2.7. How Many Iterations Do I Need?

(This part of this article is adapted from "How Many Trials Do We Need?" in the book Simulation Modeling Using @RISKby Wayne L. Winston [Duxbury, 2000].)

The attached example, ConfIntervalWidth2.xls, uses traditional Monte Carlo sampling. Let's suppose that we want to use simulation to estimate the mean of the output in cell B11 and be accurate within 5 units 95% of the time. The number of iterations needed to meet these requirements can be calculated using the following formula:

n = [ z_α/2 S / E ] ²

In this formula,

• n is the number of iterations needed.
• S is the estimated standard deviation of the output.
• E is the desired margin of error (in this case, 5 units). The width of the confidence interval is twice the margin of error.
• z_α/2 is the critical value of the normal distribution for α/2, the z value such that the area of the right-hand tail is α/2. It is the number that satisfies P(Z>z_α/2) = α/2, where Z follows a normal distribution with mean 0 and standard deviation 1. α/2 can be found by setting the desired confidence level equal to 100(1-α) and solving for α.

For a 95% confidence level, as shown in the attached example, 95 = 100(1-α).  Then α is 0.05 and α/2 is 0.025. To compute z_α/2 in Excel, use the NORMSINV function and enter =NORMSINV(1-α/2, 0, 1). Cell E13 of the attached example shows a Z value of approximately 1.96 for a 95% confidence interval.

To obtain an estimate for the standard deviation of the output, the @RISK statistics function RiskStdDev was placed in cell B14 and a simulation was run with just 100 iterations. This gave us a standard deviation of approximately 53.5. If we plug the above information into our formula, we get

n = [ 1.96 × 53.5 / 5 ] ² = 440

Thus, if you use Monte Carlo sampling, you should run at least 440 iterations to be 95% sure that your estimate of the mean of the output in cell B11 is accurate within ±5 units.