Fermi Estimation

Table of Contents

• Workflow
• Common Patterns
• Guardrails
• Quick Reference

Example

Question: How many piano tuners are in Chicago?

Decomposition:

1. Chicago ~3M people ÷ 3/household = 1M households
2. ~1 in 20 has piano = 50,000 pianos, tuned once/year
3. Tuner: 250 days/year × 4 tunings/day = 1,000/year
4. 50,000 ÷ 1,000 = ~50 piano tuners (Actual: ~80-100, within order of magnitude)

Workflow

Copy this checklist and track your progress:

Fermi Estimation Progress:
- [ ] Step 1: Clarify the question and define metric
- [ ] Step 2: Decompose into estimable components
- [ ] Step 3: Estimate components using anchors
- [ ] Step 4: Bound with upper/lower limits
- [ ] Step 5: Calculate and sanity-check
- [ ] Step 6: Triangulate with alternate path

Step 1: Clarify the question and define metric

Restate question precisely (units, scope, timeframe). Identify what decision hinges on estimate (directional answer sufficient? order of magnitude?). See resources/template.md for question clarification framework.

Step 2: Decompose into estimable components

Break unknown into product/quotient of knowable parts. Choose decomposition strategy (top-down, bottom-up, dimensional analysis). See resources/template.md for decomposition patterns.

Step 3: Estimate components using anchors

Ground estimates in known quantities (population, physical constants, market sizes, personal experience). State assumptions explicitly. See resources/methodology.md for anchor sources and calibration.

Step 4: Bound with upper/lower limits

Calculate optimistic (upper) and pessimistic (lower) bounds to bracket answer. Check if decision changes across range. See resources/methodology.md for constraint-based bounding.

Step 5: Calculate and sanity-check

Compute estimate, round to 1-2 significant figures. Sanity-check against reality (does answer pass smell test?). See resources/template.md for validation criteria.

Step 6: Triangulate with alternate path

Re-estimate using different decomposition to validate. Check if both paths yield same order of magnitude. Validate using resources/evaluators/rubric_estimation_fermi.json. Minimum standard: Average score ≥ 3.5.

Common Patterns

Pattern 1: Market Sizing (TAM/SAM/SOM)

• Decomposition: Total population → Target segment → Addressable → Reachable → Price point
• Anchors: Census data, industry reports, analogous markets, penetration rates
• Bounds: Optimistic (high penetration, premium pricing) vs Pessimistic (low penetration, discount pricing)
• Sanity check: Compare to public company revenues in space, VC market size estimates
• Example: E-commerce TAM = US population × online shopping % × avg spend/year

Pattern 2: Infrastructure Capacity

• Decomposition: Users → Requests per user → Compute/storage per request → Overhead
• Anchors: Similar services (Instagram, Twitter), known capacity (EC2 instance limits), load testing data
• Bounds: Peak (Black Friday) vs Average load, Growth trajectory (2x/year vs 10x/year)
• Sanity check: Cost per user should be < LTV, compare to public cloud bills of similar scale
• Example: Servers needed = (DAU × requests/user × ms/request) ÷ (instance capacity × utilization)

Pattern 3: Staffing/Headcount

• Decomposition: Work to be done (features, tickets, customers) → Productivity per person → Overhead (meetings, support)
• Anchors: Industry benchmarks (engineer per X users, support agent per Y customers), team velocity, hiring timelines
• Bounds: Experienced team (high productivity) vs New team (ramp time), Aggressive timeline (crunch) vs Sustainable pace
• Sanity check: Headcount growth should match revenue growth curve, compare to peers at similar scale
• Example: Engineers needed = (Story points in roadmap ÷ Velocity per engineer) + 20% overhead

Pattern 4: Financial Projections

• Decomposition: Revenue = Users × Conversion rate × ARPU, Costs = COGS + Sales/Marketing + R&D + G&A
• Anchors: Cohort data, industry CAC/LTV benchmarks, comparable company metrics, historical growth
• Bounds: Bull case (high growth, efficient scaling) vs Bear case (slow growth, rising costs)
• Sanity check: Margins should approach industry norms at scale, growth rate should follow S-curve not exponential forever
• Example: Year 2 revenue = Year 1 revenue × (1 + growth rate) × (1 - churn)

Pattern 5: Impact Assessment

• Decomposition: Total impact = Units affected × Impact per unit × Duration
• Anchors: Emission factors (kg CO2/kWh), conversion rates (program → behavior change), precedent studies
• Bounds: Conservative (low adoption, small effect) vs Optimistic (high adoption, large effect)
• Sanity check: Impact should scale linearly or sub-linearly (diminishing returns), compare to similar interventions
• Example: Carbon saved = (Users switching × Miles driven/year × Emissions/mile) - Baseline

Guardrails

1. State assumptions explicitly: Every Fermi estimate rests on assumptions. Make them visible ("Assuming 250 workdays/year", "If conversion rate ~3%"). Unstated assumptions create false precision.

