Monte Carlo Error Calculator
How simulation accuracy improves with sample count — standard error, and samples needed for a target precision.
Monte Carlo error shrinks as 1/√N — the fundamental, frustrating convergence rate. Halving the error costs 4× the simulations; one more decimal digit of precision costs 100×. This √N law governs all sampling-based estimation and is why variance-reduction techniques (importance sampling, antithetic variates) matter so much.
Formula
About Monte Carlo Error Calculator
Monte Carlo methods estimate quantities by random sampling — pricing options, estimating integrals, simulating risk — and their accuracy is governed by one unforgiving law: error shrinks as 1/√N. This calculator computes the standard error and 95% margin at a given number of simulations, and tells you how many you'd need for a target precision. The √N rule has a brutal consequence it makes vivid: halving the error requires quadrupling the runs, and each additional decimal digit of precision costs 100× more compute — which is precisely why variance-reduction techniques exist.
How to use Monte Carlo Error Calculator
- 1Enter your values into Monte Carlo Error Calculator — sensible, domain-typical defaults are pre-filled so you see a real result immediately.
- 2The result recomputes live using the formula shown on the page; there is no button to press.
- 3Adjust any input to compare scenarios, then read the worked example to see the substituted numbers.
Why use Monte Carlo Error Calculator?
- ✓Computes Monte Carlo Error instantly in your browser — no sign-up, no upload, no server round-trip.
- ✓100% free and unlimited, with the exact formula shown: standard error = σ / √N.
- ✓Runs entirely client-side, so every value you enter stays private on your device.
- ✓Live recompute as you type, with a worked example and authoritative references for trust.
Frequently asked questions
Why does Monte Carlo error shrink only as 1/√N?+
Because the standard error of a sample mean is σ/√N — a fundamental result of averaging independent random samples. The randomness averages out slowly: doubling samples reduces error by only ~30% (1/√2). This sub-linear convergence is the central cost of simulation-based estimation and why high precision can require enormous sample counts.
How many simulations do I actually need?+
Work backward from your target precision: N = (σ / target standard error)². If the quantity's standard deviation is 1 and you want a standard error of 0.01, that's 10,000 runs; for 0.001 it jumps to 1,000,000. This calculator does the arithmetic — decide the precision your decision requires, then size the simulation to it.
How do I reduce Monte Carlo error without more samples?+
Variance-reduction techniques attack σ rather than N: importance sampling (sample more where it matters), antithetic variates (paired negatively-correlated samples), control variates (subtract a known correlated quantity), and stratification. These can cut the effective variance by orders of magnitude, achieving the same precision with far fewer runs than the brute-force √N law implies.
Is the 95% margin a confidence interval?+
Yes — for a large number of independent runs, the estimate is approximately normal by the Central Limit Theorem, so ±1.96 standard errors is a 95% confidence interval around your Monte Carlo estimate. Always report it: a simulation result without its error bar hides whether you've run enough samples to trust the digits you're quoting.
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