Chapter 6 Numerical Summaries

The numerical summary must match up with the type of variable(s).

Variable Type of summary
1 Qualitative frequency table, most common category
1 Quantitative mean, median, SD, IQR etc
2 Qualitative contingency table
2 Quantitative correlation, linear model
1 Quantitative, 1 Qualitative mean, median, SD, IQR etc across categories


We’ll keep working with the mtcars dataset. So again remind yourself what it is like.

str(mtcars)
dim(mtcars)
head(mtcars)
help(mtcars)


6.1 Frequency and contingency tables

  • A frequency table summarises 1 qualitative variable.
table(mtcars$gear)
  • A contingency table summarises 2 qualitative variable.
table(mtcars$gear, mtcars$am)


6.2 Mean and median

  • The mean and median measure centre for quantitative variables. 
mean(mtcars$gear)
median(mtcars$gear)


6.3 Standard deviation (SD)

  • The standard deviation measures spread for quantitative variables.

  • The sd command calculates the sample standard deviation. The squared SD is the variance.

sd(mtcars$gear)
sd(mtcars$gear)^2
var(mtcars$gear)
  • The popsd command calculates the population standard deviation, but requires the multicon package. 
#install.packages(multicon)   # a package only needs to be installed once.
library(multicon)
popsd(mtcars$gear)

# Longer way
N = length(mtcars$gear)
sd(mtcars$gear)*sqrt((N-1)/N)
  • Note: When we model a population by the box model [Section 8 and following], we will require the population SD.


6.4 Interquartile range (IQR)

  • The quickest method is to use IQR.
IQR(mtcars$gear)
  • There are lots of different methods of working out the quartiles. We can use the quantile command, and then work out the IQR.
quantile(mtcars$gear)
quantile(mtcars$gear)[4]-quantile(mtcars$gear)[2]

What is the 50% quantile equivalent to?


6.5 Summary

  • The numerical summaries for quantitative variables can all be produced with summary, which is an expanded version of the 5 number summary. Sometimes these values will vary from using quantile as there are different conventions for calculating quartiles.
summary(mtcars$mpg)  # 1 variable
summary(mtcars)  # All variables
  • We can consider a subset of the data. Here, we choose the mpg of cars which have a weight greater or equal to 3.
summary(mtcars$mpg[mtcars$wt>=3])

Here we take all the data from mtcars dataset for a specific cylinder e.g. 6.

mtcars[ which(mtcars$cyl==6), ]


6.6 Linear Correlation

  • We can consider the linear correlation between pairs of quantitative variables.
cor(mtcars)