Statistics Continuous Normal Distribution

What Is a Normal Distribution?

The normal distribution is a continuous probability distribution shaped like a symmetric bell curve, centered on its mean. It's fully described by two parameters: the mean (μ), which sets the center, and the standard deviation (σ), which sets the spread. Heights, measurement errors, and test scores frequently follow this shape closely.

The Normal Probability Density Function

The formula is f(x) = (1 / (σ√(2π))) × e^(−(x − μ)² / (2σ²)). Because the distribution is continuous, the probability of any single exact value is technically zero — what matters is the area under the curve between two points, which represents the probability of landing in that range.

import math

mu = 175     # mean height in cm
sigma = 7    # standard deviation in cm
x = 175      # the point we're evaluating (the mean itself)

pdf = (1 / (sigma * math.sqrt(2 * math.pi))) * math.exp(-((x - mu) ** 2) / (2 * sigma ** 2))
print(f"f({x}) = {pdf:.4f}")  # about 0.0570, the peak density of the curve

Standardizing with Z-Scores

A z-score converts any normal value into units of standard deviation from the mean: z = (x − μ) / σ. Once standardized, every normal distribution can be compared against the same standard normal table (mean 0, standard deviation 1), which is what makes z-scores so useful for looking up probabilities.

import math

mu = 175
sigma = 7
x = 189

z = (x - mu) / sigma
print(f"z-score = {z}")  # 2.0

def standard_normal_cdf(z):
    return 0.5 * (1 + math.erf(z / math.sqrt(2)))

percentile = standard_normal_cdf(z)
print(f"P(X <= {x}) = {percentile:.4f}")  # 0.9772, roughly the 97.7th percentile

• About 68% of values fall within 1 standard deviation of the mean (μ ± σ).
• About 95% of values fall within 2 standard deviations (μ ± 2σ).
• About 99.7% of values fall within 3 standard deviations (μ ± 3σ).

The normal distribution underlies standardized test scoring, manufacturing tolerance checks, and any measurement process where small random errors accumulate from many independent sources, producing that familiar bell shape.