Simulating Proportion Distributions Across Sample Sizes

Why Simulate Proportion Distributions?

A sample proportion varies from sample to sample, even when every sample is drawn from the same population. A simulation makes that variation visible: repeatedly draw samples, calculate a proportion for each one, and examine the resulting collection of proportions. Repeating the process at several sample sizes helps show what changes when each sample contains more observations.

To make a fair comparison, keep the population and sampling method fixed. In this tutorial, the hypothetical population has two outcomes: success and failure. A randomly selected individual is a success with probability \(p=0.40\), and each draw is made with replacement. Thus the chance of success is the same on every draw, including when a sample is larger.

For one simulated sample, if \(X\) is the number of successes among its \(n\) observations, its sample proportion is

The simulation produces \(R\) values, which we can label \(\hat{p}_1,\hat{p}_2,\ldots,\hat{p}_R\). Their average describes the simulated center, and their sample standard deviation describes the simulated spread:

Here \(\bar{\hat{p}}\) is the average of the \(R\) simulated proportions. It is a summary of the simulation output, not the population proportion itself. In this model, the theoretical center is \(p=0.40\), and the standard deviation of the sampling distribution is \(\sqrt{p(1-p)/n}\). These provide benchmarks for judging the simulation; finite simulation results will not generally match them exactly.

Building a Simulation

One simple way to simulate a binary outcome is to generate a random number uniformly between 0 and 1 for each draw. Record a success if the number is at most 0.40, and a failure otherwise. Repeating this rule generates outcomes with the required success probability. To simulate one sample, repeat the draw rule \(n\) times; to simulate the distribution, repeat the whole sample-generation process \(R\) times.

For a comparison across sample sizes, hold \(p\), the sampling rule, and \(R\) fixed, and change only \(n\). The repeated runs are simulated versions of repeated sampling, not additional observations combined into one large sample. Each run produces its own proportion.

Summarizing Repeated Simulations

A single simulated sample shows one possible outcome, but the collection of many samples is what reveals the distribution. For example, suppose six simulated samples of size 10 produce success counts 2, 3, 3, 4, 4, and 5. Dividing each count by 10 gives six simulated proportions. We can calculate their average and spread just as we would for any list of quantitative values.

Comparing Sample Sizes

Under the fixed model with \(p=0.40\), the sampling distribution has center \(p\) and standard deviation \(\sqrt{p(1-p)/n}\). Increasing \(n\) therefore leaves the benchmark center unchanged while reducing the benchmark spread. Simulation provides a visual and numerical check of these patterns. The particular simulated values fluctuate, so the center need not equal 0.40 exactly and the simulated standard deviation need not equal its benchmark exactly.

The next example uses 500 simulated samples at each of three sample sizes. The figures are hypothetical simulation output, rounded to four significant figures. Their small differences from the benchmarks are plausible simulation variation.

Reading the Shapes of Simulated Distributions

A histogram of the simulated proportions adds information that the mean and standard deviation alone cannot show. Use the same horizontal scale when comparing sample sizes. Then a narrower distribution is visibly narrower in proportion units, rather than appearing narrow only because the graph's scale changed. Keep the population probability fixed as well, so that the comparisons isolate the effect of changing \(n\).

Sample proportions are discrete: in one sample of size \(n\), the success count must be an integer from 0 to \(n\). Therefore the possible proportions are \(0/n,1/n,\ldots,n/n\), and adjacent possible values differ by \(1/n\). For example, with \(n=10\), the grid spacing is 0.1; with \(n=40\), it is 0.025. This follows directly by subtracting proportions for consecutive success counts:

Thus, a larger sample size gives more possible proportion values and finer spacing, while its sampling distribution is also less spread out. A histogram with a small number of simulated runs may look uneven simply because there are few runs to display. Increasing \(R\) tends to make the histogram a more stable depiction of the model's sampling distribution; it does not itself make each sample proportion less variable.

For the fixed population used here, compare histograms on one common horizontal axis centered near 0.40. At \(n=10\), the proportions can only fall at tenths, so the histogram has relatively coarse possible locations. At larger \(n\), the possible locations are more closely spaced, and the distribution is more concentrated near its center. The simulated shape is still subject to random variation, especially when the number of repetitions is limited.

Why the Comparison Matters

Simulation is useful when a theoretical description feels abstract or when we want to check how the model behaves in repeated samples. It lets us see the variability that remains even when the population is fixed and the sampling process is repeated consistently. It also makes a crucial distinction visible: increasing the sample size changes the distribution of individual sample proportions, whereas increasing the number of repetitions improves how clearly the simulation displays that distribution.

Always report the model and the simulation settings alongside the output: the population success probability, how sampling was performed, the sample sizes, and the number of repetitions. Without these details, a simulated mean or histogram cannot be interpreted reliably. Simulation output is evidence about a specified model, not a substitute for stating what the model assumes.

Common Mistake

Wrong reasoning: “The simulation used 500 repetitions, so each simulated sample effectively has 500 observations and the proportions should have much smaller spread.”

Correction: Each repetition is a separate sample of size \(n\), and its proportion is calculated from that sample alone. The 500 repetitions provide 500 values with which to summarize the distribution. They do not change the sample size within a run. To reduce the model's standard deviation of sample proportions, increase \(n\); to stabilize the simulation's estimated summaries, increase \(R\).