Quantitative Insights in Economic and Statistical Sciences

A k-means cluster analysis was conducted to identify latent groupings among countries based on multiple quantitative socioeconomic indicators drawn from the Gapminder dataset. Variables were numerically encoded, and missing data was imputed using the median for each feature. Given the dataset's moderate size, all observations were included to maximize subgroup detection.

K-means partitioned countries into three clusters according to shared patterns across indicators. This approach minimizes within-cluster variation and highlights differences between subpopulations.

Results

The clustering algorithm assigned:

112 countries to Cluster 1

72 countries to Cluster 2

29 countries to Cluster 0

Summary statistics for each cluster show distinct profiles:

Clusters are differentiated by combinations of high income, technology adoption, urbanization, and health outcomes. Higher income and longer life expectancy are concentrated in Cluster 0, while lower values are observed in Cluster 2.

Interpretation

Three subgroups emerged, each characterized by similar socioeconomic profiles. Cluster analysis identified countries with higher income, better health, and more urbanized populations in one group, compared to countries with distinctly lower scores in another. The approach reveals meaningful geographic and developmental differences without relying on predefined labels.

Appendix: Python Syntax

This analysis evaluates how national indicators contribute to the classification of countries by income level. The dataset was numerically encoded, and missing values were filled by the median of each feature. The binary response variable reflects whether a country’s income per person is above the sample median.

A Random Forest classifier was fitted to the data, utilizing an ensemble of decision trees to provide robust classification and importance measures for each predictor variable.

Results

The model achieved perfect classification performance on the available data, with an accuracy score of 1.0. The confusion matrix confirms that all instances were classified correctly:

importance values, showing which variables most influenced the classification, are visualized below. Internet user rate, life expectancy, and urbanization rate had the highest impact in distinguishing between high- and low-income countries.

Interpretation

The analysis demonstrates that only a few socioeconomic features are needed to classify countries by income group with high certainty. Variables such as internet user rate and urbanization provide the most predictive power, as seen in their relative importance scores and the perfect model accuracy. Ensemble methods like Random Forests are effective at capturing feature interactions without overfitting when variables are properly preprocessed.

Appendix: Python Syntax

A quantitative analysis was conducted to investigate the association between various demographic and economic indicators and life expectancy across countries. The dataset comprises multiple predictors, including numeric variables from health, consumption, and social domains. Prior to modeling, missing values were addressed using median imputation to maintain consistency and completeness in the sample.

The approach used Lasso regression, which introduces a regularization parameter to enforce sparsity in the model, automatically reducing certain coefficients to zero and thus simplifying the set of explanatory variables. This penalty parameter was optimized via five-fold cross-validation. All available observations were included in the modeling phase without a train-test split due to the moderate data size.

Results

After regularization, only a subset of predictors were retained by the model. The cross-validated alpha parameter converged to approximately 0.26. Predictors such as armedforcesrate, breastcancerper100th, hivrate, internetuserate, polityscore, relectricperperson, suicideper100th, employrate, and urbanrate exhibited non-zero coefficients. The intercept for the optimized model reached 70.10, with an R-squared value of 0.71, indicating that the selected variables explained a substantial proportion of the variance in life expectancy.

Interpretation of the resulting model shows that internetuserate, urbanrate, and employrate impact life expectancy positively, while variables such as hivrate and suicideper100th act in the opposite direction. The Lasso regression delivers an interpretable model by excluding predictors with negligible association, focusing analysis on a limited and influential subset of features.

Appendix: Python Syntax

The dataset, comprising 213 countries, includes a broad range of socioeconomic and demographic variables, such as alcohol consumption, internet usage, life expectancy, and urbanization rates. Each record describes a single country, with all features numerically encoded and missing values imputed by the median of each respective variable to ensure continuity and minimize information loss.

After preprocessing, a binary variable was constructed to specify income group, attributing the label of high or low income depending on whether a country's income per person exceeds or falls below the sample median. All predictors except the unique country identifier and the income variable were retained for model development.

A decision tree classifier, employing the entropy criterion, was trained to distinguish between high- and low-income countries. The dataset was partitioned into training and test sets at a 70/30 ratio, and model fitting was performed on the former.

