The Normal Distribution

A model for a distribution, not a default assumption

A normal distribution is a continuous, symmetric bell-shaped probability model. Its mean μ locates the center; its positive standard deviation σ controls the spread. We write X ~ N(μ, σ²), using the second parameter for variance in this lesson. Some software instead asks for SD, so check the convention.

Measurement errors and some biological measurements can be approximately normal in a suitably defined population. Income, counts, bounded scores, waiting times, and mixtures of groups often are not. The model is useful when it fits the question and data, not because all data ought to be normal.

Probability is an area

The density is:

$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$

The curve’s total area is 1. The probability of an interval is the area over that interval, not the curve’s height at a point. A density can exceed 1 when the distribution is very concentrated; its total area still equals 1.

For a continuous model, the probability of one exact value is zero. Measurements rounded to the nearest minute represent small intervals, so a recorded value can have a nonzero observed frequency.

The empirical rule and its limits

For a normal distribution:

Region	Approximate probability
μ − σ to μ + σ	68.27%
μ − 2σ to μ + 2σ	95.45%
μ − 3σ to μ + 3σ	99.73%

The familiar 68–95–99.7 rule rounds these values. A central 95% interval uses approximately μ ± 1.9600σ, not exactly μ ± 2σ. Also distinguish an interval for individual values from a confidence interval for an estimated mean.

A dataset with the same mean and SD can have entirely different tail probabilities if its shape is not normal.

Standardize, then choose the required area

If X is normal, Z = (X − μ)/σ is standard normal with mean 0 and SD 1. The cumulative function Φ(z) = P(Z ≤ z) gives the area to the left.

$P(X\le x)=\Phi\left(\frac{x-\mu}{\sigma}\right)$

The site’s z-table uses this cumulative convention. Other printed tables may give the area between zero and z or the upper tail; read their headings.

Worked example: bottle fill volumes

Assume bottle volume is modeled as normal with mean 500 mL and SD 5 mL. These are hypothetical model inputs, not measurements from a real factory.

Below 490 mL: z = (490 − 500)/5 = −2. Thus P(X < 490) = Φ(−2) ≈ 0.0228, or 2.28%.

Above 507.5 mL: z = 1.5. Thus P(X > 507.5) = 1 − Φ(1.5) ≈ 0.0668, or 6.68%.

Between 495 and 507.5 mL: the standardized endpoints are −1 and 1.5. Subtract cumulative probabilities:

$P(495<X<507.5)=\Phi(1.5)-\Phi(-1)=0.9332-0.1587\approx0.7745$

Using unrounded cumulative values gives 0.7745375. Small last-digit differences arise when you subtract rounded table entries.

Work backward: find a cutoff

To find a value below which 90% of volumes lie, use the 90th-percentile z-score, about 1.2816:

$x_{0.90}=500+1.2816(5)=506.408\text{ mL}$

“Top 10%” means the 90th percentile, not the 10th. Check whether the requested percentage is above or below the cutoff before using an inverse normal function.

Inspect the distribution you actually need

Use a histogram to see skew, clusters, and outliers. A Q–Q plot compares observed quantiles with normal-model quantiles; strong curvature or tail departures suggest a poor normal fit. Small samples cannot establish a shape reliably, and large samples can detect departures too small to matter for a particular procedure.

The relevant normality assumption depends on the analysis:

• For probabilities of individual measurements, the outcome distribution matters.
• For a paired t-test, the distribution of within-pair differences matters.
• For regression’s small-sample t and F inference, the errors conditional on predictors matter; the predictors need not be normal.

A normality test alone is not a complete model check. Study design, independence, influential observations, and the practical consequences of a departure also matter.

Why the central limit theorem does not normalize your data

For independent identically distributed observations with finite variance, the standardized sample mean approaches a normal distribution as sample size grows. This concerns repeated-sample means, not the original observations.

The approximation’s accuracy depends on the population’s shape and tails. There is no universal sample size that makes every population safe to treat as normal. Read sampling distributions before applying normal approximations to means.

Practice

1. Under X ~ N(100, 15²), what is P(X > 130)?

Show the solution

z = (130 − 100)/15 = 2. The requested area is above the cutoff: 1 − Φ(2) ≈ 0.0228, or 2.28%.

2. A normal model has mean 20 and SD 4. Find its 95th percentile using z = 1.6449.

Show the solution

x = 20 + 1.6449 × 4 = 26.5796. About 95% of modeled values fall below this cutoff and 5% above it. It is not the upper endpoint of a central 95% interval, which uses z ≈ 1.9600.

Reference and tools

NIST’s normal distribution reference defines the density, cumulative function, and standard form. Use the z-score calculator for standardization and the z-table for areas.