (ln(GNIpc) – ln(100)) / (ln(75,000) – ln(100))**

Where:

**GNIpc**is the Gross National Income per capita.- The natural logarithm (
**ln**) is used to account for the diminishing importance of income with increasing GNI. - 100 is considered the minimum GNI per capita (PPP $).
- 75,000 is considered the maximum GNI per capita (PPP $).

This formula normalizes the GNI per capita on a logarithmic scale between 0 and 1, where $100 is the minimum and $75,000 is the maximum.

### Final HDI Calculation

After calculating the indices for health, education, and income, the HDI is computed by geometrically averaging these normalized indices:

**HDI = √[I _{Health} × I_{Education} × I_{Income}]**

This geometric mean ensures that a 1% improvement in any of the dimensions has the same impact on the HDI, promoting a balanced approach to development across all three dimensions.

## Example Calculation

Given:

**Life Expectancy at Birth**: 70 years**Mean Years of Schooling**: 10 years**Expected Years of Schooling**: 15 years**GNI per Capita (PPP)**: $15,000

1. **Life Expectancy Index**: Using the same formula, *(70 – 20) / (85 – 20) = 0.769*.

2. **Education Index**:

**MYSI**:*10 / 15 = 0.667***EYSI**:*15 / 18 ≈ 0.833***EI**:*(0.667 + 0.833) / 2 = 0.75*

3. **Income Index**: Assuming the same formula,

*(log(15,000) – log(100)) / (log(75,000) – log(100)) ≈ 0.6*.

## Recalculation of HDI

Now, with the corrected Education Index, let’s recalculate the HDI:

**HDI = ∛(0.769 × 0.75 × 0.6)**

**HDI ≈ ∛(0.34605)**

**HDI ≈ 0.703**

So, with the corrected calculation for the Education Index, the HDI in this example would be approximately 0.703.