Although reality doesn’t always match up with our estimates, using sample data to estimate a characteristic for a larger group can be very useful, especially when we attach a margin of error to the estimate. A point estimate together with its associated margin of error gives a range of values in which we can expect the actual characteristic for the larger group to be with a certain amount of confidence. 

One way to determine a margin of error is by running a simulation based on the data from a sample to create a sampling distribution. When we double the standard deviation of the sampling distribution, we can be about 95% sure that the value for the actual population characteristic will be at most that far away from our point estimate.

For example, a group goes to an island and collects a random sample of 10 lizards, finding that 5 of them are male. This random sample has a proportion of 0.5 males in the group. How close is this likely to be to the actual proportion of lizards that are male on the island? To investigate, we can simulate taking many samples from a population in which the proportion is 0.5 and see how far away from 0.5 the sample proportions tend to be. The sampling distribution of proportions from the simulated samples will give us an idea of how far off our sample estimate of 0.5 might be from the actual population value.

Suppose we simulate taking 30 random samples of 10 lizards from a population with a 0.5 probability of each one being male, and this results in a sampling distribution that has a mean of 0.503 and a standard deviation of 0.145. The sampling distribution is approximately normal in shape, so it is reasonable to think that simulated proportions from the population should be within about 2 standard deviations, or 0.290 () of the actual population mean.

Based on the simulations and analysis, we expect that the original estimate of 0.5 for the proportion of the population that is male is likely to be within 0.290 of the actual value of the population proportion of lizards that are male. The researchers should report an estimate of 0.5 for the population proportion with an associated margin of error of 0.290.

Later, another group goes to the island and collects a sample of 40 lizards, finding that 20 of them are male. After simulating 30 samples of 40 lizards with a 0.5 probability of each one being male, the sampling distribution has a mean proportion of 0.503 again, but the standard deviation is 0.062. This group should report an estimated proportion of lizards on the island that are male of 0.5 with a margin of error of 0.124 (). This means that they believe their estimate of 0.5 is within 0.124 of the actual population proportion.

In general, the standard deviation of sampling distributions tends to be less with larger samples, so the margin of error reported is less.