Question

A simple example of calculating standard dev:

d <- c(2,4,4,4,5,5,7,9)
sd(d)

yields

[1] 2.13809

but when done by hand, the answer is 2. What am I missing here?

Answer 1

Try this

R> sd(c(2,4,4,4,5,5,7,9)) * sqrt(7/8)
[1] 2
R> 

and see the rest of the Wikipedia article for the discussion about estimation of standard deviations. Using the formula employed ‘by hand’ leads to a biased estimate, hence the correction of sqrt((N-1)/N). Here is a key quote:

The term standard deviation of the
sample is used for the uncorrected
estimator (using N) while the term
sample standard deviation is used for
the corrected estimator (using N − 1).
The denominator N − 1 is the number of
degrees of freedom in the vector of
residuals, .