What a Convergence Plot Shows
A simulation estimate is built one trial at a time. If we plot the estimate after each additional trial, the result is a running-proportion plot, often called a convergence plot. It can reveal whether an estimate is still changing substantially, whether a few early outcomes had a large influence, and how its fluctuations evolve as the number of trials increases.
The word “convergence” can invite too strong a conclusion. A trace that looks nearly horizontal over its final section has not proved that the estimate is close to the probability being estimated. The plot is one view of a random simulation, not a certificate of negligible error. To read it well, we need to know what is plotted and what behavior the simulation model predicts.
The Running Event Proportion
Suppose the goal is to estimate the probability \(p\) of an event in a chance experiment. Under the model used here, each simulated trial is independent of the others, and the event has the same probability \(p\) on every trial. Let \(I_i\) equal 1 if the event occurs on trial \(i\), and 0 otherwise. After \(m\) trials, the running estimate is the proportion of trials on which the event occurred.
Each new point uses all outcomes so far, not just the latest trial or the latest block of trials. A success moves the running estimate upward; a non-success moves it downward. The size of either change generally gets smaller as \(m\) grows, because a single new outcome is being added to a larger total.
The model assumptions matter. The interpretation and fluctuation calculations below require that trials are generated according to the intended chance model, that each trial has the same event probability \(p\), and that trial outcomes are independent. A plot alone cannot verify that the simulation was coded correctly or that these assumptions hold.
Why the Trace Fluctuates
Even when the model is fixed, the simulated outcomes vary randomly. If \(p\) is the event probability, then each indicator \(I_i\) has expected value \(p\) and variance \(p(1-p)\). Independence means that the variance of the sum of \(m\) indicators is the sum of their variances. Dividing that sum by \(m\) gives the variance of the running proportion:
The standard deviation of the running estimate, also called its Monte Carlo standard error under this model, is therefore \(\sqrt{p(1-p)/m}\). This is a typical fluctuation scale across repeated simulations of the same size. It is not a guaranteed maximum error for any one run. As the number of trials grows, the scale decreases in proportion to \(1/\sqrt{m}\): multiplying the run size by four halves this scale, rather than making it four times smaller.
This result also explains why a trace often seems to settle down. The effect of an individual outcome becomes smaller relative to the accumulated total, and the typical variation in the estimate decreases. But neither fact says that the current estimate must be close to \(p\). Randomness can produce an estimate that remains noticeably high or low for a while.
Worked Examples
Worked Example: A Fair-Coin Event
Imagine a hypothetical simulation of repeated fair-coin tosses, with “heads” as the event. Each trial is independent, and the event probability is \(p=0.5\). Suppose the running successes and proportions at selected trial counts are as follows.
| Trials \(m\) | Heads so far | Running proportion \(\hat{p}_m\) |
|---|---|---|
| 100 | 54 | 0.540 |
| 250 | 129 | 0.516 |
| 500 | 258 | 0.516 |
| 1,000 | 523 | 0.523 |
For instance, the point at 1,000 trials is calculated as \(523/1{,}000=0.523\). The estimate did not move steadily toward 0.5: it fell from 0.540 to 0.516, stayed at 0.516 at the next listed checkpoint, and then rose to 0.523. Such movement is compatible with random fluctuation.
At 1,000 trials, the model-based standard error is $$ SE=\sqrt{\frac{p(1-p)}{m}} =\sqrt{\frac{0.5(0.5)}{1{,}000}} =\sqrt{0.00025} \approx 0.0158. $$ The estimate is \(0.523-0.5=0.023\) above the model probability, about \(0.023/0.0158\approx1.46\) standard errors. That comparison gives a scale for the difference; it does not mean the estimate is exact or that all other runs will be similarly close. A plot ending at 0.523 can look reasonably settled while still ending above the known probability.
Worked Example: A Stable-Looking Rare-Event Estimate
Consider a hypothetical simulation in which an event has probability \(p=0.10\) on each independent trial. Suppose the running count is 153 events after 1,500 trials and 204 after 2,000 trials. The running proportions are $$ \hat{p}_{1{,}500}=\frac{153}{1{,}500}=0.102,\qquad \hat{p}_{2{,}000}=\frac{204}{2{,}000}=0.102. $$
The two plotted points are identical. In fact, the added block contains \(204-153=51\) events in \(2{,}000-1{,}500=500\) trials, also a proportion of \(51/500=0.102\). A viewer might describe the plotted segment as flat. Yet at 2,000 trials, the model-based standard error is $$ SE=\sqrt{\frac{0.10(0.90)}{2{,}000}} =\sqrt{0.000045} \approx 0.00671. $$ The estimate is \(0.102-0.100=0.002\) above the model probability, much smaller than this standard-error scale. That is consistent with a good estimate in this particular run, but the flat segment itself did not establish that fact. It only showed that the cumulative proportion did not change between those two checkpoints.
