Calculator Inputs
Choose raw simulation outputs or summary statistics. Results appear above this form after submission.
Convergence Plot
The graph shows how error changes as simulation count grows. Stable decline generally suggests improving Monte Carlo precision.
Example Data Table
This sample illustrates a small run of Monte Carlo outputs and the resulting running precision pattern.
| Iteration | Simulation Output | Running Mean | Approx. Running Error |
|---|---|---|---|
| 1 | 12.400 | 12.400 | 0.0000 |
| 2 | 11.800 | 12.100 | 0.3000 |
| 3 | 12.100 | 12.100 | 0.1732 |
| 4 | 12.700 | 12.250 | 0.1936 |
| 5 | 11.900 | 12.180 | 0.1655 |
| 6 | 12.300 | 12.200 | 0.1366 |
| 7 | 12.000 | 12.171 | 0.1190 |
| 8 | 12.500 | 12.213 | 0.1109 |
Formula Used
Monte Carlo error measures the uncertainty in a simulation-based estimate. The most common quantity is the Monte Carlo standard error.
Mean = (x1 + x2 + ... + xn) / n
s = √[ Σ(xi - x̄)2 / (n - 1) ]
MCSE = s / √n
Margin = z × MCSE
x̄ ± z × MCSE
n = (z × s / target error)2
Lower MCSE means higher precision. Doubling accuracy often requires many more iterations because error falls roughly with 1 / √n.
How to Use This Calculator
- Choose Raw simulation outputs when you already have generated values.
- Choose Summary statistics when you only know mean, standard deviation, and run count.
- Set the confidence level to match your reporting standard.
- Enter a target margin of error to estimate required iterations.
- Optionally provide a reference value to inspect simulation bias.
- Press Calculate Error to show results above the form.
- Review the plot to judge convergence and declining error.
- Export the summary using the CSV or PDF buttons.
Frequently Asked Questions
1) What does Monte Carlo error mean?
Monte Carlo error is the uncertainty caused by using a finite number of simulation draws. It tells you how much the estimate may vary purely from random sampling.
2) Is Monte Carlo error the same as model bias?
No. Monte Carlo error reflects random simulation noise. Bias reflects systematic difference between the simulation estimate and a true or reference value.
3) Why does error shrink slowly?
Error usually decreases with the square root of sample size. To cut error in half, you often need about four times as many iterations.
4) When should I use raw outputs?
Use raw outputs when you have individual simulated values. This allows richer diagnostics such as percentiles, minimum, maximum, and running convergence behavior.
5) What is a good Monte Carlo standard error?
A good MCSE depends on the decision context. It should be small relative to the estimate or smaller than your acceptable practical tolerance.
6) Why include a confidence interval?
The interval translates sampling uncertainty into a range around the estimated mean. It helps communicate precision in a familiar and interpretable format.
7) Can this calculator estimate required iterations?
Yes. If you provide a target margin of error, the calculator estimates how many iterations are needed to meet that precision level.
8) What does the convergence plot show?
The plot tracks how estimated error changes with more iterations. A steady downward trend usually indicates that the simulation estimate is stabilizing.