Model Results

Classification performance was evaluated on the test set using several complementary approaches. The confusion matrix below illustrates the correspondence between actual labels and model predictions.

Further performance metrics, including precision, recall, and F1-score, are as follows:

The model attained an overall accuracy of approximately 80%. While the approach exhibited a slight bias toward high-income countries in terms of precision, the recall was higher for low-income instances. The distribution of misclassifications, predominantly involving false negatives, is visible in the confusion matrix.

A visual representation of the decision tree underscores the variables most involved in determining income group. Attributes such as internet user rate, urbanization, female employment rate, and CO₂ emissions contributed repeatedly to splits distinguishing the two categories.

Inspection of the tree visualization reveals clear threshold-based rules segmenting countries, with splits on internetuserate and urbanrate typically featuring near the root node. This suggests these features play a significant role in differentiating income groups.

Discussion

Employing a simple decision tree enables interpretability and highlights the non-linear interactions that govern the income class assignment of countries. The model captures relevant socioeconomic determinants and exposes the hierarchical manner in which features contribute to the final classifications. This approach thus facilitates both predictive accuracy and the extraction of actionable insights.

Appendix: Python Code

A cross-country dataset was analyzed to assess the association between urbanization rate and the probability of having above-median life expectancy. The binary outcome variable was defined as 1 for countries whose life expectancy exceeded the median value and 0 otherwise. After converting the relevant columns to numeric and removing missing values, the final sample allowed a straightforward logistic regression with urbanization rate as sole predictor.

A logistic regression model was fitted, providing the following parameter estimates: the regression coefficient for urbanrate was 0.052, translating into an odds ratio of 1.054. The model intercept was estimated at -3.221. These results indicate that for each percentage point increase in urbanization, the odds of belonging to the “high life expectancy” group increase by approximately 5%. The uncertainty in coefficient estimation and odds ratio implies that urbanization has a meaningful positive association with binary life expectancy status.

Model performance was evaluated using the area under the ROC curve (AUC), which reached 0.795, reflecting strong discrimination between the two categories. The ROC curve visualizes the tradeoff between true positive and false positive rates, and the confusion matrix quantifies classification accuracy at the conventional threshold.

The analysis supports the hypothesis that countries with higher urbanization rates are more likely to exhibit higher life expectancy. All diagnostic steps confirmed stable estimation, and no major limitations arose in residuals or classification metrics.

Regression Results Table

Figure 1. ROC curve for logistic regression model

Figure 2. Confusion matrix at threshold 0.5

Appendix: Code

A cross-country dataset was analyzed to investigate how urbanization and average income per person are associated with life expectancy. The original data featured 213 countries and 16 variables—of which the relevant fields were extracted, transformed to numeric format, and restricted to complete cases, resulting in 176 valid observations. Urbanization rate and income per person were both mean-centered to facilitate interpretation and reduce multicollinearity, confirmed by post-centering means effectively equating to zero.

A multiple linear regression was fitted with life expectancy as the outcome and the centered urbanization rate and centered income per person as predictors. The model accounts for 46% of the variance in life expectancy. The regression equation estimates an intercept of 69.65 years; each additional percentage point of urbanization increases life expectancy by 0.17 years, and each additional unit increase in income per person contributes 0.00033 years to life expectancy. All coefficients were statistically significant at conventional thresholds.

To assess confounding, separate models were fit for each predictor individually and compared with the full model. The coefficient for urbanization decreased by 36% and for income by 40% when both predictors were included together, indicating substantial confounding and interaction between the effects of urbanization and income.

Examination of diagnostic plots confirmed that standard model assumptions were reasonably satisfied. The Q–Q plot showed the residuals to be approximately normally distributed with minor tail deviations. Standardized residuals displayed homoscedastic scatter and no systematic patterns, and leverage diagnostics indicated few influential observations with no extreme outliers. The histogram of residuals further corroborated the near-normality assumption.

Both urbanization and income remain independently associated with higher life expectancy in the joint model, supporting the initial hypothesis. These findings suggest each factor contributes through different mechanisms to longevity gradients observed across countries. The model’s robustness is upheld by diagnostic checks, though moderate collinearity and non-causal design remain important limitations.

