Variability

• Also called dispersion, variability refers to “how spread out” data are.
• Range is one measure of variability, but it doesn’t capture how data is dispersed within the range.
• Variance is better measure of variability.
  • It measures how far the values are dispersed around the mean.
• Standard deviation is an even better measure.
  • It's like variance but in the original scale of the variable, so one can relate it to real values.

Variance = \( s^2 \)

• Variance is the mean of the squared deviations from the mean.
• A deviation from the mean is: \( Y_i - \bar{Y} \)
• The simple sum of deviations from the mean is worthless:
  • \( \sum\limits_{i=1}^N Y_i - \bar{Y} = 0 \)
• That’s why we square the deviations first.
• Add them up, and divide by 1 - n (or divide by 1/1-n)
  • \( \frac{1}{n-1} \sum\limits_{i=1}^N (Y_i - \bar{Y})^2 \)

Standard deviation = s

• Standard deviation, then, is just the square root of the variance.
• \( \sqrt{ \frac{1}{n-1} \sum\limits_{i=1}^N (Y_i - \bar{Y})^2 } \)

summary(cars)

##      speed           dist    
##  Min.   : 4.0   Min.   :  2  
##  1st Qu.:12.0   1st Qu.: 26  
##  Median :15.0   Median : 36  
##  Mean   :15.4   Mean   : 43  
##  3rd Qu.:19.0   3rd Qu.: 56  
##  Max.   :25.0   Max.   :120

You can also embed plots, for example:

plot(cars)