1.3 Estimates of Variability

• Variability (aka dispersion) = are values clustered or spread out?

1.3.1 SD & Friends

• Variance = average of squared deviations, \(s^2 = \frac{\sum_{i=1}^{n}{(x_{1}-\bar{x})^2}}{n-1}\)

s_squared <- var(dataset)
s_squared

## [1] 12.5

• Standard deviation = square root of variance, \(s = \sqrt{variance}\)

s <- sd(dataset)
s

## [1] 3.535534

s == sqrt(s_squared)

## [1] TRUE

• Median absolute deviation from the median (MAD) is robust to outliers.

mad(dataset)

## [1] 1.4826

Wait, why did that return the standard scale factor?

• dataset is c(1, 2, 3, 4, 10)
• The difference between any 2 values is 1 (except the outlier)
• 1 * 1.4826 = 1.4826

1.3.2 Percentiles & Friends

• Percentiles = quantiles, \(P\%\) of values are \(<= x\)

x <- sample(1:100, 100, replace = TRUE)
y <- rnorm(100, mean = 50, sd = 20)
quantile(x, probs = seq(0, 1, 0.1))

##    0%   10%   20%   30%   40%   50%   60%   70%   80%   90%  100% 
##   2.0  13.0  21.0  27.0  38.0  45.5  55.8  66.3  81.6  96.0 100.0

quantile(y, probs = seq(0, 1, 0.1))

##         0%        10%        20%        30%        40%        50%        60% 
##  -1.830337  29.526910  36.006697  40.274329  44.657078  48.180893  54.405362 
##        70%        80%        90%       100% 
##  58.980133  65.088256  73.732659 100.371937

quantile(x) # quartile

##     0%    25%    50%    75%   100% 
##   2.00  25.00  45.50  73.25 100.00

IQR(x) # They introduce this later but I like it here.

## [1] 48.25