Normal Distribution: Z-Scores, Probabilities, and Limits

Mean controls location; standard deviation controls spread

A normal distribution has a bell-shaped density that is symmetric around its mean. Its mean, median, and mode coincide. The standard deviation must be positive and determines how spread out the distribution is. NIST provides the formal density, cumulative distribution, and parameter definitions. ^[1]

The vertical height of the density curve is not the probability of one exact measurement. For a continuous model, probability corresponds to area over an interval. The area under the entire curve is one.

Real observations can be rounded, censored, bounded, skewed, or mixtures of different groups. A bell-shaped drawing is a model choice, not evidence that those observations are normal.

Convert a value to a z-score

For a value x, population mean μ, and standard deviation σ:

z = (x − μ) ÷ σ

A positive z-score lies above the mean; a negative one lies below it. A z-score of 2 means two standard deviations above the mean, regardless of the original measurement unit.

Suppose an illustrative measurement follows a normal model with mean 100 and standard deviation 15. A value of 130 has z = (130 − 100) ÷ 15 = 2. Under that model, approximately 97.72% of values are at or below 130, while approximately 2.28% are above it.

Use the 68–95–99.7 rule carefully

The familiar rule describes approximate probability within one, two, and three standard deviations of the mean for a normal distribution.

These are two-sided central areas. “Within two standard deviations” is not the same as “below two standard deviations above the mean.” The latter includes the entire lower tail and is approximately 97.72%.

Do not transfer the rule automatically to income, waiting times, or any other dataset simply because a mean and standard deviation can be calculated. Every finite dataset has descriptive statistics; not every dataset has a normal shape.

Calculate an interval probability

Standardize both endpoints, then subtract their cumulative probabilities. With the illustrative mean of 100 and standard deviation of 15, the interval from 85 to 130 corresponds to z-scores −1 and 2.

The modeled probability is Φ(2) − Φ(−1), approximately 0.97725 − 0.15866 = 0.81859, or 81.86%. Here Φ is the cumulative distribution function of the standard normal distribution.

Cumulative probabilities can be obtained from a normal-distribution table or a numerical approximation. They describe a stated model, not observed frequencies or proof that a sample fits the model.

Individual values and sample means are different

The standard deviation of observations describes their spread. The standard error of a sample mean describes the sampling variability of the mean under the relevant assumptions.

For independent observations with a common finite variance, the standard error is often modeled as σ ÷ √n, using an appropriate estimate when σ is unknown. Do not divide by the square root of sample size when the question concerns the spread of individual observations.

Large sample sizes can make the distribution of a properly standardized sample mean approximately normal under suitable conditions. That does not make the original observations normal or repair biased sampling and dependent measurements.

Check the model before relying on its tails

Inspect a histogram, a normal probability plot where appropriate, and the way data were collected. Look for skew, outliers, multiple populations, and boundaries. Tail probabilities are particularly sensitive to a poor distributional assumption.

For an engineering tolerance or risk decision, state why the model is suitable and how its parameters were estimated. A small calculated probability can look precise while being driven mostly by an unsupported model choice.

Use this guide thoughtfully. Examples illustrate a calculation method, not a guaranteed outcome. The usefulness of any result depends on the definitions, measurements, and assumptions used.