Results Table

Figure 1: Q–Q plot of residuals

Figure 2: Standardized residuals vs fitted values

Figure 3: Leverage plot

Figure 4: Histogram of residuals

Appendix: Analysis Code

A dataset containing demographic indicators for 213 countries was utilized to assess the relationship between urbanization and life expectancy. The initial stage involved converting the variables “urbanrate” (percentage of the population residing in urban areas) and “lifeexpectancy” (average lifespan in years) into numeric formats. To ensure meaningful analysis, only complete observations were retained; this reduced the sample to 188 countries.

To enhance the interpretability of the analysis, the urbanization rate was centered by subtracting its mean value (55.93%) from each observation. This transformation guarantees that a zero value corresponds to the global average, and it allows the regression intercept to be interpreted as the expected life expectancy for a country with average urbanization. The procedure was verified by recalculating the mean after centering, yielding essentially zero (0.00), confirming correct implementation.

A linear regression model was then fitted, setting life expectancy as the dependent variable and the centered urban rate as the independent variable. Regression results indicate that the intercept is 69.60 years, representing the predicted life expectancy at a mean urbanization level. The estimated slope is 0.26, meaning that each additional percentage point in urbanization is associated with an increase of approximately 0.26 years in life expectancy (p < 0.001). The urban rate demonstrated a robust association, explaining 38.3% of the variance in life expectancy, as measured by the R^2 statistic. The overall model fit is supported by an F-statistic of 115.4 (p < 0.001), indicating the explanatory variable contributes significantly to predicting the outcome.

Throughout the analysis, key statistical outputs—including coefficients, p-values, and fit metrics—were reported and interpreted. The procedures conducted, and the criteria for coding and verifying the explanatory variable, adhere to quantitative standards for linear modeling. Figure 1 visually corroborates the numerical findings by displaying a scatter plot of centered urban rate against life expectancy, overlaid with the fitted regression line.

These findings suggest a clear positive association: increases in urbanization correspond to measurable gains in life expectancy at the national level. Such patterns plausibly reflect factors linked to greater urbanization, such as infrastructure, healthcare access, and socioeconomic conditions. It is important to recognize that this analysis focuses on association rather than causation.

Mean of Centered Urbanization Rate: 0.00

Regression Results: - Intercept (β_0): 69.60 years (p < 0.001) - Slope (β_1): 0.26 years per 1% increase in urban rate (p < 0.001) - Coefficient of Determination (R^2): 0.383 - F-statistic: 115.4 (p < 0.001)

Figure 1. Scatter plot and regression line visualizing the relationship between urban rate (centered) and life expectancy.

Appendix: Code

Recent analysis of country-level data makes it possible to highlight connections among economic resources, urban forms, and life expectancy worldwide. Drawing from a harmonized dataset spanning 176 countries, key measures—income per person, expected lifespan, and the degree of urbanization—were addressed utilizing complete reported records.

Each national entry is grounded in official statistical releases, enabling broad-based comparisons across regions. After thorough screening for completeness and reliability, the cleaned analytical sample captures essential economic and demographic features. Per capita income allows direct evaluation of available resources. Average years of life at birth reflect aggregate health achievement. Urban population rates offer insight into the scale and diversity of living environments.

The joint distribution of these metrics reveals distinct contrasts. Economic affluence and longer life expectancy are strongly linked, and this association varies along the spectrum of urbanization. Higher income correlates with increased longevity, although the rate of increase diminishes in highly urbanized settings. Additional descriptive results underscore the diversity present within present-day global development.

Statistical modeling confirms these patterns. A linear model establishes both income and urbanization as independent positive predictors of longevity. Introducing an interaction term, the results indicate that greater urbanization tempers the effect of increased income, with the coefficient revealing a significant negative adjustment. Countries with lower urban population shares stand to benefit more, on average, from economic growth in terms of gains in health outcomes.

Visualizations further illustrate these relationships. Scatter plots reveal the basic structure of economic and demographic variation, while fitted regression lines clarify the extent to which these predictors account for inter-country differences. Residual plots and model comparison metrics add perspective regarding relative model fit and explanatory power.

