Gini Index Calculation for Salary Distribution

Problem Context and Distribution Data

• The provided text presents a statistical problem regarding the distribution of salaries in a company and asks for the calculation of the Gini Index ($I_G$).
• The company's salary structure is divided into four intervals with associated frequencies representing the number of employees in each bracket.
• Salary Distribution Table:
  • Interval 1: $[1000, 2000)$, Frequency ($n_1$): $30$
  • Interval 2: $[2000, 3000)$, Frequency ($n_2$): $45$
  • Interval 3: $[3000, 4000)$, Frequency ($n_3$): $20$
  • Interval 4: $[4000, 5000)$, Frequency ($n_4$): $5$

Theoretical Foundations of the Gini Index

• Definition: The Gini Index is a measure of statistical dispersion intended to represent the income or wealth inequality within a nation or any group of people. In this context, it measures the inequality of salary distribution among the company's employees.
• Range: The index ranges from $0$ to $1$.
  • A value of $0$ represents "Perfect Equality" (everyone earns the same salary).
  • A value of $1$ (or $100\%$) represents "Perfect Inequality" (one person earns everything, and others earn nothing).
• Geometry: It is based on the Lorenz Curve, which plots the cumulative proportion of total income on the y-axis against the cumulative proportion of the population on the x-axis. The Gini Index is the ratio of the area between the line of perfect equality and the Lorenz curve to the total area under the line of perfect equality.
• Calculation Formula: For discrete data organized into intervals, the Gini Index can be calculated using the formula:   $I_G = \frac{\sum_{i=1}^{k-1} (p_i - q_i)}{\sum_{i=1}^{k-1} p_i}$   Where:
  • $p_i$: Cumulative relative frequency of the population (percentage of employees).
  • $q_i$: Cumulative relative frequency of the total salary (percentage of total wealth).
  • $k$: Number of intervals.

Step-by-Step Calculation Procedure

Step 1: Determination of Midpoints ($x_i$)

Since the data is presented in intervals, we must use the midpoint of each class as the representative salary value for the calculation.

• $x_1 = \frac{1000 + 2000}{2} = 1500$
• $x_2 = \frac{2000 + 3000}{2} = 2500$
• $x_3 = \frac{3000 + 4000}{2} = 3500$
• $x_4 = \frac{4000 + 5000}{2} = 4500$

Step 2: Population Analysis ($n_i$, $N_i$, and $p_i$)

• Total Population ($N$):   $N = 30 + 45 + 20 + 5 = 100$
• Cumulative Frequencies ($N_i$):
  • $N_1 = 30$
  • $N_2 = 30 + 45 = 75$
  • $N_3 = 75 + 20 = 95$
  • $N_4 = 95 + 5 = 100$
• Cumulative Relative Frequencies ($p_i = \frac{N_i}{N}$):
  • $p_1 = \frac{30}{100} = 0.30$
  • $p_2 = \frac{75}{100} = 0.75$
  • $p_3 = \frac{95}{100} = 0.95$
  • $p_4 = \frac{100}{100} = 1.00$

Step 3: Salary/Wealth Analysis ($u_i$, $U_i$, and $q_i$)

• Salary per Category ($u_i = x_i \times n_i$):
  • $u_1 = 1500 \times 30 = 45000$
  • $u_2 = 2500 \times 45 = 112500$
  • $u_3 = 3500 \times 20 = 70000$
  • $u_4 = 4500 \times 5 = 22500$
• Total Payroll ($U$):   $U = 45000 + 112500 + 70000 + 22500 = 250000$
• Cumulative Salaries ($U_i$):
  • $U_1 = 45000$
  • $U_2 = 45000 + 112500 = 157500$
  • $U_3 = 157500 + 70000 = 227500$
  • $U_4 = 227500 + 22500 = 250000$
• Cumulative Relative Salary ($q_i = \frac{U_i}{U}$):
  • $q_1 = \frac{45000}{250000} = 0.18$
  • $q_2 = \frac{157500}{250000} = 0.63$
  • $q_3 = \frac{227500}{250000} = 0.91$
  • $q_4 = \frac{250000}{250000} = 1.00$

Summarized Calculation Table

Final Evaluation and Solution

• To find the value of the Gini Index, we sum the differences between $p_i$ and $q_i$ for all intervals excerpt the last one, and divide by the sum of the $p_i$ values for those same intervals.

• Numerator Calculation:   $\sum_{i=1}^{3} (p_i - q_i) = (0.30 - 0.18) + (0.75 - 0.63) + (0.95 - 0.91)$$\sum_{i=1}^{3} (p_i - q_i) = 0.12 + 0.12 + 0.04 = 0.28$

• Denominator Calculation:   $\sum_{i=1}^{3} p_i = 0.30 + 0.75 + 0.95 = 2.00$

• Final Result:   $I_G = \frac{0.28}{2.00}$$I_G = 0.140$

• Conclusion: The computed value of the Gini Index is $0.140$. Comparing this result with the provided multiple-choice options:

  • a. 0,280
  • b. 0,140
  • c. 0,093
  • d. Ninguna de las respuestas anteriores es correcta

• The correct option is b. 0,140.