Estimation Fermi

---
name: estimation-fermi
description: Decomposes complex unknowns into estimable components to produce rapid order-of-magnitude answers with bounded uncertainty. Use when making quick estimates (market sizing, resource planning, feasibility checks), bounding unknowns with upper/lower limits, sanity-checking strategic assumptions, or when user mentions Fermi estimation, back-of-envelope calculation, order of magnitude, ballpark estimate, or triangulation.
---
# Fermi Estimation

## Table of Contents
- [Workflow](#workflow)
- [Common Patterns](#common-patterns)
- [Guardrails](#guardrails)
- [Quick Reference](#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](resources/template.md#clarification-template) 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](resources/template.md#decomposition-strategies) 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](resources/methodology.md#anchoring-techniques) 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](resources/methodology.md#bounding-techniques) 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](resources/template.md#sanity-check-template) 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](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](resources/template.md)**: Clarification framework, decomposition strategies, estimation template, sanity-check criteria
- **[resources/methodology.md](resources/methodology.md)**: Anchoring techniques, bounding methods, triangulation approaches, calibration exercises
- **[resources/evaluators/rubric_estimation_fermi.json](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