This page explains how our calculators work: the formulas they use, how we test them, and what their limits are. We publish it so that students, teachers and anyone who relies on our results can check our work.
Formulas We Use
Proportion: a/b = c/d, solved by cross-multiplication, a × d = b × c. Proportional relationship: y = kx, with k = y ÷ x. Direct proportion: y = kx. Inverse proportion: xy = k. Percent proportion: part/whole = percent/100. Scale: drawing = real ÷ N for a scale of 1:N. Golden ratio: φ = (1 + √5) ÷ 2 ≈ 1.6180339887. One-proportion z-test: z = (p̂ − p0) ÷ √(p0(1 − p0) ÷ n). Two-proportion z-test: z = (p̂1 − p̂2) ÷ √[p̂(1 − p̂)(1/n1 + 1/n2)] with pooled p̂. Confidence interval (one proportion): p̂ ± z* × √(p̂(1 − p̂) ÷ n). Confidence interval (two proportions): (p̂1 − p̂2) ± z* × √[p̂1(1 − p̂1)/n1 + p̂2(1 − p̂2)/n2]. Sample size: n = z*² × p(1 − p) ÷ E², rounded up.
How We Test Our Calculators
Before publishing, we test every calculator with known answers: 3/4 = x/20 must give x = 15; 7/8 = 14/16 must be confirmed true with cross products 112 = 112; a 95% interval for 60 of 100 must be (0.504, 0.696); and a sample size for 95% confidence with a ±5% margin of error and p = 0.5 must be 385. We also test edge cases: fractions, mixed numbers, decimals, negatives and zero denominators, which are rejected with a clear message. The normal-distribution p-values use the standard Abramowitz–Stegun approximation, accurate to about 0.0000001.
Rounding and Precision
Where an exact fraction exists, we show it — for example, 1/3 rather than 0.333333. Decimals are shown to four to six places, and sample sizes are always rounded up to the next whole person. A confidence interval is an estimate, not a guarantee: it describes what repeated sampling would produce, and it depends on the conditions stated on each tool page (random sampling, enough successes and failures, and the 10% condition).
Sources and Limits
Our formulas follow standard introductory statistics and algebra textbooks, including the AP Statistics framework for inference about proportions. Our calculators are educational tools: they do not replace professional statistical advice, and they should not be used for medical or legal decisions without expert review. If you find an error, please report it on the Contact page — verified errors are logged on the Corrections page.