How to Use VSTACK and LAMBDA to Author Dynamic Bonferroni Inequality Index Models in Microsoft Excel

While the Gini coefficient is the standard metric used in macro-economic inequality studies, it is well documented that the Gini index is most sensitive to income transfers around the middle of the distribution (the modal income range). When studying extreme poverty, minimum wage policies, or vulnerable lower-tier socioeconomic strata, economists frequently prefer the Bonferroni Inequality Index (B). The Bonferroni index compares the cumulative average income of the poorest k individuals to the overall population mean, placing superior weighting on deficits at the bottom tail. By combining modern dynamic array functions—specifically VSTACK, HSTACK, LAMBDA, LET, SORT, and SCAN—analysts can construct dynamic Bonferroni inequality models in Microsoft Excel without legacy VBA macros.

Mathematical Foundations of the Bonferroni Index

For an income distribution of N individuals sorted in ascending order y₁ ≤ y₂ ≤ … ≤ y_N with overall mean income μ, let m_k denote the partial mean of the lowest k individuals: m_k = (1 / k) * ∑_{i=1}^k y_i.

The Bonferroni Inequality Index B is defined as the average deficit ratio between the overall mean μ and the cumulative partial mean m_k across all ranks k = 1, …, N:

B = (1 / N) * ∑_{k=1}^N (1 - m_k / μ) = 1 - (1 / N) * ∑_{k=1}^N (m_k / μ)

The index ranges between 0.0 (perfect equality where every individual earns exactly μ) and 1.0 (pure inequality where one individual holds the entire income pool). Because each partial mean m_k includes the poorest individual y₁, changes in the lowest incomes exert a continuous cumulative influence on the entire sum.

Creating the Reusable LAMBDA Function in Name Manager

In Microsoft Excel (Microsoft 365 or Excel 2024), navigate to Formulas > Name Manager. Click New, enter the name BONFERRONI_INDEX, and define the formula as follows:

=LAMBDA(incomes,
    LET(
        clean, FILTER(incomes, (incomes > 0) * ISNUMBER(incomes)),
        sorted, SORT(clean, 1, 1),
        n, ROWS(sorted),
        mu, AVERAGE(sorted),
        cum_sums, SCAN(0, sorted, LAMBDA(acc, val, acc + val)),
        k_seq, SEQUENCE(n, 1, 1, 1),
        partial_means, cum_sums / k_seq,
        mean_ratios, partial_means / mu,
        bonferroni, 1 - AVERAGE(mean_ratios),
        bonferroni
    )
)

This dynamic formula dynamically sorts observations in ascending order, calculates running cumulative sums via SCAN, derives partial means for every rank, and resolves the Bonferroni index.

Constructing the Full Dynamic Scorecard with VSTACK

Suppose household survey incomes are listed in cells A2:A20. In cell C2, insert this self-contained reporting formula:

=LET(
    inc, A2:A20,
    b_val, BONFERRONI_INDEX(inc),
    clean_inc, FILTER(inc, (inc > 0) * ISNUMBER(inc)),
    sorted_inc, SORT(clean_inc, 1, 1),
    n, ROWS(sorted_inc),
    mu, AVERAGE(sorted_inc),
    
    // Gini calculation for comparison
    gini_val, (2 * SUM(SEQUENCE(n, 1, 1, 1) * sorted_inc) / (n * SUM(sorted_inc))) - ((n + 1) / n),
    
    header, HSTACK("Welfare Inequality Metric", "Computed Value", "Analytical Assessment"),
    r1, HSTACK("Bonferroni Index (B)", ROUND(b_val, 4), IF(b_val > 0.45, "Severe Lower-Tail Deprivation", "Moderate Dispersion")),
    r2, HSTACK("Standard Gini Index (G)", ROUND(gini_val, 4), IF(gini_val > 0.4, "High National Inequality", "Moderate Inequality")),
    r3, HSTACK("Lower-Tail Sensitivity Premium (B - G)", TEXT(b_val - gini_val, "+0.0000;-0.0000;0.0000"), "Quantifies bottom-decile deficit beyond median spread"),
    r4, HSTACK("Mean Income Benchmark (μ)", TEXT(mu, "£#,##0"), "Overall distribution mean"),
    
    VSTACK(header, r1, r2, r3, r4)
)

Analytical Advantages in Poverty and Welfare Economics

1. Enhanced Lower-Tail Sensitivity: Because B ≥ G holds across all positive distributions, the spread (B - G) specifically highlights how deeply the lower deciles lag behind the overall economic standard of living.
2. Instant Dynamic Recalculation: Updating sample income figures automatically updates partial means, ratios, and the Bonferroni index across spilled cells with zero manual recalculation.
3. 100% Macro-Free: Operates natively in Excel Online, iOS, and desktop without requiring macro security exceptions or add-ins.