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.
- Answer the prediction question to open Experiment 1.
- Move the score and read the z, the percentile and the shaded area.
- Move σ and watch the peak drop as the curve spreads.
- 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.
Experiment 1 — one distribution, one score
The horizontal axis is the raw score and the vertical axis is probability density — the height of the curve, not a probability. The shaded region is the probability, and the whole area under the curve is always exactly 1 whatever μ and σ are set to.
Key terms
- z-score
- How many standard deviations a raw score sits from its own mean.
- Percentile
- The percentage of the distribution falling below a score.
- Tail area
- The share of the curve lying beyond a score. It is a third statement, distinct from the z and from the percentile.
- μ and σ
- The mean and the standard deviation. Between them they fix the whole curve, because its total area is always 1.
Two numbers make the whole curve
μ slides it sideways; σ stretches it out and, because the area is fixed at 1, flattens it at the same time.
The curve
| Quantity | Value |
|---|
Saying it properly
The same score on a standardised axis
| z | Raw score here | Area below |
|---|
Experiment 2 — one score, two distributions
The same fictional student sits two tests and gets the same raw mark on both. Commit to a judgement before the z-scores appear.
The same mark, twice
Test A has a mean of 50 and a standard deviation of 6. Set up Test B, decide which score is more unusual, then reveal.
Two curves, one vertical line
| Test | Mean | SD | z | Percentile |
|---|
Challenge — what standardising does and does not do
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.
For teaching elsewhere: take this activity as one self-contained block of HTML, on the clipboard or as a file. Either way it is styled so that it will not disturb the page you put it into.