Research & insights

Source-backed reporting, original comparisons and practical analysis across our eight subject areas.

October 19, 2026 · Products

Original explanatory diagram separating the solid-object position, water position and product-specific conditions of an IP claim.

IP Ratings: Read Water and Dust Protection as Separate Claims

Read IP water and dust claims separately, check product conditions and avoid unsupported waterproof promises.

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October 19, 2026 · Business & Industry

Invented cohort revenue bridge: $10,000 less $1,000 churn and $500 contraction plus $2,000 expansion equals $10,500.

Customer Retention: Follow the Same Cohort Before Celebrating Growth

Calculate customer and revenue retention for the same cohort and separate expansion from acquisition.

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October 19, 2026 · Countries & Cities

Fictional apartments: A costs $1,400 plus $200 utilities; B costs $1,550 including those utilities.

Comparing City Rents: Asking Rent, Contract Rent and Gross Rent Answer Different Questions

Compare asking, contract and gross rents without mixing utilities, populations, time windows or affordability measures.

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October 19, 2026 · Economy & Finance

Fictional economy: GDP growth 5%, population growth 4%, and GDP per capita growth approximately 0.96%.

GDP per Capita: Why Economic Growth Can Outrun the Gain per Person

Calculate GDP per capita growth correctly and distinguish output per resident from income, productivity and welfare.

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October 19, 2026 · Technology & AI

Calculated chunk counts for a 10,000-token teaching document: 20, 25 and 33 windows with 0, 100 and 200 tokens of overlap.

RAG Chunking: Preserve the Evidence Before You Tune the Answer

Test RAG chunking by complete evidence coverage, source context, permissions and version handling rather than token count alone.

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October 18, 2026 · Healthcare & Public Data

Hypothetical diagnostic-test populations with 90% sensitivity and 95% specificity; no real disease or test represented.

Sensitivity, Specificity and Predictive Value: Read a Test Result Without Reversing the Probability

Understand sensitivity, specificity and predictive values through hypothetical population tables and careful clinical boundaries.

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October 18, 2026 · Sports & Statistics

Fictional records 8–2, 40–10 and 18–12 produce winning percentages of 80%, 80% and 60%.

Winning Percentage: Compare the Record, the Sample and the Rules

Calculate winning percentage correctly and retain game counts, weighting and sport-specific definitions.

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October 18, 2026 · Real Estate

Fictional monthly cash budgets comparing the specified ownership costs with $1,700 rent plus $20 renters insurance.

Rent vs Buy: Compare Cash, Equity and Risk on the Same Timeline

Compare rent and ownership using cash budgets, equity, transaction costs, uncertain assumptions and a shared timeline.

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October 18, 2026 · Products

Idealized 60 Wh energy calculation: 6, 4 and 3 hours at constant average draws of 10, 15 and 20 W.

Laptop Battery Life: Compare Watt-Hours, Workloads and Battery Health

Compare battery energy, workload, health and charging without mistaking advertised hours for a universal test.

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October 18, 2026 · Business & Industry

Original break-even teaching model: price $50, variable cost $30 and fixed cost $12,000; intersection at 600 units.

Break-Even Analysis: Test Price, Costs and Capacity Before Trusting the Sales Target

Test break-even sales against contribution, price changes, returns, sales mix, cash flow and capacity.

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October 18, 2026 · Countries & Cities

Fictional places with 100,000 residents each have densities of 2,000 and 500 per square kilometer because their areas differ.

Population Density: Why a City Average Can Hide How People Actually Live

Understand how boundaries, land area and weighting change population-density comparisons.

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October 18, 2026 · Economy & Finance

Fictional five-household incomes: only the highest income rises from 70 to 270 thousand dollars.

Mean vs Median Income: The Average Can Rise While Most Households Stand Still

Compare mean and median income with distribution examples, consistent population units and appropriate inflation adjustments.

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