Comparing Data Using z-scores
Learn how to compare data using z-scores for NSW Year 12 Mathematics Standard 2. When scores come from different datasets — two exams, two subjects or two squads — you standardise each one into a z-score so they sit on one common scale, then the larger z-score is the relatively higher result.
This Standard 2 topic shows you how to standardise and compare two scores, rank several results by their z-scores, and interpret relative standing — including the case where a smaller value is better (like race times), where the more negative z-score wins. It is a core part of the normal-distribution work in the NSW Year 12 course.
Theory
Comparing data using z-scores lets you fairly compare scores from different datasets in NSW Year 12 Mathematics Standard 2. Standardise each score with \(z=\dfrac{x-\mu}{\sigma}\), then the larger z-score is the relatively higher result — with the more negative z-score winning when a smaller value is better.
Scores from different datasets — two exams, two subjects or two squads — sit on different scales, so you cannot compare the raw numbers directly. Comparing data using z-scores means standardising each score first.
The z-score \(z=\dfrac{x-\mu}{\sigma}\) tells you how many standard deviations a score is above or below its own dataset's mean. Once every score is a z-score they share one common scale, so they can be compared.
This Year 12 Standard 2 (NSW) skill is used to decide which of two results is relatively higher, to rank several standardised scores, and — where a smaller value is better, like race times — to see that the more negative z-score is the better result.
To compare scores from different datasets, standardise each one. The z-score counts how many standard deviations a score sits above (or below) its own mean:
where \(x\) is the score, \(\mu\) the dataset mean and \(\sigma\) its standard deviation. Once every score is a z-score they share one scale, so you compare them directly:
The gap between two results on the common scale is the difference of their z-scores:
How to compare scores with z-scores
- List each score \(x\) with its own dataset's mean \(\mu\) and standard deviation \(\sigma\).
- Standardise each score: \(z=\dfrac{x-\mu}{\sigma}\).
- Compare the z-scores on the common scale — the larger z-score is relatively higher.
- Interpret in context. To rank, order the z-scores; the difference \(z_1-z_2\) is the gap. If a smaller raw value is better, the more negative z-score wins. State the answer in words.
Standardise each mark, then compare the z-scores.
| \(z_{\text{Bio}}\) | \(=\) | \(\dfrac{80-66}{7}=2\) |
| \(z_{\text{Chem}}\) | \(=\) | \(\dfrac{85-70}{10}=1.5\) |
| \(2\) | \(\gt\) | \(1.5\) |
Biology is relatively stronger — \(z=2\) beats \(z=1.5\).
Standardise Zara's mark, then compare with Leo's z-score.
| \(z_{\text{Zara}}\) | \(=\) | \(\dfrac{80-64}{8}=2\) |
| \(z_{\text{Leo}}\) | \(=\) | \(1.8\) |
| \(2\) | \(\gt\) | \(1.8\) |
Zara did relatively better — \(z=2\) is higher than \(1.8\).
Standardise each mark, then order the z-scores.
| Strand | Mark | Mean | SD |
|---|---|---|---|
| Reading | 88 | 70 | 9 |
| Writing | 63 | 54 | 6 |
| Numeracy | 46 | 40 | 12 |
| \(z_{\text{Reading}}\) | \(=\) | \(\dfrac{88-70}{9}=2\) |
| \(z_{\text{Writing}}\) | \(=\) | \(\dfrac{63-54}{6}=1.5\) |
| \(z_{\text{Numeracy}}\) | \(=\) | \(\dfrac{46-40}{12}=0.5\) |
Highest to lowest: Reading, Writing, Numeracy \((2 \gt 1.5 \gt 0.5)\).
Standardise each time. A lower time gives a more negative z-score.
| \(z_{\text{Ana}}\) | \(=\) | \(\dfrac{54-62}{4}=-2\) |
| \(z_{\text{Bec}}\) | \(=\) | \(\dfrac{51-58}{5}=-1.4\) |
| \(-2\) | \(\lt\) | \(-1.4\) |
Ana was faster — her time is further below her squad's mean, so \(z=-2\) (more negative) is the better result.
Common pitfalls
Frequently asked questions
How do you compare marks from two different tests?
Standardise each mark into a z-score using z equals x minus the mean, divided by the standard deviation. This puts both marks on the same scale. The mark with the larger z-score is relatively higher, because it sits more standard deviations above its own test's mean.
What does a higher z-score mean when comparing scores?
A higher z-score means the score is further above its dataset's mean, so it is relatively better. For example, a z-score of 2 is two standard deviations above the mean and beats a z-score of 1.5, even when the raw marks came from different tests.
Can you compare test scores without using z-scores?
Not fairly, if the tests have different means or standard deviations. A raw mark of 80 on an easy test is not the same achievement as 80 on a hard one. Standardising with z-scores removes the effect of the different means and spreads so the comparison is fair.
What if a lower score is better, like race times?
Then the more negative z-score is the better result. A lower time is further below the mean, giving a negative z-score, so the most negative z-score is the fastest relative to its own group. Always check whether larger or smaller raw values are 'better' before you decide.
Does the bigger raw mark always win?
No. Because each test has its own mean and standard deviation, a smaller raw mark can have a larger z-score. Always compare the z-scores, not the raw marks, when the scores come from different datasets.