scale() — Standardize (z-scores)

Changes each number into how far it is from the middle.

scale(x)

Key idea

• scale turns a value into a z-score: how many standard deviations (spread-steps) it sits from the mean (average). (e.g. (14 - 10) / 2 = 2)
• The recipe is (value minus the mean) divided by the standard deviation.
• A positive z is above the average and a negative z is below it. (e.g. z = -1.5 is below)
• A value equal to the mean has a z-score of zero. (e.g. (10 - 10) / 2 = 0)
• Z-scores let you fairly compare numbers measured on different scales.

Definition

$$z_i=\frac{x_i-\bar{x}}{s}$$

$x$

one value

$\bar{x}$

the sample mean

$s$

the sample standard deviation

$z$

the z-score — steps from the mean

$$z=\frac{x-\bar{x}}{s}=\frac{190-170}{10}=2$$

In plain words

For each number it asks: how many steps away from average is this? That step count is the z-score. It lets you compare things measured in different ways using one shared ruler.

Where you'd use it

• You score 80 in math (class mean 70, sd 5) and 85 in English (class mean 80, sd 10). scale turns these into z-scores of +2 and +0.5, so even though English looks higher, math was your more impressive result.
• A player is 190 cm tall (team mean 180, sd 5) and weighs 78 kg (team mean 75, sd 6). Their z-scores are +2 for height and +0.5 for weight, showing they stand out far more for being tall than heavy.
• Run scale on a list of exam marks and the result tells you how many sds above or below average each student is; a z-score of +3 would flag the single most surprising, unusually high result in the class.

Frequently asked questions

What does scale() do?

It centres each column (subtracts the mean) and divides by the sample SD, giving z-scores.

Why don’t my values match a hand z-score?

scale uses the sample SD (÷ n − 1); a population z-score uses σ (÷ n).