Normal Distribution

The normal distribution, often called the bell curve, is one of the most important probability models in all of statistics. It describes how many real-world quantities—such as heights of people, measurement errors, and test scores—tend to cluster symmetrically around a central value, tapering off gradually toward the extremes. Its elegant, symmetric shape emerges naturally whenever a measured quantity is the result of many small, independent random effects added together, a fact guaranteed by the Central Limit Theorem. Mathematically, the normal distribution is completely determined by just two parameters: its mean (μ), which locates the center of the distribution, and its standard deviation (σ), which controls how spread out the values are. A larger σ produces a wider, flatter bell; a smaller σ produces a taller, narrower one. Every normal distribution, no matter its mean or standard deviation, shares the same fundamental shape and the same proportional rules about how data are spread around the center. Understanding the normal distribution is essential for making predictions, constructing confidence intervals, and performing hypothesis tests. By learning to calculate the proportion of data that falls within any given range—using z-scores and standard normal tables—you gain a powerful tool for analyzing everything from SAT scores to manufacturing tolerances to biological measurements. This article will walk you through the key properties, formulas, and practical techniques for working with normal models.

Key takeaways
  • The Bell Shape and Symmetry
  • The Empirical Rule (68-95-99.7 Rule)
  • Standardizing with z-Scores
  • The Standard Normal Distribution N(0, 1)
  • Finding Areas Between Two Values

The Bell Shape and Symmetry

A normal distribution is perfectly symmetric about its mean μ. The curve rises smoothly from near zero on the left, reaches its peak at exactly x = μ, and falls symmetrically back toward zero on the right. Because of this symmetry, the mean, median, and mode are all equal to μ. The curve never actually touches the x-axis—it is asymptotic—meaning that, in theory, values can be arbitrarily large or small, though extremely far-out values are exceedingly rare.

The Empirical Rule (68-95-99.7 Rule)

One of the most practical facts about any normal distribution is the Empirical Rule. Approximately 68% of all values fall within one standard deviation of the mean (between μ − σ and μ + σ). About 95% of values fall within two standard deviations (μ − 2σ to μ + 2σ). Nearly 99.7% of all values—almost everything—fall within three standard deviations (μ − 3σ to μ + 3σ). This rule lets you make quick, powerful estimates about how data is spread without needing a calculator.

Standardizing with z-Scores

To find the exact proportion of data below or above any value x, we convert x to a z-score: z = (x − μ) / σ. The z-score tells you how many standard deviations x lies above (positive z) or below (negative z) the mean. Once you have the z-score, you look up the corresponding cumulative probability in a Standard Normal Table (z-table), which gives the proportion of the distribution to the left of that z-value. For example, z = 1.50 corresponds to a cumulative probability of 0.9332, meaning 93.32% of the distribution falls below that point.

The Standard Normal Distribution N(0, 1)

The standard normal distribution is the special case where μ = 0 and σ = 1, written N(0, 1). Every normal distribution can be transformed into the standard normal by the z-score conversion. This is extremely useful because z-tables are built for N(0, 1), so a single table covers all possible normal distributions. The total area under any normal curve equals exactly 1, corresponding to 100% of probability.

Finding Areas Between Two Values

To find the probability that a value falls between two points a and b, compute z-scores for both: z₁ = (a − μ)/σ and z₂ = (b − μ)/σ. Look up the cumulative probabilities P(Z < z₂) and P(Z < z₁) in the z-table, then subtract: P(a < X < b) = P(Z < z₂) − P(Z < z₁). To find the probability above a value b, use P(X > b) = 1 − P(Z < z₂), because the total area under the curve is 1.

The Key Equation

f(x)=1σ2π e−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\dfrac{(x-\mu)^2}{2\sigma^2}}
  • f(x) — Probability density (per unit of x): The height of the normal curve at value x; area under the curve gives probability.
  • \mu — Mean (population) (same as x): The center of the distribution; the peak of the bell curve occurs at x = μ.
  • \sigma — Standard deviation (same as x): Controls the spread of the distribution; larger σ means wider and flatter bell.
  • e — Euler's number (dimensionless): The mathematical constant ≈ 2.71828, base of the natural logarithm.
  • \pi — Pi (dimensionless): The mathematical constant ≈ 3.14159; appears here as part of the normalization constant.

Worked Example

Problem. The heights of adult women in a certain country are approximately normally distributed with a mean of μ = 165 cm and a standard deviation of σ = 7 cm. (a) What percentage of women are shorter than 172 cm? (b) What percentage are between 151 cm and 172 cm?

P(X<172)=0.8413;P(151<X<172)=0.8413−0.0228=0.8185P(X < 172) = 0.8413 \quad;\quad P(151 < X < 172) = 0.8413 - 0.0228 = 0.8185

Answer. 84.13% of women are shorter than 172 cm; 81.85% have heights between 151 cm and 172 cm.

Figure
Diagram
Normal Distribution N(μ, σ) — Empirical Ruleμμ−σμ+σμ−2σμ+2σ68%13.5%13.5%2.35%2.35%Total area under curve = 1 (100% of probability)
Origins · 1 minCarl Friedrich Gauss and how this was discovered
CF
Carl Friedrich Gauss
1777–1855 · German

Gauss formalized the normal distribution in his 1809 work on the theory of celestial body motion, using it to model astronomical measurement errors. He showed that if errors are normally distributed, the method of least squares gives the best possible estimates—a foundational result connecting the normal curve to statistics and data analysis. The normal distribution is sometimes called the Gaussian distribution in his honor. Gauss is said to have been a child prodigy who reportedly corrected his father's payroll arithmetic at age 3. He contributed to number theory, algebra, physics, and astronomy, and Germany honored him by putting his portrait—along with the normal curve—on the 10 Deutsche Mark banknote.

  • Z-Scores and Standardization — Z-scores are the key tool for converting any normal distribution value into a standard normal value, enabling use of the z-table to find probabilities.
  • The Empirical Rule (68-95-99.7 Rule) — The Empirical Rule is the most-used shortcut for estimating probabilities within a normal distribution without needing a full table lookup.
  • Central Limit Theorem — The Central Limit Theorem explains why the normal distribution appears so widely: sample means from any distribution approach normality as sample size grows.
  • Mean and Standard Deviation — Mean (μ) and standard deviation (σ) are the two parameters that fully define every normal distribution, making them essential prior knowledge.
  • Probability Distributions — The normal distribution is one example of a broader class of probability models; understanding the general concept helps place the normal model in context.