Introduction: In this Math HL Internal Assessment (IA), I have conducted an income survey in three towns within my country. The primary goal is to construct Lorenz curves and calculate the Gini coefficients for each town. Despite a survey sample size of 200 people per town, ensuring meticulous data evaluation and reflection is crucial for a successful presentation of the results.

Background

Income inequality is a significant issue across many countries, and understanding its intricacies is crucial for policy-making and social development. The Lorenz curve and the Gini coefficient are fundamental statistical tools to analyze and quantify income distribution. The Lorenz curve graphically represents income distribution, and the Gini coefficient is a measure of statistical dispersion intended to represent income inequality within a nation or a social group.

Methodology

To execute this analysis, I conducted a survey of 200 people in each of the three towns. This sample size is substantial enough to provide a meaningful insight but not so large that it becomes overly burdensome. The following steps were taken to ensure the integrity of the data:

Survey Design

Sample Selection: Stratified random sampling was used to ensure a representative sample from each town. Stratification by demographic factors such as age, occupation, and education level was performed to control for potential biases. Data Collection: Data collection was completed using a structured questionnaire. The survey included questions regarding household income, assets, and consumption patterns.

Statistical Analysis

Constructing the Lorenz Curve: Once the data was collected, software tools like Microsoft Excel or specialized statistical software such as R or Python were used to plot the Lorenz curves for each town. The Lorenz curve illustrates the cumulative share of income earned by the corresponding cumulative share of the population. Calculating the Gini Coefficient: The Gini coefficient is calculated as 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. This measure ranges from 0 (perfect equality) to 1 (perfect inequality).

Addressing Statistical Insignificances

Rigorous Evaluation: Given the survey sample size, a rigorous evaluation of statistical insignificances became essential. To address this, I employed statistical methods such as the Central Limit Theorem (CLT), which justifies the use of sample means to estimate population parameters when sample sizes are large enough. Confidence Intervals: Confidence intervals were calculated to provide a range of values within which the true Gini coefficient is likely to lie. This approach helps in understanding the precision of the estimate. Bias Correction: Potential sources of bias were identified, such as non-response bias and measurement errors. To mitigate these biases, I employed adjustments and corrections in the survey design and data analysis process.

Reflection and Critical Thinking

The core of any successful Math HL IA lies in critical thinking and the ability to diagnose sources of inaccuracy. Here are some reflections and considerations:

Sample Size: While a sample size of 200 per town is considerable, it may not be sufficient for absolute precision. The Central Limit Theorem provides a theoretical basis, but practical considerations such as variability in income distributions necessitate a larger sample size to improve accuracy. Data Quality: Ensuring high data quality is vital. Consistency and reliability in survey questions and data entry can significantly impact the results. Adhering to strict data validation and cleaning protocols is essential. Variable Considerations: Other variables such as wealth distribution, education levels, and employment status may influence income inequality. Accounting for these variables in the analysis is crucial for a comprehensive understanding of the results.

Conclusion

In conclusion, the exploration and analysis of income inequality in three towns using Lorenz curves and Gini coefficients provide valuable insights into the distribution of income. The rigorously evaluated and reflected statistical methods employed ensure that the findings are robust and reliable. By addressing statistical insignificances and considering biases, the study contributes to a more nuanced understanding of the socio-economic landscape in these towns.

Through this project, I have not only learned about sophisticated mathematical methods but also developed critical thinking skills essential for meaningful data analysis. The integration of these methods and a structured approach to addressing potential biases has laid a solid foundation for a successful Math HL IA submission.