Converting and comparing with z-scores

Converting a z-score back to a raw score, comparing performances across different tests using z-scores, and finding percentages from whole-number z-scores.

Mia got $72$ in English and $80$ in Maths. Her parents congratulate her on Maths. But the Maths test was easy (mean $74$) and the English test was hard (mean $60$). Raw marks from different tests can't be compared directly. z-scores put them on the same scale.

Comparing results

To compare scores from different datasets, convert each one to a z-score. The higher z-score is the better result relative to the group.

Example 1●●

Mia's results are below. In which subject did she do better compared with her class?

Subject	Mia's mark	Class mean	Standard deviation
English	72	60	8
Maths	80	74	6

$z_{\text{Eng}} = \dfrac{72 - 60}{8} = \dfrac{12}{8} = 1.5$

z-score for English: subtract the mean, then divide by s.

$z_{\text{Maths}} = \dfrac{80 - 74}{6} = \dfrac{6}{6} = 1$

z-score for Maths, using the Maths mean and s.

$1.5 \gt 1$, so Mia did better in English relative to her class, even though her raw mark was lower.

The larger z-score is further above the class mean.

For negative z-scores the same rule holds: $-0.5$ is a better result than $-1.2$, because it's closer to (less far below) the mean. Careful with times in a race, though. There a lower time is better, so the lower z-score is the better performance.

Converting a z-score to a raw score

Rearrange the z-score formula to go backwards:

Raw score from a z-score

$$x = \bar{x} + z\,s$$

This just says: start at the mean, then move $z$ standard deviations. A negative $z$ moves down.

Example 2●

An exam has mean $65$ and $s = 12$. Students with a z-score of $2$ or more get a merit certificate. What mark is needed?

$x = 65 + 2 \times 12$

Start at the mean and move 2 standard deviations up.

$x = 65 + 24 = 89$

Multiply before adding.

A mark of $89$ or more earns a merit certificate. About $2.5\%$ of students will get one.

Answer in context.

z-scores and percentages

For whole-number z-scores, the 68–95–99.7 rule gives percentages directly. About $68\%$ of scores have $z$ between $-1$ and $1$, $95\%$ between $-2$ and $2$, and $99.7\%$ between $-3$ and $3$. The table from section 8B works the same way with z-values marked on the curve instead of raw scores.

InteractiveCompare two exams

Try this

Each exam has its own mean and standard deviation. Set a mark for each and see both z-scores side by side on matching bell curves.

Find two marks where the raw scores say one thing but the z-scores say the other. Then press New challenge and decide which result is better before the answer shows.

Your turn

Jai ran $100$ m in $12.4$ s (his age group: mean $13.0$ s, $s = 0.4$ s) and threw a discus $38$ m (mean $32$ m, $s = 4$ m). Which was his better performance?

Show the answer

Sprint: $z = \dfrac{12.4 - 13.0}{0.4} = \dfrac{-0.6}{0.4} = -1.5$. A faster time is better, so being $1.5$ standard deviations below the mean time is a strong result.

Discus: $z = \dfrac{38 - 32}{4} = 1.5$, so $1.5$ standard deviations above the mean distance.

Both are $1.5$ standard deviations better than average, so the two performances are equally good relative to the group.

Flashcards

Raw score from a z-score

$x = \bar{x} + zs$. Start at the mean and move $z$ standard deviations.

Comparing two results from different tests

Find each z-score. The higher one is better relative to its group (lower if smaller is better, like race times).

Percentage with $z$ between $-2$ and $2$

About $95\%$.

Practice

A test has mean $60$ and $s = 7$. Find the score with a z-score of $2$.
A test has mean $54$ and $s = 6$. Find the score with a z-score of $-1.5$.
Sam's z-score in Biology is $0.8$ and in Chemistry is $1.3$. Relative to the class, Sam did better in:
Which z-score is the better result in a test?
Ella scored $68$ in Test A (mean $60$, $s = 4$) and $75$ in Test B (mean $65$, $s = 10$). Which is true?
What percentage of normally distributed scores have a z-score between $-2$ and $2$?
What percentage of normally distributed scores have a z-score less than $1$?
In a swimming race, lower times are better. Which z-score for a swimmer's time shows the best performance?
Exam style: In a Year 12 English exam, the mean was $66$ and the standard deviation $6$. Priya's z-score was $-1.5$. What was her mark?
Exam style: Liam scored $81$ in Legal Studies (mean $72$, $s = 6$) and $84$ in Business Studies (mean $78$, $s = 3$). In which subject did he perform better relative to his classes?