- Scatter plots of relationships among income, urbanization, and life expectancy

- Regression prediction plots (main effects and with interaction)

Appendix: Analysis Code

The association between national income and average life expectancy is well established, but the extent of this effect may vary across countries with differing degrees of urbanization. This analysis investigates whether urbanization moderates the link between economic prosperity and longevity.

Data and Methods

A global dataset comprising GDP per capita, life expectancy, and urbanization rates for over 170 countries was analyzed. Both bivariate correlation and moderation analyses were conducted to examine these relationships.

Results

Correlation Analysis

Analysis revealed a moderate and highly significant positive correlation between income per person and life expectancy:

Pearson correlation coefficient: 0.60

p-value: < 0.001

Consequence: Approximately 36% of the variance in life expectancy across countries is explained by national income.

Moderation Analysis: Urban Rate

Regression modeling incorporating an interaction term was used to evaluate whether the strength of the income-life expectancy relationship differs according to urbanization rate.

Main effect (Income): An increase of $1 in per capita income is associated with a 0.00066 year rise in life expectancy (p < 0.001).

Main effect (Urbanization): Each 1% increase in urban rate corresponds to a 0.11 year increase in life expectancy (p < 0.001).

Interaction effect: The negative coefficient ($$-0.000014$$, p < 0.001) indicates that the effect of income on life expectancy is diminished in more urbanized countries.

Model fit: The regression model accounts for approximately 50.6% of the variance in life expectancy.

Interpretation

The results indicate that economic gains have a larger impact on life expectancy in countries with lower urbanization rates. In highly urbanized contexts, baseline infrastructure and services are more established, resulting in diminishing marginal returns for additional increases in national income. Conversely, in less urbanized settings, improvements in income are more strongly associated with advances in population health and longevity.

Implications

These findings suggest that interventions aimed at increasing income will have a pronounced effect on public health primarily in countries with lower urbanization rates. For countries with widespread urban infrastructure, further increases in economic prosperity may yield comparatively smaller enhancements in life expectancy.

Appendix: Analysis Code

Across the world, economic prosperity and health outcomes often go hand-in-hand. To better understand this relationship, I analyzed data from 176 countries, focusing on how income per person (GDP per capita) relates to average life expectancy.

After cleaning the numbers and removing countries lacking data, the analysis showed a clear, upward tendency: nations with higher income per person tend to have longer-lived populations. The association is not just a vague trend—it’s backed by statistics.

Key findings:

Pearson correlation coefficient (r): 0.60

Interpretation: This indicates a moderately strong and positive relationship.

R² value: 0.36

Interpretation: About 36% of the variation in life expectancy between countries can be explained by differences in income per person; the rest is shaped by health systems, policies, culture, and education.

(Scatter plot showing upward trend: higher income, higher life expectancy)

The scatter plot visualizes this pattern. For most countries, wealth is linked with longer life. Still, there are exceptions—some countries achieve higher life expectancy than their wealth alone would predict, highlighting the role of social, healthcare, and educational factors.

The relationship is highly statistically significant (p < 0.001). In other words, it’s extremely unlikely that this pattern is due to random chance. It reflects a true global trend.

But, numbers alone never tell the full story. Correlation is not causation! While richer countries generally enjoy longer lives, this doesn’t guarantee that raising incomes will automatically lengthen life expectancy. The path runs through many channels—healthcare quality, policy choices, cultural practices, and more.

In summary: The global data reveals a meaningful and robust link between income and life expectancy. As nations grow richer, their people tend to live longer, but the puzzle of public health is influenced by much more than economic indicators. For policymakers and citizens alike, the message is clear: prosperity helps, but investment in health, education, and infrastructure is crucial for longer, healthier lives.

Appendix: Python Code to Reproduce This Analysis

A cross-sectional look at worldwide economic development often suggests profound differences across continents. To investigate whether a country's continent is linked to its income status, I examined the relationship between continent and national income level for over 130 countries, using a robust statistical approach.

