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)
## 'data.frame':    32 obs. of  11 variables:
##  $ mpg : num  21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
##  $ cyl : num  6 6 4 6 8 6 8 4 4 6 ...
##  $ disp: num  160 160 108 258 360 ...
##  $ hp  : num  110 110 93 110 175 105 245 62 95 123 ...
##  $ drat: num  3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
##  $ wt  : num  2.62 2.88 2.32 3.21 3.44 ...
##  $ qsec: num  16.5 17 18.6 19.4 17 ...
##  $ vs  : num  0 0 1 1 0 1 0 1 1 1 ...
##  $ am  : num  1 1 1 0 0 0 0 0 0 0 ...
##  $ gear: num  4 4 4 3 3 3 3 4 4 4 ...
##  $ carb: num  4 4 1 1 2 1 4 2 2 4 ...
dim(mtcars)
## [1] 32 11
head(mtcars)
##                    mpg cyl disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
## Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1
help(mtcars)
## starting httpd help server ... done


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)