In this post we explore the formula of the Human Development Index (HDI).
Context
The Human Development Index (HDI) is a composite statistic used to rank and gauge the social and economic development of nations, created by the United Nations Development Programme (UNDP).
HDI
The HDI consists of the combination from three dimensions:
Health
Education
Economy
In this article we will cover the post 2010 version of the metric, given by:
\[ \begin{align} \mathrm{HDI} &= \sqrt[3]{\mathrm{LEI}\times \mathrm{EI} \times \mathrm{II}},\\ \end{align} \]
where:
\(\mathrm{LEI}\) = Life expectancy index;
\(\mathrm{EI}\) = Economy index;
\(\mathrm{II}\) = Income index.
Next, we will show how each index it is calculated.
Health
To measure Health the metric choosen was the life expectancy index (LEI), given by:
\[ \begin{align} \mathrm{LEI} &= \frac{\mathrm{LE}-20}{65},\\ \end{align} \tag{1}\]
where:
\(\mathrm{LE}\) = Life expectancy at birth, in years;
20 = Minimum life expectancy threshold, historically observed;
85 = Maximum life expectancy threshold, historically observed.
Given the Equation 1, a LE of 85 years would mean a LEI of 1, and if the LE is 20 the LEI is zero.
Education
To measure Education the metric choosen was the education index (EI), to understand it, first we need to explore two other metrics.
First, the Mean Years of Schooling Index (MYSI) given by:
\[ \begin{align} \mathrm{MYSI} &= \frac{\mathrm{MYS}}{15},\\ \end{align} \tag{2}\]
where
\(\mathrm{MYS}\) = Mean Years of Schooling, i.e., refers to the average number of completed years of formal education by people aged 25 and older;
15 is the projected maximum of this indicator for 2025.
Next, the Expected Years of Schooling Index (EYSI), given by:
\[ \begin{align} \mathrm{EYSI} &= \frac{\mathrm{EYS}}{18},\\ \end{align} \tag{3}\]
where
\(\mathrm{EYS}\) = Expected Years of Schooling, i.e., refers to the number of years a child of school entrance age is expected to spend at school, or university, including years spent on repetition.
18 is equivalent to achieving a master’s degree in most countries.
Finally, the EI is given by the mean of MYSI and EYSI:
\[ \begin{align} \mathrm{EI} &= \frac{\mathrm{MYSI}+\mathrm{EYSI}}{2},\\ \end{align} \tag{4}\]
where:
\(\mathrm{MYSI}\) = Mean Years of Schooling Index;
\(\mathrm{EYSI}\) = Expected Years of Schooling Index.
Economy
To measure Economy the metric choosen was the income index (II), given by:
\[ \begin{align} \mathrm{II} &= \frac{\ln(\mathrm{GNI_{pc}})-\ln(100)}{\ln(75000)-\ln(100)},\\ \end{align} \]
where:
\(\ln\) = natural logarithm function;
\(\mathrm{GNI_{pc}}\) = Gross national income at purchasing power parity per capita, in dollars.
So, the Income Index is 1 when \(\mathrm{GNI_{pc}}\) is US$75,000 and 0 when is US$100.
Final considerations
The HDI provides a broad measure of development by incorporating health, education, and living standards, moving beyond purely economic indicators like GDP. Its simplicity and comparability make it a valuable tool for identifying disparities and and raising awareness about global development challenges.
However, the HDI has limitations, such as ignoring inequality within countries, oversimplifying complex issues, and excluding critical factors like environmental sustainability, political freedoms, and cultural diversity. Its reliance on national averages can obscure significant regional or demographic disparities, and the quality of its insights depends on reliable data, which may be lacking in some contexts. While the HDI is a useful starting point, it should be complemented with other metrics to fully capture the multifaceted nature of human development.