# Math for Statistics and Unit Economics > The school math behind every formula in our statistics, finance and product-analytics courses, explained for people who forgot it. Percent and percentage points, rounding and quick estimates, the sum sign, averages and weights, squares and roots, lines, compounding and discounting, probability and the normal curve, then unit economics from scratch. No proofs and no calculus: read a formula, say what each part does and compute it by hand or in a spreadsheet. - Source: https://productovi.com/courses/math-basics - Free, no sign-up needed to read. ## Modules ### Module 1. Numbers and shares Fractions, ratios and the order of operations; percent vs percentage points and relative change; rounding at the end and rounding up; quick estimates and orders of magnitude. - [Order of operations, ratios and multiples](https://productovi.com/courses/math-basics/order-of-operations-and-ratios.md): Read a business formula in the right order, tell a share from a ratio, and turn a multiple into a value. - [Percent vs percentage points](https://productovi.com/courses/math-basics/percent-vs-percentage-points.md): Absolute and relative change, new ÷ old − 1, lift in an A/B test, and why "+1%" can mean two different things. - [Rounding: at the end, and sometimes up](https://productovi.com/courses/math-basics/rounding-and-rounding-up.md): Compute from unrounded values, round once at the end, and round up whenever you need enough: break-even, staff, test length. - [Back-of-the-envelope estimates](https://productovi.com/courses/math-basics/back-of-the-envelope-estimates.md): Check any number in ten seconds: round to one significant figure, count the zeros, and give a range instead of false precision. ### Module 2. The language of formulas Notation and Greek letters, the sum sign with its limits and double indices, rearranging a formula to find the unknown, inequalities, intervals and formulas with cases. - [Notation: indices, bars, hats and Greek letters](https://productovi.com/courses/math-basics/formula-notation.md): How to read x_i, sorted values, x̄ and ŷ, the Greek letters μ, σ, α, β and λ, Δ for a change, and sample vs population. - [The Σ sign: sums, limits and running totals](https://productovi.com/courses/math-basics/the-sum-sign.md): Read Σ with its limits, see why deviations from the mean add up to zero, use sums of products, running totals and double indices, and two shortcuts: n(n + 1)/2 and k(k − 1)/2. - [Rearranging a formula: solve for what you need](https://productovi.com/courses/math-basics/rearranging-a-formula.md): Do the same to both sides and undo in reverse order: break-even Q = F/c, runway, the regression intercept, sample size, a fraction inside a denominator, and why a gross-up divides by (1 − f). - [Inequalities, intervals and formulas with cases](https://productovi.com/courses/math-basics/inequalities-and-intervals.md): <, ≤, ± and intervals, "if and only if", min and max, formulas with cases, fences, payback as the first month that clears a bar, and tiered rates with effective vs marginal. ### Module 3. Averages and shares of shares Mean, median and trimmed mean; weighted means and mix; products of shares in funnels and metric trees; choosing the right denominator. - [Typical value and position: mean, median, quartiles](https://productovi.com/courses/math-basics/mean-median-trimmed-mean.md): Mean, median, mode and trimmed mean, quartiles, percentiles and the IQR (two ways to compute them), a seasonal index, and the cumulative share curve. - [Weighted mean and mix](https://productovi.com/courses/math-basics/weighted-mean-and-mix.md): Σwx/Σw, why a pooled rate is not the mean of rates, mix-adjusted rates, pooled variance and pooled p̄, the cost of capital, PERT estimates and weighted scoring. - [Products of shares: funnels and metric trees](https://productovi.com/courses/math-basics/products-of-shares.md): Overall conversion as a product of step rates, working a funnel backwards from a goal, ARPU = paying share × ARPPU, revenue per visitor, stickiness, diluted effects, ownership dilution and RICE. - [What do we divide by? Denominator traps](https://productovi.com/courses/math-basics/choosing-the-denominator.md): ARPU vs ARPPU, LTV per cohort user vs per survivor, win rate with or without "no decision", gross vs net margin, concentration with the item in the total, and incremental results against a control group. ### Module 4. Squares, roots and distances Why statistics squares deviations, the square root and √n, errors that add like the sides of a right triangle, and standardizing with z. - [Why we square: variance, sums of squares and n − 1](https://productovi.com/courses/math-basics/why-we-square.md): Why spread formulas square deviations, variance s² and σ², sums of squares split into between and within, λ as a sum of squared loadings, and the idea behind n − 1 and degrees of freedom. - [Square roots and √n: SD and the standard error](https://productovi.com/courses/math-basics/square-roots-and-root-n.md): The standard deviation as √s², the standard error s/√n and why 4× the data halves it, the standard error of a proportion, the minimum detectable effect, the paired t and the finite-population correction. - [Pythagoras in statistics: distances and errors in quadrature](https://productovi.com/courses/math-basics/pythagoras-in-statistics.md): Euclidean and Manhattan distance, why independent errors add as squares, the pooled SE of a z-test vs the unpooled SE of an interval, and how to read the Wilson and Newcombe interval formulas. - [Standardizing: z, CV, effect size and r](https://productovi.com/courses/math-basics/standardizing.md): Put different scales side by side: z-scores, the coefficient of variation, t as signal over noise, Cohen's d as an effect in SD units, and correlation as a covariance scaled