2. Aim for order of magnitude, not precision: Goal is 10^X, not X.XX. Round to 1-2 significant figures (50 not 47.3, 3M not 2,847,291). If the decision needs precision, get real data instead.

3. Decompose until components are estimable: Break down until you reach quantities you can estimate from knowledge/experience. If a component is still "how would I know that?", decompose further.

4. Use multiple paths (triangulation): Estimate same quantity via different decompositions (top-down vs bottom-up, supply-side vs demand-side). If paths agree within factor of 3, confidence increases. If they differ by 10x+, investigate which decomposition is flawed.

5. Bound the answer: Calculate optimistic and pessimistic cases to bracket reality. If the decision holds across the range, bounds matter less. If the decision flips, invest in a better estimate.

6. Sanity-check against reality: Compare to known quantities, use dimensional analysis (units should cancel correctly), and check extreme cases (what if everyone did X? does it break physics?).

7. Calibrate on known problems: Practice on questions with verifiable answers to identify personal biases (overestimate? underestimate? anchoring?).

8. Acknowledge uncertainty ranges: Express estimates as ranges when appropriate ("10-100k users", "likely $1-5M").

Common pitfalls:

• ❌ Anchoring on the wrong number: Using irrelevant or biased starting point. If someone says "Is it 1 million?" you anchor there even if no reason to.
• ❌ Double-counting: Including same quantity twice in decomposition (counting both businesses and employees when businesses already includes employees).
• ❌ Unit errors: Mixing per-day and per-year, confusing millions and billions, wrong currency conversion. Always check units.
• ❌ Survivor bias: Estimating based on successful cases (average startup revenue from unicorns, not including failures).
• ❌ Linear extrapolation: Assuming linear growth when exponential (or vice versa). Growth rates change over time.
• ❌ Ignoring constraints: Physical limits (can't exceed speed of light), economic limits (market can't grow faster than GDP forever).

Quick Reference

Key resources:

• resources/template.md: Clarification framework, decomposition strategies, estimation template, sanity-check criteria
• resources/methodology.md: Anchoring techniques, bounding methods, triangulation approaches, calibration exercises
• resources/evaluators/rubric_estimation_fermi.json: Quality criteria for decomposition, assumptions, bounds, sanity checks

Common Anchors:

Demographics:

• US population: ~330M, Households: ~130M, Labor force: ~165M
• World population: ~8B, Urban: ~55%, Internet users: ~5B

Business:

• Fortune 500 revenue: $100k to $600B (median ~$30B)
• Startup valuations: Seed ~$5-10M, Series A ~$30-50M, Unicorn >$1B
• SaaS metrics: CAC ~$1-5k, LTV/CAC ratio >3, Churn <5%/year

Technology:

• AWS EC2 instance: ~10k requests/sec, S3 storage: $0.023/GB/month
• Mobile app: ~5-10 screens/day per user, 50-100 API calls/session
• Website: ~2-3 pages/session, 1-2min session duration

Physical:

• Person: ~70kg, 2000 kcal/day, 8 hours sleep
• Car: ~25 mpg, 12k miles/year, $30k new, 200k mile lifetime
• House: ~2000 sq ft, $300k median US, 30-year mortgage

Conversion factors:

• 1 year ≈ 250 workdays ≈ 2000 work hours
• 1 million seconds ≈ 11.5 days, 1 billion seconds ≈ 32 years
• 1 mile ≈ 1.6 km, 1 kg ≈ 2.2 lbs, 1 gallon ≈ 3.8 liters

Decomposition Strategies:

• Top-down: Start with total population, filter down (US population → Car owners → EV buyers)
• Bottom-up: Start with unit, scale up (1 store revenue × Number of stores)
• Rate × Time: Flow rate × Duration (Customers/day × Days/year)
• Density × Area/Volume: Concentration × Space (People/sq mile × City area)
• Analogous scaling: Known similar system, adjust for size (Competitor revenue × Our market share)

Typical estimation time:

• Simple question (1-2 levels of decomposition): 3-5 minutes
• Market sizing (3-4 levels): 10-15 minutes
• Complex business case (multiple metrics, triangulation): 20-30 minutes

When to escalate:

• Decision requires precision (< factor of 2 uncertainty)
• Estimate spans >2 orders of magnitude even with bounds
• No reasonable decomposition path (too many unknowns)
• Stakeholders need confidence intervals and statistical rigor → Invest in data collection, detailed modeling, expert consultation

Inputs required:

• Question (what are we estimating? units? scope?)
• Decision context (what decision hinges on this estimate? required precision?)
• Known anchors (what related quantities do we know?)

Outputs produced:

• estimation-fermi.md: Question, decomposition, assumptions, calculation, bounds, sanity check, triangulation, final estimate with confidence range