Normal Distribution

The probability density function

The normal distribution with mean μ and variance σ² has density:

f(x) = (1 / (σ · √(2π))) · exp(−(x − μ)² / (2σ²))

The bell curve is symmetric about μ, with width controlled by σ. The factor 1/(σ·√(2π)) ensures the integral over all x is 1 (a valid probability density).

The standard normal distribution N(0, 1) — mean 0, variance 1 — has the simpler form:

φ(x) = (1/√(2π)) · exp(−x²/2)

Any normal X ~ N(μ, σ²) can be transformed to standard normal Z by:

Z = (X − μ) / σ

Z follows the standard normal distribution. This transformation is called standardization or computing a Z-score.

The 68-95-99.7 rule

For a normal distribution, the proportion of values within k standard deviations of the mean is approximately:

Range	Probability	Useful for
Within 1σ (μ − σ, μ + σ)	68.3%	"Typical" range
Within 2σ	95.4%	"Almost all" — common alpha = 0.05 cutoff
Within 3σ	99.7%	Six Sigma quality (rare deviations)
Within 4σ	99.9937%	Very rare — 1 in 16,000
Within 5σ	99.99994%	Particle physics discovery threshold
Within 6σ	99.9999998%	1 in 500 million — Six Sigma quality

Useful for quick mental statistics. If you know μ and σ, you can estimate the probability of any value range.

Why the normal appears everywhere

The Central Limit Theorem — sums of many independent random variables tend toward normal, regardless of the originating distributions (provided they have finite mean and variance).

Concrete examples:

• Heights. A height is determined by many genes plus environmental factors. The combined effect, by CLT, is approximately normal. Adult human heights follow a normal distribution closely.
• Test scores. A score is a sum of many question results. Each is binary; the sum becomes normal for many questions. Standardized tests are designed to produce normal-shaped score distributions.
• Measurement errors. Each measurement error is a sum of many small independent factors. Their cumulative effect is normal — Gauss derived the formula assuming this.
• Brownian motion. A particle's position is the sum of many small random kicks (collisions with surrounding molecules). Position over time is normally distributed.
• Sample means. Even from a non-normal underlying distribution, the average of n samples is approximately normal for n large enough (n ≥ 30 is the rule of thumb). This is why hypothesis testing for sample means uses normal-based math.

Worked examples

Example 1 — IQ scores (μ=100, σ=15)

P(IQ > 130)? IQ 130 is 2σ above mean. P(Z > 2) = (1 − P(Z ≤ 2)) ≈ (1 − 0.9772) = 0.0228 = 2.28%.

P(IQ < 70)? Symmetrically, 2σ below — same 2.28%.

P(IQ in [85, 115])? Within 1σ — 68.3%.

Example 2 — Quality control

A factory produces bolts with mean diameter 10 mm, σ = 0.1 mm. Bolts outside 9.7 to 10.3 mm fail inspection. What fraction fail?

9.7 is 3σ below; 10.3 is 3σ above. By the 99.7% rule, 99.7% pass. So 0.3% fail.

If you tightened to ±2σ (9.8 to 10.2), 95.4% pass — 4.6% fail. Tighter standards mean more rejects.

Example 3 — Z-score in hypothesis testing

Average human heart rate is 72 bpm. Standard deviation 12 bpm. Someone has heart rate 90 bpm. How unusual?

Z = (90 − 72)/12 = 1.5σ above mean. P(Z > 1.5) ≈ 0.0668 = 6.68%. Not extreme; about 1 in 15 people have heart rates above 90 bpm.

JavaScript — normal distribution functions

// Probability density function
function normalPDF(x, mu = 0, sigma = 1) {
  const z = (x - mu) / sigma;
  return Math.exp(-z*z/2) / (sigma * Math.sqrt(2 * Math.PI));
}

// Cumulative distribution function — using error function approximation
function normalCDF(x, mu = 0, sigma = 1) {
  const z = (x - mu) / sigma;
  return 0.5 * (1 + erf(z / Math.SQRT2));
}

// Error function — approximation accurate to ~6 decimals
function erf(x) {
  const sign = x >= 0 ? 1 : -1;
  x = Math.abs(x);
  const t = 1 / (1 + 0.3275911 * x);
  const a1 =  0.254829592;
  const a2 = -0.284496736;
  const a3 =  1.421413741;
  const a4 = -1.453152027;
  const a5 =  1.061405429;
  const y = 1 - (((((a5*t + a4)*t) + a3)*t + a2)*t + a1)*t * Math.exp(-x*x);
  return sign * y;
}

// Generate samples — Box-Muller transform
function normalSample(mu = 0, sigma = 1) {
  const u1 = Math.random();
  const u2 = Math.random();
  const z = Math.sqrt(-2 * Math.log(u1)) * Math.cos(2 * Math.PI * u2);
  return mu + sigma * z;
}

// Verify 68-95-99.7
const samples = Array.from({length: 100000}, () => normalSample(0, 1));
const within1 = samples.filter(z => Math.abs(z) < 1).length / samples.length;
const within2 = samples.filter(z => Math.abs(z) < 2).length / samples.length;
const within3 = samples.filter(z => Math.abs(z) < 3).length / samples.length;
console.log(within1, within2, within3);  // ≈ 0.683, 0.954, 0.997

When to use the normal distribution

• Sums of many factors. Heights, weights, test scores, measurement errors, sample means — anything that's a sum of many small contributions tends toward normal.
• Hypothesis testing. Z-tests, t-tests, ANOVA — all assume normality (or normality of the test statistic via the Central Limit Theorem).
• Quality control. Process variation modeling, control charts, Six Sigma. Manufacturing aims for normal-distributed deviations from target.
• Finance. Daily stock returns are approximately normal in central regions (though heavy tails complicate this; modern finance uses fat-tailed models).
• Machine learning. Many algorithms (linear regression assumptions, Gaussian mixtures, Gaussian processes) are normal-based.

When NOT to use — heavy-tailed distributions (income, file sizes), bounded distributions (probabilities, percentages), counts (use Poisson), small sample sizes where the normality assumption is unclear.

Common mistakes

• Assuming normality without checking. Many real distributions look "normal-ish" but aren't. Check with a histogram, Q-Q plot, or Shapiro-Wilk test before applying normal-based methods.
• Confusing standard deviation with standard error. SD describes spread of individual values. SE describes spread of the sample mean (= SD/√n). For confidence intervals on means, you use SE; for individual prediction intervals, SD.
• Treating the 68-95-99.7 rule as a 100% guarantee. For non-normal distributions, the rule doesn't hold. Heavy-tailed distributions have more extreme values than normal predicts.
• Wrong interpretation of "outside 3σ is rare." Rare under normal assumption. Real-world distributions often have many more extreme values than normal predicts. Black Swan events (Taleb) are about the heavy tail of the actual distribution vs the normal idealization.
• Using normal CDF without standardizing. The CDF table is for the standard normal N(0, 1). For X ~ N(μ, σ²), compute Z = (X−μ)/σ first, then look up Φ(Z).
• Forgetting that variance is in squared units. Var(height in meters) is in m². For interpretation, use σ = √Var (in meters). Confusing the two leads to wrong magnitudes.