Data Preparation & Variables

Using the Gapminder dataset, I categorized countries into "High Income" or "Low Income" groups based on the median income value. The analysis considered five continents: Africa, Americas, Asia, Europe, and Oceania.

Frequency Table

The table below presents the observed counts of countries by both continent and income group:

Total sample size: 137 countries.

Chi-Square Test Results

The statistical test revealed a clear pattern: - Chi-square statistic: 41.50 - p-value: < 0.001 - Degrees of freedom: 4

The results suggest a very strong association between continent and income status; the observed pattern is highly unlikely to have arisen by chance.

The expected frequencies if the two variables were truly independent are provided below:

Interpretation

There is a pronounced imbalance, particularly in Africa and Europe. While only 16.3% of African countries are classified as "High Income," a striking 82.4% of European countries fall into that group. The Americas show 69.6% high-income, Asia is more evenly distributed (37.1% high-income), and Oceania is too small a sample to draw firm conclusions but consists entirely of high-income entries here.

Post Hoc Comparisons

Since the overall association was strong and involved several continental groups, I compared continents pairwise for differences in income distribution, applying a conservative correction for multiple tests (Bonferroni). Three continent pairs were distinctly different:

Europe vs Africa: Europe (82.4% High Income), Africa (16.3% High Income).

Africa vs Americas: Africa (16.3% High Income), Americas (69.6% High Income).

Europe vs Asia: Europe (82.4% High Income), Asia (37.1% High Income).

No other pairs displayed strong enough differences at the stricter significance threshold.

Summary

This analysis demonstrates a robust link between a country's continent and its income status. Europe's and the Americas' economies appear much more developed compared to Africa and parts of Asia, while subtle differences emerge between all regions. The Chi-Square Test and subsequent pairwise contrasts highlight persistent geographic disparities in global wealth distribution.

Appendix: Core Python Code

Introduction

Understanding how national income relates to life expectancy provides a useful lens on population health and development. This analysis examines differences in life expectancy across four income groups derived from per-capita income. The goal is to quantify whether average life expectancy varies across these groups and to identify which groups differ from one another.

Data and Variables

Outcome: life expectancy (years), treated as a quantitative variable.

Grouping factor: income quartile, a categorical variable with four ordered levels defined from per-capita income (Q1-Low, Q2-MedLow, Q3-MedHigh, Q4-High).

Dataset: country-level observations.

Methods

Descriptive statistics summarize life expectancy within each income quartile to provide initial context.

A one-way analysis of variance (ANOVA) evaluates whether mean life expectancy differs across the four income groups.

If differences are detected, pairwise comparisons are performed to locate which groups diverge from each other.

Basic diagnostic checks assess distributional assumptions for residuals and the comparability of variances across groups.

Results Descriptive Patterns

A systematic increase in average life expectancy is observed as income quartiles rise from Q1 to Q4, with variability differing by group.

Statistical Testing

The omnibus test indicates that mean life expectancy is not equal across all groups. Follow-up pairwise comparisons point to clear differences involving the highest income group relative to others, while contrasts among the lower three groups are more modest.

Assumptions

Residual distribution and variance equality diagnostics are reported. Where diagnostics flag concerns, results are interpreted conservatively with attention to potential robustness considerations.

Interpretation

Higher national income is associated with higher average life expectancy. The pattern is not uniform across all adjacent groups—differences are most pronounced when comparing the highest-income group to the rest. This aligns with the expectation that resource availability, infrastructure, and access to care scale with national income.

Conclusion

Average life expectancy differs across income quartiles, with the highest-income countries enjoying substantially longer life spans. These findings reinforce the link between national resources and population health.

Appendix: Python Code (Reproducible Workflow)

You can run this end-to-end block in a single notebook. It generates descriptives, the omnibus test, pairwise comparisons, diagnostics, and figures.

Here, the Gapminder dataset is used to explore patterns in three major variables: income per person, life expectancy, and urban population rate. Through univariate and bivariate analyses, fundamental trends and relationships among these indicators are made intelligible both for researchers and general audiences.

Data Preparation and Cleaning

The analysis begins with importing and preparing the data. Three core variables—incomeperperson (GDP per capita), lifeexpectancy, urbanrate—were selected and converted to numeric format. To ensure validity, only countries with complete data for these metrics were included.

