var() — Variance

Average squared distance from the mean — dividing by n−1 or n.

var(x, sample = TRUE)

Key idea

• Variance is the average of the squared distances of each value from the mean (the average).
• Divide by n (how many values) for the POPULATION variance, when your data is the whole group. (e.g. 40 ÷ 5 = 8)
• Divide by n − 1 for the SAMPLE variance, when your data is just a sample of a bigger group (the usual default). (e.g. 40 ÷ 4 = 10)
• Using n − 1 (called Bessel's correction) fixes the slight under-estimate a sample makes.
• It is the standard deviation squared; the gap between the two divisors shrinks as n grows. (e.g. variance 9 -> sd 3)
• Quicker by hand: average the SQUARES, then subtract the average squared. (e.g. σ² = (Σx²)/n − μ²)

n or n − 1: which divisor?

1. Find the mean, then add up every squared distance from it — that total is the same for both versions.
2. Whole population (you have every member, e.g. all 30 students in the class): divide by n → the POPULATION variance σ².
3. A sample (your data is a subset used to estimate a bigger group): divide by n − 1 → the SAMPLE variance s². This is the default in R and on most calculators (numpy needs ddof=1).
4. Dividing by n − 1 (Bessel’s correction) nudges the estimate up a little, because a sample’s points hug their own mean and would otherwise under-state the true spread.

Definition

$$\begin{array}{rl}{\sigma }^2& =\frac{1}{n}\sum _{i=1}^n(x_i-\mu {)}^2{s}^2=\frac{1}{n-1}\sum _{i=1}^n(x_i-\bar{x}{)}^2\\ {\sigma }^2& =\frac{\sum x^2}{n}-{\mu }^2(\text{quicker to compute})\end{array}$$

$x_i$

each value

$\bar{x},\mu$

the mean

$n$

how many values

$s^2$

sample variance (÷ n−1)

${\sigma }^2$

population variance (÷ n)

$$s^2=\frac{40}{5-1}=10{\sigma }^2=\frac{40}{5}=8$$

Where you'd use it

• Two funds both return 5% on average, but one swings wildly year to year. Computing var on each year’s returns gives a bigger number for the riskier fund — say 40 versus 4 — flagging it as the more volatile bet.
• You measure the same object 5 times in the lab and get slightly different readings. var turns those tiny differences into one number for the scatter; a variance near 0 means your instrument is very precise.
• When a statistics formula asks for the squared spread (like combining errors), you feed it var rather than sd because variances add up cleanly, whereas standard deviations do not.

Frequently asked questions

Why is the variance so large?

It’s in squared units. Take the square root (SD) for data units.

Does var use n or n − 1?

n − 1 — the sample variance.