zscore() — Z-scores

How many standard deviations a value sits from the mean.

zscore(value, mean, sd)

Key idea

• A z-score is (value − mean) ÷ sd: how many standard deviations (typical gaps) the value sits from the mean. (e.g. (90 − 75) / 10.8 = 1.39)
• z = 0 is the mean; a positive z is above it, a negative z is below. (e.g. z = −2 -> two SDs below)
• It puts different scales on one ruler, so values can be fairly compared. (e.g. height vs exam score)
• On a normal (bell-shaped) curve about 95% of values have |z| < 2; a |z| > 3 is a rare outlier.

Definition

$$z=\frac{x-\mu }{\sigma }$$

$x$

the value you are scoring

$\mu$

the mean

$\sigma$

the standard deviation

$z$

the z-score (number of SDs from the mean)

$$z=\frac{90-75}{10.8}=1.39\Rightarrow 1.39\text{ SDs above the mean}$$

In plain words

Instead of saying "90 marks", a z-score says "1.4 standard deviations above average" — a fairer way to compare across different tests.

Where you'd use it

• Comparing a 1.5 z-score in maths with a 0.8 z-score in English shows the student was more exceptional in maths, even if the raw marks differ.
• Spotting outliers in sensor data: any reading with |z| > 3 is flagged as suspicious.
• Standardizing features before training a model so a column measured in thousands does not swamp one measured in tenths.

Frequently asked questions

How do I read a z-score?

It’s how many SDs a value sits above (+) or below (−) the mean.

Is |z| = 2 unusual?

Roughly the outer 5% — fairly unusual. |z| > 3 is rare.