# Import libraries and read the data

import pandas as pd import numpy as np import matplotlib.pyplot as plt import seaborn as sns

# Set visualization style

sns.set_style('whitegrid') plt.rcParams['figure.figsize'] = (10, 6)

# Load the dataset

data = pd.read_csv('gapminder.csv', low_memory=False)

# Convert selected columns to numeric

for col in ['incomeperperson', 'lifeexpectancy', 'urbanrate']: data[col] = pd.to_numeric(data[col], errors='coerce')

# Subset and drop missing values

subset = data[['country', 'incomeperperson', 'lifeexpectancy', 'urbanrate']].dropna() print(f"Working with {len(subset)} countries after removing missing values") print(subset[['incomeperperson', 'lifeexpectancy', 'urbanrate']].describe())

Working with 176 countries after removing missing values Basic statistics: incomeperperson lifeexpectancy urbanrate count 176.000000 176.000000 176.000000 mean 7327.444414 69.654733 55.566364 std 10567.304022 9.729521 23.225708 min 103.775857 47.794000 10.400000 25% 702.366463 63.041500 36.685000 50% 2385.184105 73.126500 56.970000 75% 8497.779228 76.569500 73.465000 max 52301.587179 83.394000 100.000000

Univariate Analysis

Income per Person

To visualize the economic landscape, a histogram was plotted for income per person across countries.

plt.figure(figsize=(12, 6)) plt.hist(subset['incomeperperson'], bins=30, color='steelblue', edgecolor='black', alpha=0.7) plt.xlabel('Income Per Person (USD)', fontsize=12) plt.ylabel('Number of Countries', fontsize=12) plt.title('Distribution of Income Per Person Across Countries', fontsize=14, fontweight='bold') plt.grid(axis='y', alpha=0.3) plt.tight_layout() plt.show()

This distribution is right-skewed: most countries have relatively low GDP per capita, with a minority accounting for much higher levels. The shape of the histogram underscores the breadth of global inequality, as the majority cluster below $10,000 while a handful exceed $30,000. Histograms are an ideal choice here, as they show both the central tendency and spread for continuous variables.

Life Expectancy

A boxplot summarizes life expectancy, highlighting the range and potential outliers.

plt.figure(figsize=(12, 6)) plt.boxplot(subset['lifeexpectancy'], vert=False, widths=0.5, patch_artist=True, boxprops=dict(facecolor='lightcoral', alpha=0.7), medianprops=dict(color='darkred', linewidth=2), whiskerprops=dict(linewidth=1.5), capprops=dict(linewidth=1.5)) plt.xlabel('Life Expectancy (years)', fontsize=12) plt.title('Distribution of Life Expectancy Across Countries', fontsize=14, fontweight='bold') plt.yticks([]) plt.grid(axis='x', alpha=0.3) plt.tight_layout() plt.show()

Life expectancy is relatively normally distributed, with a median near 73 years and most values between 63 and 77. Outliers at the lower end (below 50 years) reflect underlying differences in health, resources, and other country-specific challenges. Boxplots clearly delineate medians, quartiles, and outliers, giving a holistic view of the distribution.

Urban Rate

Urbanization was explored with another histogram.

plt.figure(figsize=(12, 6)) plt.hist(subset['urbanrate'], bins=25, color='forestgreen', edgecolor='black', alpha=0.7) plt.xlabel('Urban Rate (% population urban)', fontsize=12) plt.ylabel('Number of Countries', fontsize=12) plt.title('Distribution of Urbanization Rate Across Countries', fontsize=14, fontweight='bold') plt.grid(axis='y', alpha=0.3) plt.tight_layout() plt.show()

Urban rate displays a fairly uniform spread, with a slight concentration around 50–60%. Countries range from highly rural to nearly entirely urbanized, indicating diverse development pathways. This visualization captures the variability in global urbanization and the absence of a single dominant pattern.

