sd() — Standard deviation

Tells you how far the numbers usually sit from the average.

sd(x)

Key idea

• The standard deviation measures the typical distance of values from the mean (the average).
• It is in the same units as your data, so it is easy to read. (e.g. heights in cm -> sd in cm)
• An sd of 0 means every value is identical, with no spread at all. (e.g. 5,5,5 -> sd 0)
• The standard deviation is just the square root of the variance (the squared spread). (e.g. variance 9 -> sd 3)
• A larger sd means the data is more spread out from the average.

Definition

$$s=\sqrt{\frac{1}{n-1}\sum _{i=1}^n(x_i-\bar{x}{)}^2}$$

$x_i$

each value

$\bar{x}$

the mean

$n$

how many values

$s$

the standard deviation

$$s=\sqrt{\frac{(-4{)}^2+(-2{)}^2+0^2+2^2+4^2}{5-1}}=\sqrt{\frac{40}{4}}=3.16$$

Where you'd use it

• Two classes both average 70 on a test, but class A’s scores cluster from 65–75 while class B ranges 40–100. sd might give A about 4 and B about 20, instantly showing B’s results are far more spread out.
• A player scores 14, 15, 16, 15, 15 points per game. Their sd is only about 0.7 points, telling you they are remarkably steady from game to game.
• Two factories make bolts meant to be 10 mm. The one whose lengths have an sd of 0.1 mm is much more consistent than one with an sd of 0.5 mm.

Frequently asked questions

Sample or population SD?

R’s sd() is the sample SD (÷ n − 1). Multiply by √((n−1)/n) for the population SD.

What units is SD in?

The same units as the data (variance is in squared units).