Research Methods · Simplified
The Same Mark, Two Different Distributions
A score on its own says almost nothing. What it is worth depends entirely on the distribution it sits in.
The curve
One mark, held still, while the distribution moves
The score axis below never changes and neither does the mark. Move the mean and the spread, and watch the same 62 turn from unremarkable into exceptional and back again.
Key terms
- z-score
- How many standard deviations a score sits above or below the mean. A z of 0 is exactly average; a z of +2 is two standard deviations above it.
- Percentile
- The percentage of the distribution that falls below the score. The shaded area on the figure.
- Density
- The height of the curve. It is not a probability; the area underneath is. Spreading a distribution out lowers the peak, because the total area has to stay at one.
The shaded area is everyone scoring below the mark. The lower axis is the very same axis as the upper one, relabelled in standard deviations from the mean: that relabelling is all standardising does.
What this shows
Standardising relabels the axis
The mark never moved. Everything that changed about what it is worth came from the distribution around it.
Key idea: A raw score is not interpretable on its own. A z-score answers the only question that can be asked of a single mark: how far from the middle is it, measured in the units this particular distribution actually varies by. That is why the same 62 can be thoroughly ordinary in one cohort and near the top of another, and why moving the standard deviation changes a z-score just as decisively as moving the mean does. Standardising does not change the distribution. It relabels the axis.
Two limits worth keeping in view. The percentile and the tail areas here are read off a normal curve, so they are properties of that model rather than of any real data: if a set of marks is skewed or has a ceiling, as coursework marks usually do, the model gives the wrong answer and the familiar 68, 95 and 99.7 figures go with it. And a z-score says where a score sits, not how good, how meaningful or how reliable it is. A z of +2 on a poorly made measure is still a score on a poorly made measure.
The longer version adds a second experiment placing one mark on two distributions at once, the distinction between one-tailed and two-tailed areas worked through in detail, and a set of read-the-curve challenges. It is at The Normal Curve and z-Scores in the main collection.