A different run could also have a flat segment while sitting further from 0.10. The shape of one short section of one path is not enough to certify accuracy. If the true probability were unknown, the same plot would not reveal its exact distance from the estimate.
Worked Example: More Trials Reduce Typical Variation, Not Every Error
Suppose a simulation estimates an event with probability \(p=0.25\). In one hypothetical run, the running proportions are 0.310 after 100 trials, 0.2625 after 400 trials, and 0.2531 after 1,600 trials. These correspond to 31, 105, and 405 events, respectively, so each point is consistent with the cumulative counts.
The model-based standard errors at the three run sizes are $$ SE_{100}=\sqrt{\frac{0.25(0.75)}{100}}\approx0.0433, $$ $$ SE_{400}=\sqrt{\frac{0.25(0.75)}{400}}\approx0.0217, \qquad SE_{1{,}600}=\sqrt{\frac{0.25(0.75)}{1{,}600}}\approx0.0108. $$ The run size quadruples from 100 to 400 and again from 400 to 1,600, while the standard-error scale halves each time. The observed estimate also moves closer to 0.25 in this particular run: its differences from 0.25 are 0.060, 0.0125, and approximately 0.0031.
This is an illustration, not a rule that every running proportion must move closer to \(p\) as trials accumulate. A later point can be farther from \(p\) than an earlier one. More trials reduce the typical variability across runs of a given size; they do not force a particular single trace to improve at every step.
How to Read the Shape Without Overclaiming
A running-proportion plot is most useful when read as a record of random behavior over time. Look at the vertical scale as well as the apparent smoothness: a small-looking movement on a wide axis can still represent a meaningful change in estimated probability. Note whether the trace continues to make sizeable changes late in the run, and whether early swings shrink in relative importance. These observations describe the path; they do not, by themselves, measure its distance from the unknown probability.
If \(p\) is known from an exact calculation, a horizontal reference line at \(p\) can help show whether the estimate is above or below that value and how the gap changes. When \(p\) is unknown, there is no such reference line. Repeatedly seeing the trace near one level can be reassuring about its apparent stability, but visual stability alone cannot determine whether that level is accurate.
The distinction is important: stability concerns how much the estimate changes as more simulated trials are added; accuracy concerns how close it is to the target probability. A slowly changing estimate can be inaccurate, and an estimate that is currently close can move away after additional trials. Use the run size and the model-based fluctuation scale to put the plot in context. The earlier material on Monte Carlo error gives a fuller account of using that scale to plan a run.
Common Mistake: Treating a Plateau as Proof
A common mistake is to say, “The graph has flattened out, so simulation error is negligible.” The reasoning is not valid: a plateau shows only that the running proportion changed little over the displayed segment or at the plotted resolution. It does not show that the running estimate is close to \(p\), and it does not rule out a later change. The correction is to describe the visible behavior cautiously, check the number of trials, and compare the estimate with a justified fluctuation scale or an exact probability if one is available.
Another visual trap is expecting the line to approach its limit smoothly. Because each new result is random, the running proportion may rise and fall. Under the independent, equal-probability model, the law of large numbers says that the running proportion gets close to \(p\) with probability approaching 1 as the number of trials grows without bound. This long-run statement is not a guarantee about how quickly any finite run settles, nor does a finite plot prove that the limit has effectively been reached.
Check Your Understanding
Answer these questions using the definitions and model assumptions in this tutorial.
- In a running-proportion plot, what does one point at trial count \(m\) represent?
- Under independent trials with the same event probability \(p\), write the formula for the variance of \(\hat{p}_m\).
- If the run size is multiplied by four, what happens to the model-based standard-error scale?
- A trace is flat between two displayed checkpoints. What can you conclude, and what can you not conclude, from that segment alone?
- Why can a running proportion move farther from the true probability after more trials have been added?