Free lesson · Statistics

Law of large numbers: what averaging can and cannot establish

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Start here · the same idea in everyday words

Unpack a technical sentence

Technical wording: The sample mean is consistent under iid integrability.

Read it as: If repeated observations follow the required stable sampling rule and have a finite average absolute size, their average becomes more likely to be close to the true mean as the sample grows.

A slower worked example · one step at a time

1. A synthetic trade pays either $3 or −$1 with equal probabilities. Its population mean is .5 times 3 plus .5 times −1, or $1 per trade.
2. A sample of two wins has mean $3. A sample of one win and one loss has mean $1. Both are possible from the same unchanged process.
3. For a sample [3,−1,3,−1], add the outcomes to get $4 and divide by four trades to get $1 per trade.
4. Add another loss. The new sum is $3 across five trades, so the sample mean becomes $.60 per trade. More observations did not move this particular sample closer to $1.
5. The theorem concerns the probability of an estimation error over repeated samples as size grows. It does not say each added observation must improve the estimate. Real data also require checks for dependence and changing distributions.

Study loop: predict the result, work through the arithmetic, run the Python, then explain which assumption would make the result unreliable.

• Start with repeated net P&L observations measured in dollars per trade. The arithmetic mean adds them and divides by the number observed.
• The weak law of large numbers says that for independent, identically distributed observations with a finite absolute first moment, the sample mean converges in probability to the population mean. Convergence in probability means the chance of a fixed-size estimation error tends to zero.
• It does not promise a particular finite-sample error or a monotone path toward the answer. Dependent data need a different set of conditions, such as an appropriate ergodic theorem. Repeating a biased measurement does not remove its bias.

An analogy to remember

Repeated measurements can reduce sampling noise around the same target; a miscalibrated ruler keeps its systematic error.

How this becomes a building block

A simulation engine uses repeated scenario values to estimate an expectation. Consistency belongs to the sampling procedure and assumptions.

Python · run and reuse

Architecture pseudocode

# Python 3.10+; standard library only unless NumPy is imported.
# Synthetic teaching inputs; no historical performance claim.
from math import isfinite

def sample_mean(values):
    values = list(values)  # Same horizon and units for every observation.
    if not values or not all(isfinite(x) for x in values):
        raise ValueError("Nonempty finite sample required")
    return sum(values)/len(values)

print(sample_mean([-2,1,3,2]))

# Replay the slower, synthetic example above. Units are stated in its steps.
def plain_language_example():
    from math import exp, e  # Standard library; no extra dependency.
    return {
        "population": .5*3+.5*(-1),
        "four_mean": (3-1+3-1)/4,
        "five_mean": (3-1+3-1-1)/5,
    }

print(plain_language_example())

# Python 3.10+; standard library only unless NumPy is imported.
# Synthetic teaching inputs; no historical performance claim.
from math import isfinite

def sample_mean(values):
    values = list(values)  # Same horizon and units for every observation.
    if not values or not all(isfinite(x) for x in values):
        raise ValueError("Nonempty finite sample required")
    return sum(values)/len(values)

print(sample_mean([-2,1,3,2]))

# Replay the slower, synthetic example above. Units are stated in its steps.
def plain_language_example():
    from math import exp, e  # Standard library; no extra dependency.
    return {
        "population": .5*3+.5*(-1),
        "four_mean": (3-1+3-1)/4,
        "five_mean": (3-1+3-1-1)/5,
    }

print(plain_language_example())