What happened

Across 20,000 independent samples at each size, relative error shrank as sample size increased. The CSV reports count and proportion deviations separately.

For fair tosses, the heads count has standard deviation √N/2, while the heads proportion has standard deviation 1/(2√N). More observations can therefore increase typical absolute count error and decrease typical relative error simultaneously.

**Figure 1.** 20,000 independent samples at each size. The observed mean absolute error in a proportion is distinct from its theoretical standard deviation; both shrink on this logarithmic display. The samples are separate replicates, not a single monotone trajectory.
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Method and benchmark

Generate 20,000 independent binomial counts at each sample size. Report mean absolute deviations and theoretical standard deviations separately; they are different summaries.

The observed summaries are mean absolute deviations. The theoretical reference columns are standard deviations. These are different measures of spread, so the plotted curves are not expected to lie exactly on top of one another.

Reproduction record

Python 3.12.14 · NumPy 2.3.5 · NumPy PCG64 · seed 20260932. Each experiment starts a separate stream. Code and all datasets are linked below.

What this does not establish

Different sample sizes use independent replicates, not one continuing trajectory. Average relative error shrinks; no individual path is forced to improve at every step.

Samples at each size are independently generated, not nested portions of a single continuing path. The experiment illustrates scaling across repeated samples. It does not show that every next toss improves the running proportion or that the future compensates for a past imbalance.

These are original educational simulations prepared for RoomForChance. They are not historical lottery records, a physical-machine audit, an external peer review or evidence of a prediction advantage.

Inspect the data

Download law-large-numbers.csv · All results and metadata (JSON) · Download the complete Python program · Download figure-generation code

Published aggregate results · synthetic data
tosses	mean absolute count error	mean absolute proportion error	count standard deviation	proportion standard deviation
20	1.7653	0.088265	2.23606798	0.111803399
100	4.012	0.04012	5	0.05
1,000	12.63405	0.01263405	15.8113883	0.0158113883
10,000	40.0128	0.00400128	50	0.005

Reproduce the experiment

Download the Python program to an empty working folder. Use the recorded environment for an exact replay. The program runs all eleven studies and creates a lab-data folder containing the result files. The largest study performs ten million draws; allow time for it to finish.

python -m pip install numpy==2.3.5
python run_experiments.py

Expected CSV SHA-256: 848d08f460183469f29ec9ad040c8e54b8ff7840e71b25cf19f3a1b93d5d2bcf. A matching seed alone is insufficient if you change the implementation or call sequence. The CSV files use CC BY 4.0; the original code uses the MIT license included with the downloads.

Read the reasoning

The Law of Large Numbers Does Not Make a Losing Number Due
See how sample proportions stabilize without future compensation, and why absolute deviations can grow while relative errors shrink.

Random Does Not Mean Even: Why Frequencies Fluctuate
Why fair random numbers form unequal counts, how variation scales with sample size, and why the law of large numbers does not force compensation.

Technical references

NumPy PCG64 documentation describes the generator family. Harvard Stat 110 provides university-level probability background. The model-specific derivation is linked above.

Closer in proportion, further in count: the law of large numbers