Normal Distribution Properties Explained with Examples

The normal distribution, also called the Gaussian distribution, is the famous bell-shaped curve that describes countless natural measurements: heights of adults, measurement errors, test scores and blood pressure readings. It is symmetric about its centre, with most values near the middle and fewer far away.

It matters more than any other continuous distribution because the Central Limit Theorem pushes averages toward normality, because many statistical tests assume it, and because its probabilities are completely determined by just two numbers, the mean μ and the standard deviation σ.

Key Concepts

• Parameters: X ~ N(μ, σ²). The mean μ sets the centre; the standard deviation σ sets the width. Variance is σ².
• Symmetry: the curve mirrors around μ, so the mean, median and mode are all equal to μ.
• Empirical (68-95-99.7) rule: about 68% of values lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
• Density: f(x) = (1/(σ√(2π))) e^(-(x-μ)²/(2σ²)). The height never touches zero but the total area is exactly 1.
• Linearity: sums and averages of independent normal variables stay normal, which is why it survives aggregation.

Formulas

Density f(x) = (1/(σ√(2π))) × e^(-(x-μ)²/(2σ²)). Standardisation: Z = (X - μ)/σ converts any normal variable to the standard normal N(0, 1), so probabilities come from the Z table: P(a < X < b) = P((a-μ)/σ < Z < (b-μ)/σ). Assumptions: the variable is continuous, symmetric, and extreme values get rarer smoothly. It fits poorly for skewed data like incomes or counts bounded at zero.

Worked Example

Exam scores are normal with μ = 70 and σ = 10. (a) Probability a random student scores above 85: Z = (85 - 70)/10 = 1.5. From the standard normal table, P(Z > 1.5) = 1 - 0.9332 = 0.0668, about 6.7%. (b) 95% of scores lie within μ ± 2σ = 70 ± 20, i.e. between 50 and 90. (c) Score for the top 5%: need Z = 1.645, so X = 70 + 1.645 × 10 = 86.45. A score above about 86.5 is in the top 5%.

Common Misconceptions

• Normal does not mean "everything": incomes, waiting times and city populations are skewed or bounded and are not normal. Always check shape before assuming normality.
• The 68-95-99.7 rule is approximate: the exact figures are 68.27%, 95.45% and 99.73%. Using 2σ for a 95% interval is a convenient approximation, not an exact law.
• σ is spread, not error: a small σ means values cluster tightly around μ; it does not mean the data are "wrong". Normality and accuracy are different ideas.

Real-World Applications

• Measurement science: repeated measurements scatter normally around the true value; σ quantifies instrument precision.
• Quality control: part dimensions are checked against μ ± 3σ limits to catch defective production.
• Finance and testing: modelling returns, setting exam grade boundaries, and building confidence intervals.

The normal distribution is the bell curve defined by μ and σ: standardise with Z = (X - μ)/σ, read probabilities from the standard table, and use the 68-95-99.7 rule for quick estimates. Whenever data pile symmetrically around a centre, this is the first and best model to reach for.