Estimating Uncertainty and Confidence Intervals
When you use simulation to estimate outcomes, you are not just interested in the average or expected value—you also want to understand how much those results might vary. This is called uncertainty estimation. In Monte Carlo simulation, every run produces slightly different results due to randomness. To quantify how much you can trust your simulation result, you use a confidence interval. A confidence interval gives you a range of values that likely contains the true mean (or another parameter) you are estimating, with a specified level of confidence, such as 95%. This helps you express not just the result, but also the reliability of your simulation.
123456789101112131415161718192021222324252627import numpy as np from scipy import stats # Simulate rolling a fair six-sided die 100 times, repeated for 1000 simulations num_simulations = 1000 num_rolls = 100 means = [] for _ in range(num_simulations): rolls = np.random.randint(1, 7, size=num_rolls) means.append(np.mean(rolls)) # Calculate the mean of means sample_mean = np.mean(means) # Calculate the standard error of the mean sem = stats.sem(means) # Compute the 95% confidence interval confidence = 0.95 h = sem * stats.t.ppf((1 + confidence) / 2, num_simulations - 1) ci_lower = sample_mean - h ci_upper = sample_mean + h print(f"Estimated mean of dice rolls: {sample_mean:.3f}") print(f"95% confidence interval: ({ci_lower:.3f}, {ci_upper:.3f})")
To calculate a confidence interval for your simulation results, you first collect the mean outcome from each simulation run. The standard error of the mean (SEM) measures how much these means vary from one simulation to another. Using the SEM and the t-distribution, you calculate a margin of error, which is added and subtracted from your overall sample mean to create the confidence interval. In the code above, after performing 1000 simulations of 100 dice rolls each, you find the mean of all simulation means. The 95% confidence interval provides a range that, if you repeated the entire simulation process many times, would contain the true average dice roll about 95% of the time. This interval reflects the uncertainty in your estimate due to the random nature of simulation, so you can report not just a single number but a range that quantifies your confidence in the result.
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Can you explain why the t-distribution is used instead of the normal distribution?
How would the confidence interval change if I increased the number of simulations?
What does the standard error of the mean tell us in this context?
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When you use simulation to estimate outcomes, you are not just interested in the average or expected value—you also want to understand how much those results might vary. This is called uncertainty estimation. In Monte Carlo simulation, every run produces slightly different results due to randomness. To quantify how much you can trust your simulation result, you use a confidence interval. A confidence interval gives you a range of values that likely contains the true mean (or another parameter) you are estimating, with a specified level of confidence, such as 95%. This helps you express not just the result, but also the reliability of your simulation.
123456789101112131415161718192021222324252627import numpy as np from scipy import stats # Simulate rolling a fair six-sided die 100 times, repeated for 1000 simulations num_simulations = 1000 num_rolls = 100 means = [] for _ in range(num_simulations): rolls = np.random.randint(1, 7, size=num_rolls) means.append(np.mean(rolls)) # Calculate the mean of means sample_mean = np.mean(means) # Calculate the standard error of the mean sem = stats.sem(means) # Compute the 95% confidence interval confidence = 0.95 h = sem * stats.t.ppf((1 + confidence) / 2, num_simulations - 1) ci_lower = sample_mean - h ci_upper = sample_mean + h print(f"Estimated mean of dice rolls: {sample_mean:.3f}") print(f"95% confidence interval: ({ci_lower:.3f}, {ci_upper:.3f})")
To calculate a confidence interval for your simulation results, you first collect the mean outcome from each simulation run. The standard error of the mean (SEM) measures how much these means vary from one simulation to another. Using the SEM and the t-distribution, you calculate a margin of error, which is added and subtracted from your overall sample mean to create the confidence interval. In the code above, after performing 1000 simulations of 100 dice rolls each, you find the mean of all simulation means. The 95% confidence interval provides a range that, if you repeated the entire simulation process many times, would contain the true average dice roll about 95% of the time. This interval reflects the uncertainty in your estimate due to the random nature of simulation, so you can report not just a single number but a range that quantifies your confidence in the result.
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