to between −1 and 1. ### Module 5. Lines and models The linear function, slope as a marginal quantity, models with several variables, and splitting a total into parts. - [The linear function](https://productovi.com/courses/math-basics/the-linear-function.md): Read y = a + bx in business formulas: profit = c·n − F, trend lines, a tax with a fixed part, the design effect, where two lines cross, and two relations that are not lines. - [Slope as a marginal quantity](https://productovi.com/courses/math-basics/slope-as-a-marginal-quantity.md): What one more unit adds: marginal vs average CAC and cost, operating leverage as 1 ÷ margin of safety, and price elasticity as a ratio of relative changes. - [Several variables](https://productovi.com/courses/math-basics/several-variables.md): Formulas with several inputs: ŷ = b₀ + Σ bⱼxⱼ, 0/1 variables and interactions, a driver-based forecast, ±10% sensitivity and scenarios. - [Splitting a total](https://productovi.com/courses/math-basics/splitting-a-total.md): Split a total into parts that add up exactly: sums of squares and R², plan vs actual as volume + mix + price, and a change in savings as contributions + growth. ### Module 6. Growth and time Compounding and average growth rates, discounting and NPV, geometric series behind virality and LTV, and e, ln, odds and the S-curve. - [Compounding](https://productovi.com/courses/math-basics/compounding.md): Growth factors, why −50% then +50% loses a quarter, (1 + r)ⁿ, simple vs compound interest, CAGR with a fractional power, and monthly ↔ yearly rates and churn. - [Discounting](https://productovi.com/courses/math-basics/discounting.md): Compounding in reverse: PV = FV ÷ (1 + r)ᵗ, NPV as a sum of discounted cash flows, IRR found by trial, the hurdle rate, and payback with linear interpolation. - [Geometric series](https://productovi.com/courses/math-basics/geometric-series.md): Sums that shrink by the same factor: seed ÷ (1 − k) for referrals, LTV ≈ margin ÷ churn and lifetime 1 ÷ churn, the finite sum for annuities, capped LTV and a savings pot. - [e, ln, odds and the sigmoid](https://productovi.com/courses/math-basics/e-ln-odds-sigmoid.md): ln undoes a power, so it solves for time; odds = p ÷ (1 − p), log-odds and the sigmoid p = 1 ÷ (1 + e⁻ᶻ), eᵇ as an odds multiplier, and payback with churn. ### Module 7. Probability and distributions p and 1 − p, conditional probability and the 2×2 table, expected value, many tests at once, the normal curve, and the other curves with power. - [p and 1 − p](https://productovi.com/courses/math-basics/p-and-1-minus-p.md): Probability as a share, its complement 1 − p, retention as 1 − churn, and why p(1 − p) peaks at 0.5. - [Conditional probability and the 2×2 table](https://productovi.com/courses/math-basics/conditional-probability.md): P(A | B) as a share inside a row, precision, recall and accuracy, relative risk and the odds ratio, and why the direction of a condition matters. - [Expected value and scenarios](https://productovi.com/courses/math-basics/expected-value.md): Weighting outcomes by their probabilities, deal pipelines and earn-outs, expected loss against an insurance premium, and risk exposure in matching units. - [Many tests at once](https://productovi.com/courses/math-basics/many-tests-at-once.md): The complement rule, multiplying independent chances, the chance of at least one false alarm, counting pairs and the Bonferroni split. - [The normal curve](https://productovi.com/courses/math-basics/the-normal-curve.md): Area as probability, the 68–95–99.7 rule, where 1.96 comes from, one- and two-tailed p-values, and the z-values for α and power. - [Other curves and power](https://productovi.com/courses/math-basics/other-curves-and-power.md): t, chi-square and F as the same idea with another table, degrees of freedom, expected counts and Σ(O − E)²/E, and computing the power of a test. ### Module 8. Unit economics, the bridge Contribution margin, break-even and the discount break-even; flows of customers and cash, runway, CAC payback and incrementality. The business reading lives in the finance and product-analytics courses. - [Contribution margin and break-even](https://productovi.com/courses/math-basics/contribution-and-break-even.md): The unit-economics formulas as pure math: CM = P(1 − f) − T − V, CM%, break-even ⌈F ÷ c⌉, margin of safety, step costs, the volume a discount needs, gross-up and ROI. - [Flows: customers and cash](https://productovi.com/courses/math-basics/flows-customers-and-cash.md): One rule for every stock, end = start + in − out: MRR, churn, NRR and GRR; profit vs cash, working capital, DSO and runway; CAC payback and incrementality against a holdout. ### Module 9. Money over time Simple interest and day counts, loans and amortisation, real and nominal rates, taxes and fees. - [Simple interest, days and annualizing](https://productovi.com/courses/math-basics/simple-interest-and-day-counts.md): Interest on the original amount, counting days, the cost of paying late or skipping an early-payment discount, and turning a period rate into a yearly one. - [Loans and amortization](https://productovi.com/courses/math-basics/loans-and-amortisation.md): How a payment splits into interest and principal, months to repay, a flat fee against interest on the balance, and the real annual cost of a loan as its internal rate. - [Real and nominal rates](https://productovi.com/courses/math-basics/real-and-nominal-rates.md): Inflation and purchasing power, the real rate as a ratio of growth factors, why n − i is only a shortcut, deflating a future sum, and exchange spreads. - [Taxes and fees](https://productovi.com/courses/math-basics/taxes-and-fees.md): The after-tax rate r(1 − t), the tax-equivalent rate r ÷ (1 − t), a flat tax plus a fixed contribution and its effective rate, and how a yearly fee compounds against you.