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

Many modern IQ scales are normed to mean 100 and standard deviation 15 in a reference population. Norming and score transformations can make the central distribution approximately normal, but observed scores are discrete, bounded by the instrument, and not guaranteed to follow an exact normal distribution. 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 heights within a narrowly defined sex, age, cohort, and population are often approximately normal in the center. Mixtures, secular trends, measurement protocols, and tails matter. Any Korean mean and standard deviation must be tied to a named survey and year; a CLT story is a motivation, not proof of the observed shape.

Given a particular reference mean and standard deviation, these heights can be converted to z-scores; percentiles should then come from that population's empirical distribution or a justified model.

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 reports empirical percentile ranks for defined cohorts. Those percentiles do not have to equal the CDF of an exact normal distribution.

When tests are not designed for this — informal classroom tests, ad-hoc surveys — assuming the bell can mislead. A z-score remains arithmetic, but its percentile meaning can be badly wrong without a justified reference distribution.

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.