Bivariate Analysis

Income vs. Life Expectancy

A scatter plot elucidates the relationship between GDP per capita and life expectancy.

plt.figure(figsize=(12, 8)) plt.scatter(subset['incomeperperson'], subset['lifeexpectancy'], alpha=0.6, s=80, c='purple', edgecolors='black', linewidth=0.5) plt.xlabel('Income Per Person (USD)', fontsize=12) plt.ylabel('Life Expectancy (years)', fontsize=12) plt.title('Relationship Between Income Per Person and Life Expectancy', fontsize=14, fontweight='bold') plt.grid(True, alpha=0.3) plt.tight_layout() plt.show() # Calculate correlation corr = subset[['incomeperperson', 'lifeexpectancy']].corr().iloc[0, 1] print(f"Correlation coefficient: {corr:.3f}")

The analysis reveals a strong positive correlation (0.60): as income rises, life expectancy increases—though the effect tapers at high income levels. This non-linear (logarithmic) pattern suggests that basic economic resources have large impacts on health, but marginal gains diminish at greater wealth. Scatter plots are valuable here, displaying both strength and character of the association.

Life Expectancy Across Income Quartiles

Countries were divided into quartiles by income, then life expectancy compared across groups using boxplots.

subset['income_quartile'] = pd.qcut(subset['incomeperperson'], 4, labels=['Q1: Lowest', 'Q2: Low-Mid', 'Q3: Mid-High', 'Q4: Highest']) plt.figure(figsize=(12, 8)) subset.boxplot(column='lifeexpectancy', by='income_quartile', patch_artist=True, grid=False) plt.suptitle('') plt.xlabel('Income Quartile', fontsize=12, fontweight='bold') plt.ylabel('Life Expectancy (years)', fontsize=12) plt.title('Life Expectancy Distribution Across Income Quartiles', fontsize=14, fontweight='bold', pad=20) plt.xticks(rotation=15) plt.grid(axis='y', alpha=0.3) plt.tight_layout() plt.show()

Lower income quartiles show the greatest variability and lowest median life expectancy. As income rises, median and lower bounds increase, while the spread narrows dramatically. The highest quartile features consistently high life expectancy (75–83 years), with minimal variation. This comparison highlights how economic disparity translates directly into health outcomes—wealthier nations having both higher and more consistent life expectancies.

Summary

Visualizing key socioeconomic indicators reveals pronounced patterns in global development:

Income per person is heavily concentrated at low values, with high inequality.

Life expectancy distributions reflect substantial differences rooted in health and wealth.

Urbanization is diverse, without a dominant global trend.

Country-level wealth correlates strongly with life expectancy, though increases plateau in richer countries.

Income quartile stratification demonstrates the extent to which wealth drives population health stability.

The initial stage includes importing necessary libraries, examining dataset columns, converting relevant variables to numeric types, and quantifying missing data. This process ensures that subsequent analysis is both valid and interpretable. Missing values are handled via exclusion to avoid bias, while quartile binning is employed for income per person, offering an equal representation across the economic spectrum. Urban rate receives categorical binning for nuanced interpretability, and life expectancy is scrutinized for outliers using statistical methods. Each step pursues methodological rigor and clarity of exposition.

Python Code for Data Handling and Visualization

# Import necessary libraries import pandas as pd import numpy as np import matplotlib.pyplot as plt # Read Gapminder dataset data = pd.read_csv('gapminder.csv') # Select variables and convert to numeric selected = ['incomeperperson', 'urbanrate', 'lifeexpectancy'] df = data[selected].copy() for col in selected: df[col] = pd.to_numeric(df[col], errors='coerce') # Quartile binning for income per person df['income_quartile'] = pd.qcut(df['incomeperperson'], 4, labels=['Q1-Low', 'Q2-MedLow', 'Q3-MedHigh', 'Q4-High']) quartile_ranges = ['Q1\nLow\n($104-$744)', 'Q2\nMedLow\n($760-$2,550)', 'Q3\nMedHigh\n($2,557-$9,244)', 'Q4\nHigh\n($9,425-$105,147)'] freq = df['income_quartile'].value_counts().sort_index().values # matplotlib bar chart fig, ax = plt.subplots(figsize=(12,7)) colors = ['#FF6B6B', '#FFA500', '#4ECDC4', '#45B7D1'] bars = ax.bar(quartile_ranges, freq, color=colors, edgecolor='black', linewidth=1.5, alpha=0.8) for i, bar in enumerate(bars): height = bar.get_height() ax.text(bar.get_x() + bar.get_width()/2., height + 0.5, f'{freq[i]} countries\n({freq[i]/df["income_quartile"].count()*100:**.1f}%)', ha='center', va='bottom', fontsize=11, fontweight='bold') ax.set_xlabel('Income Quartile (Income per Person in USD)', fontsize=13, fontweight='bold') ax.set_ylabel('Number of Countries', fontsize=13, fontweight='bold') ax.set_title('Distribution of 190 Countries by Income Per Person Quartiles\n(Gapminder Dataset)', fontsize=15, fontweight='bold', pad=20) ax.set_ylim(0, 55) ax.yaxis.grid(True, linestyle='--', alpha=0.3) ax.set_axisbelow(True) plt.tight_layout() plt.show()

