• The variance has one problem: it is measured in units squared
• This isn’t a very meaningful metric so we take the square root value
• This is the standard deviation (\(s\), sometimes \(sd\)):

\[s = \sqrt{\frac{\sum(x_i-\bar{x})^2}{n-1}} = \sqrt{\frac{5.2}{4}} = 1.14\] NB: mostly, the population standard deviation is called \(s\), while the sample standard deviation is called \(\sigma\)

In R:

friends = c(1, 2, 3, 3, 4)
sd(friends)
[1] 1.140175

Consider this example:

x = c(10, 20)
y = c(5, 18, 22, 13, 9, 23)
sd(x)
[1] 7.071068
sd(y)
[1] 7.238784

So: the standard deviation does not indicate how well we can estimate the mean, for this purpose, we use the standard error of the mean (note sd = s = standard deviation): \[s.e. = \frac{s}{\sqrt{n}}\]

sd(x)/sqrt(2)
[1] 5
sd(y)/sqrt(6)
[1] 2.955221

• The variance and standard deviation represent the same thing:
  • The spread in a variable, how much variability there is
  • The higher the value, the higher the variability
  • With increasing sample size, we achieve a more precise estimate for the variability
• The standard error
  • measures how well we estimate the mean of the population
  • decreases with the number of observations because we gain more confidence in the estimate of the mean

Calculating the standard error in R:

friends <- c(1, 2, 3, 3, 4)
sd(friends)/sqrt(5) #or:
[1] 0.509902
sd(friends)/sqrt(length(friends)) #which of the two is better?
[1] 0.509902