Coaching Centre
z-scores
Calculating a z-score as the number of standard deviations a score is from the mean, interpreting positive, negative and zero z-scores, and identifying unusual scores.
You scored $74$ on a test. Is that good? It depends. If the class mean was $60$ and nearly everyone scored between $55$ and $65$, you did brilliantly. If the mean was $70$ and marks were all over the place, you were only a bit above average. A z-score answers the question by measuring your distance from the mean in units of standard deviation.
Calculating a z-score
Here $x$ is the score, $\bar{x}$ is the mean and $s$ is the standard deviation. The top line asks "how far from the mean?" and dividing by $s$ asks "how many standard deviations is that?". This formula is on the reference sheet.
- $z \gt 0$: the score is above the mean.
- $z \lt 0$: the score is below the mean.
- $z = 0$: the score equals the mean.
A maths test has mean $62$ and standard deviation $8$. Find the z-score for (a) $74$ and (b) $50$.
A z-score of $66.25$ should ring alarm bells anyway. Almost all normal data has $z$ between $-3$ and $3$.
How unusual is a score?
Combine z-scores with the 68–95–99.7 rule:
| z-score | Share of data this close to the mean or closer | How unusual? |
|---|---|---|
| $-1 \le z \le 1$ | about $68\%$ | typical |
| $-2 \le z \le 2$ | about $95\%$ | a $z$ beyond $\pm 2$ is unusual |
| $-3 \le z \le 3$ | about $99.7\%$ | a $z$ beyond $\pm 3$ is very rare, a possible outlier |
Two number lines are linked: raw scores on top and z-scores underneath. Drag the marker along either one and watch the other move.
Now change the standard deviation with the slider. Why does the same raw score get a smaller z-score when $s$ is bigger? Find a score with $z = -2$.
Your turn
The masses of newborn babies at a Sydney hospital are normal with mean $3.4$ kg and $s = 0.5$ kg. Find the z-score of a baby with mass $2.3$ kg, and decide whether this mass is unusual.
Show the answer
$z = \dfrac{2.3 - 3.4}{0.5} = \dfrac{-1.1}{0.5} = -2.2$. The baby is $2.2$ standard deviations below the mean. Since $z \lt -2$, the mass is outside the middle $95\%$, so it is unusual.
A dataset has mean $40$ and $s = 5$. Find the z-score of $50$.
$z = \dfrac{50 - 40}{5} = \dfrac{10}{5} = 2$.
A dataset has mean $80$ and $s = 4$. Find the z-score of $75$.
$z = \dfrac{75 - 80}{4} = \dfrac{-5}{4} = -1.25$.
A z-score of $0$ means the score is:
$z = 0$ means $x - \bar{x} = 0$, so $x = \bar{x}$.
A z-score of $-1.8$ means the score is:
A z-score counts standard deviations, not marks, and the negative sign means below.
Reaction times have mean $250$ ms and $s = 30$ ms. Find the z-score of $265$ ms.
$z = \dfrac{265 - 250}{30} = \dfrac{15}{30} = 0.5$.
Which z-score shows the most unusual score?
The most unusual is the one furthest from $0$ in either direction. $|-3.2| = 3.2$ is the largest, beyond $3$ standard deviations.
Two scores in the same test have z-scores $1$ and $-1$. Which is true?
Both are $1$ standard deviation from the mean, one above and one below. Within $\pm 1$ is typical, not unusual.
The daily maximum temperatures in Canberra in July are normal with mean $12$°C and $s = 2.5$°C. Find the z-score of a $6$°C day.
$z = \dfrac{6 - 12}{2.5} = \dfrac{-6}{2.5} = -2.4$.
Exam style: The heights of a group of netball players have mean $178$ cm and $s = 6$ cm. One player is $196$ cm tall. Which statement is correct?
$z = \dfrac{196 - 178}{6} = \dfrac{18}{6} = 3$. A score $3$ standard deviations above the mean is outside the middle $99.7\%$, so it is very rare.
Exam style: A test has mean $70$. A score of $100$ has a z-score of $2$. What is the standard deviation?
$2 = \dfrac{100 - 70}{s}$, so $2s = 30$ and $s = 15$.