# Business maths _Know what the numbers mean before you act on them._ --- The board pack shows revenue up 12%. The sales dashboard shows 8%. Finance has a third number that doesn't match either. Someone asks what's really going on, and the next twenty minutes disappear into reconciliation. This happens constantly, and it's almost always a [[Definitions|definitions problem]]. Revenue can mean bookings, billings, recognised revenue, or cash collected. Each is correct for its purpose; collectively they are useless without context. --- **Most numbers go nowhere, even with shared definitions.** Dashboards proliferate, reports multiply, everyone claims to be "data-driven", but ask what the team will do differently if churn rises or NPS drops and the room goes quiet. [[Decision to action|Data only becomes information when it passes through a question you are trying to answer.]] If the monthly variance review ends with "thoughts?" instead of clear actions, the numbers got presented and nothing was decided. --- **How much precision does the metric need?** A business case shows 127% ROI. Not "roughly 100%" or "between 80% and 150%", exactly 127%. The model has five assumptions, each estimated within plus or minus 20%. That 127% might really sit anywhere from 60% to 200%. [[Confidence|Point estimates hide distributions]], and collapsing a range to a single number loses the information you actually need: how wide is the uncertainty, and what happens at the tails? --- **A good metric at the right precision can still mislead.** Last month's revenue was £1.2m. This month it's £1.1m. Problem? It depends. [[Variance|Common cause variation]] is the natural fluctuation in any process, present even when nothing has changed. Special cause variation means something specific happened. Reacting to noise is expensive (you investigate phantoms, "fix" things that weren't broken, add process in response to randomness). Ignoring signal is dangerous (by the time the pattern is undeniable, you are months behind). A single data point tells you almost nothing; three months of decline starts to look like a trend. --- **Small samples make it worse.** A product team runs twenty user interviews. Twelve prefer option A, eight prefer option B. The readout: "60% preferred A." With twenty people, that split could easily be 50/50 with a different sample. [[Samples|At n=20, the qualitative signal is more reliable than the quantitative noise.]] The specific feedback, how people reasoned, what language they used, how strongly they felt, carries information regardless of sample size. --- **Some things resist measurement entirely.** A company rolls out quarterly engagement surveys to "measure culture." Scores edge up each quarter. Leadership cites the improvement in their board report. Meanwhile, the best people are leaving, conversations are guarded, and the team treats the survey as a compliance exercise, giving safe answers to avoid follow-up meetings. The survey didn't measure culture. It measured willingness to perform culture for a questionnaire. Trust, morale, judgement, and potential are [[Unknown and unknowable|unmeasurable by nature]]. Quantifying them changes what you're observing. --- **The same business tells different stories depending on the lens.** A £50m revenue company generating £5m EBITDA looks like a 10% margin business, tight and possibly struggling. But if it runs on £3m of working capital, it earns nearly 1.7x return on operating assets, which is excellent. And if it turns those assets four times a year, it's an operational machine despite thin margins. Same company, three [[Ratios|ratios]], three stories. Margin ratios reveal pricing power, return ratios reveal capital efficiency, and turnover ratios reveal operational speed. [[Scale|Reference points]] sharpen this further. Revenue per employee in software runs £200-500k. In professional services, £100-200k. A "software" company at 40% gross margin is really a services business. These benchmarks catch errors that spreadsheets miss, because most serious mistakes are orders of magnitude wrong, not decimal places wrong. If the model says 10x what your intuition expects, one of them is broken. --- **Retention, not sales, limits the size of a recurring revenue business.** A company adding £1m of new customers per year with 10% churn hits a ceiling at £10m. At 5% churn, £20m. At 1%, £100m. Same sales engine, completely different business. [[The churn ceiling]] works through the full arithmetic. --- **Accounting profit follows rules about when to recognise revenue, how to allocate costs, when to depreciate.** Those rules are consistent and auditable, but they are still choices, and the choices shape what you see. [[Cash and profit]] strips those choices away. You either collected the cash or you didn't. Every number in the board pack passed through definitions, precision limits, sample sizes, and accounting conventions before it reached you. The decisions that go wrong usually start with the right number and the wrong question. ---