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The decline in overall capital approval rates from 85% to 82%, despite individual product approval rates remaining constant or increasing, is a classic example of Simpson's Paradox. This statistical phenomenon occurs when a trend appears in several different groups of data but disappears or reverses when these groups are combined.
Individual Product Trends:
Overall Trend Reversal:
Weighted Average Impact:
Change in Product Mix:
Let:
The overall approval rate is calculated as: p=n∑i=14(ni⋅pi)
A change in ni towards products with lower pi, like Product 2, can lower p even if individual pi remain constant or increase.
This paradox highlights the importance of understanding distribution and weighting in data analysis. When evaluating overall trends, it's crucial to consider how changes in the composition of data groups can affect aggregate statistics. Understanding Simpson's Paradox helps in making more informed decisions and avoiding misleading conclusions drawn from aggregated data.