Interpreting Income Distribution and Methodological Choices

The improved visualization above utilizes quartile binning to group 190 countries with valid income data into four equally sized categories. This approach ensures fair representation of each economic group, allowing for unbiased comparative analysis. Each bar in the chart clearly states the number of countries and the income ranges for that group, making the disparities starkly evident.

The data reveals an immense spread in per-person income:

- Poorest quartile (Q1): $104–$744 per person annually

- Richest quartile (Q4): $9,425–$105,147 per person

Such a wide gap—over a thousandfold difference—reflects the underlying mechanisms of global economic inequality and resource concentration.

The quartile-based binning was specifically selected for its ability to normalize group sizes across a skewed distribution, offering both statistical robustness and clarity. The use of categorical labeling and direct annotation ensures that each finding is immediately interpretable, an essential quality in scientific reporting.

Frequency distributions are presented for three key variables from the Gapminder dataset: income per person, life expectancy, and urban rate. These indicators provide insights into economic status, health conditions, and urbanization patterns among countries worldwide.

Selected Variables

Income per Person: Average economic output or earnings per individual in each country.

Life Expectancy: Average expected lifespan at birth, reflecting population health outcomes.

Urban Rate: Percentage of population living in urban areas, representing settlement and development patterns.

Python Code Data Preparation

# Import necessary libraries import pandas as pd import plotly.express as px import plotly.graph_objects as go from plotly.subplots import make_subplots # Load the Gapminder dataset data = pd.read_csv('gapminder.csv')

Frequency Table and Missing Data Count

variables = ['incomeperperson', 'lifeexpectancy', 'urbanrate'] titles = ['Income per Person', 'Life Expectancy', 'Urban Rate'] colors = ['#005F73', '#EE9B00', '#0A9396'] # Convert to numeric (to prevent issues if the columns were loaded as strings) data_clean = data.copy() for var in variables: data_clean[var] = pd.to_numeric(data_clean[var], errors='coerce') # Create the subplots (3 side-by-side histograms) fig = make_subplots( rows=1, cols=3, subplot_titles=titles, horizontal_spacing=0.06 ) for i, var in enumerate(variables): fig.add_trace( go.Histogram( x=data_clean[var].dropna(), nbinsx=30, marker_color=colors[i], name=var, opacity=0.75, showlegend=False ), row=1, col=i+1 ) fig.update_layout( title_text="Frequency Distribution of Selected Variables (Countries)", font_family="Georgia", font_color="#333333", plot_bgcolor='white', height=400, width=980 ) for idx in range(1, 4): fig.update_xaxes(title_text=titles[idx-1], row=1, col=idx) fig.update_yaxes(title_text="Count", row=1, col=idx) fig.show()

Interpretation

Income per Person: The distribution is right-skewed, indicating most countries have relatively low average incomes. High-income countries are uncommon, highlighting global economic inequalities.

Life Expectancy: The majority of countries report values between 55 and 80 years. Lower life expectancy reflects challenges in healthcare and living conditions, while higher values are achieved primarily in developed regions.

Urban Rate: Urbanization values span a wide range. Some countries are highly urbanized (over 80%), while others remain predominantly rural, demonstrating diverse development models.

Conclusion