Sigma in Practice: IQ, Heights, Test Scores

"The σ lens is most powerful where the bell-curve assumption is closest to being true. IQ, body measurements, and well-designed test scores are the textbook canon for a reason."

IQ: The Classic Calibrated Bell

IQ scores are constructed, by design, to follow a normal distribution with mean 100 and σ = 15. The construction is the point — psychometricians chose normalizations that produce this shape. It is not accidental that the IQ distribution looks like a bell; it is engineered to look like one.

What this gives you:

IQ 115 → z = +1 → top ~16% of the population.
IQ 130 → z = +2 → top ~2.5% — typical 'gifted' threshold.
IQ 145 → z = +3 → top ~0.13% — very rare.
IQ 160 → z = +4 → top ~0.003% — about 1 in 30,000.

The bell-curve assumption is not perfectly true for IQ at the extreme ends (the tails depart from normality), but for the range −2σ to +2σ it is close enough that the rule-of-thumb interpretations work.

Adult Heights: A Naturally Bell-Shaped Quantity

Adult human heights within a single population, single sex, single age range are remarkably normal. Korean adult male heights, for instance, hover around a mean of ~174cm with σ ≈ 7cm. The bell is not engineered here — it is the natural consequence of human height being the sum of many small genetic and developmental factors. The CLT is doing its job in biology.

So a Korean man at 188cm is z = +2 — clearly tall, top ~2.5% of the population. At 195cm, z = +3 — visibly unusual. At 210cm (basketball-tall), z ≈ +5 — exceptionally rare.

Test Scores: Engineered to Be Read in σ

Well-designed standardized tests (SAT, GRE, TOEFL, college entrance exams) are deliberately constructed so the raw scores can be converted into normalized scores with a known mean and σ. The point is to make 'top X%' interpretations meaningful. The SAT, for instance, is reported on a scale where the percentile rank of a score corresponds directly to the bell's CDF.

When tests are not designed for this — informal classroom tests, ad-hoc surveys — assuming the bell can mislead. The σ lens still works as a rough heuristic, but the precision degrades.

The Common Thread

The σ lens is calibrated for quantities whose distribution is genuinely bell-shaped — either by natural aggregation (heights) or by deliberate engineering (IQ, standardized tests). For these, σ-based interpretations are operationally meaningful. For quantities whose distribution is NOT bell-shaped (income, asset returns, file sizes, etc.), the same σ vocabulary applies arithmetically but the rule-of-thumb interpretations break. The first lens-skill is knowing which kind of quantity you're holding.

Code

σ-based percentile reading for three calibrated domains·python

import numpy as np
from math import erf, sqrt

# Percentile of a normal value given (mean, sigma).
def pct_below(value, mean, sigma):
    z = (value - mean) / sigma
    return 0.5 * (1 + erf(z / sqrt(2)))

# Convert real values to percentiles in their respective distributions.
cases = [
    ("IQ",            (100, 15),  [115, 130, 145, 160]),
    ("Korean male height (cm)", (174, 7), [181, 188, 195, 210]),
    ("SAT total score", (1050, 200), [1250, 1450, 1600]),
]

for label, (mu, sigma), values in cases:
    print(f"\n{label}: mu={mu}, sigma={sigma}")
    for v in values:
        p = pct_below(v, mu, sigma)
        z = (v - mu) / sigma
        print(f"  value={v:>6}  z={z:+.2f}  percentile≈{p*100:6.3f}%")

# Same conversion machinery; different domains.
# The σ lens is universal for any quantity whose distribution is approximately
# normal. For non-normal distributions, the percentile→z translation fails and
# you need a different lens (Track 05's t-distribution, or non-parametric tools).

Exercise

Take one achievement you've quantified for yourself (a personal best, a test score, a competition placement). Estimate the mean and σ of that quantity for a reasonable peer group. Compute your z-score. Then translate it into a 'top X%' statement. The exercise is meant to feel a little awkward — most people never think of their own accomplishments in σ-units, but doing so makes peer comparison instantly tractable.

Hint

If you don't know the σ exactly, estimate it from extremes you've seen. The width between the best and the median in a peer group is often around 2-3σ.