Research Methods

The Normal Curve and z-Scores

A normal distribution is two numbers and nothing else. Standardising a score is one subtraction and one division. Everything people find hard about it is really about area.

Simulated — illustrative distributions, no real test data

Learning objective

By the end you should be able to say what a z-score does to a raw score, and why a raw score means nothing on its own.

About 20 minutes. Nothing you do here is saved or sent anywhere.

The rule is about a model, not the world

The 68–95–99.7 figures are properties of the normal distribution. They are not properties of data. Reaction times, incomes, symptom counts and most questionnaire totals are not normal, and applying the rule to them gives answers that are confidently wrong. Everything on this page is exact for the model being drawn and approximate for anything real.

  1. Answer the prediction question to open Experiment 1.
  2. Move the score and read the z, the percentile and the shaded area.
  3. Move σ and watch the peak drop as the curve spreads.
  4. In Experiment 2, put one raw score on two distributions and commit before revealing.

First, a prediction

A fictional test is normally distributed with a mean of 50 and a standard deviation of 10.

What proportion of scores lie above 70?

Challenge — what standardising does and does not do

Which statements are correct? Select all that apply — three of the seven are.

What this demonstrates

Two numbers, and the area does the rest

A normal distribution is completely specified by μ and σ. The mean slides the curve along the axis; the standard deviation stretches it. Because the total area is fixed at 1, stretching it necessarily flattens it — which is why the peak falls as σ rises, and why the height of the curve is not a probability. Probabilities are areas, and the shaded region on screen is the only thing that answers "how many people?".

A raw score means nothing on its own

Leaving the score at 70 and dragging the mean takes its percentile from almost nothing to almost everything, without the student answering a single extra question. That is the argument for standardising: z = (x − μ) / σ puts the score in units of the distribution it came from, so that two scores from two different distributions can be compared at all.

z, percentile and tail area are three different statements

The z-score says how many standard deviations from the mean. The percentile says what share of the distribution is below. The tail area says what share is beyond, and needs you to say beyond in which direction — the difference between 2.3% above z = 2 and 4.6% outside ±2 is where most arithmetic errors live. All three are the same information; only the last two are probabilities.

The 68–95–99.7 rule belongs to the model

Those percentages are constants of the normal curve, which is why the highlighted band's area never moves however far you drag σ. They are not facts about data. Applied to a skewed or bounded or bimodal variable — reaction times, income, symptom counts, most questionnaire totals — they give answers that are precise and wrong. The first question about any real variable is whether the model is even roughly appropriate, and the answer is often no.