Section 16 Single Variable: Summary Statistics

16.1 Descriptive statistics

Function	Explanation
length	Number of elements in a vector
sum	Sum of the values in a vector
min	Minimum of a vector
max	Maximum of a vector
range	Range (min, max) of a vector
mean	Mean of the values in a vector
median	Median of the values in a vector
var	Variance
sd	Standard deviation
cov	Covariance of two vectors
cor	Pearson correlation between two vectors

16.2 Measures of average

• Mean
• Median
• Mode
• Quartiles
• Median Absolute Deviation (MAD)

x <- c(0, 2, 4, 6, NA, 8, 4, 5, 15, 11, 4, 7)

mean(x, na.rm=TRUE)
median(x, na.rm=TRUE)

Mode <- function(x) {
  ux <- unique(x)
  # tab <- ux[which.max(tabulate(match(x, ux)))]
  tab <- tabulate(match(x, ux))
  modex <- ux[tab == max(tab)]
  return(modex)
}

Mode(x)

quantile(x=x, probs=c(0.25,0.50,0.75), na.rm=TRUE)

mad(x, na.rm=TRUE)

fivenum(x, na.rm = TRUE)

summary(x)

16.3 Measures of dispersion

• Range
• Inter-quartile range
• Standard Deviation
• Coefficient of variation

x <- c(0, 2, 4, 6, NA, 8, 4, 5, 15, 11, 4, 7)

range(x, na.rm=TRUE)

diff(range(x, na.rm=TRUE))

var(x, na.rm=TRUE)
sd(x, na.rm=TRUE)

IQR(x=x, na.rm=TRUE)

qx <- quantile(x=x, probs=c(0.25,0.50,0.75), na.rm=TRUE)
str(qx)
unname(qx[3] - qx[1])


# Coefficient of variation
cvx <- sd(x, na.rm=TRUE) / mean(x, na.rm=TRUE)

16.4 Measures of shape

• Skewness
• Kurtosis

x <- c(0, 2, 4, 6, NA, 8, 4, 5, 15, 11, 4, 7)
x <- x[!is.na(x)]
n <- length(x)

# Skewness
skx <- (sum((x - mean(x))^3)/n)/((sum((x - mean(x))^2)/n)^(3/2))

# Kurtosis
kurtx <- (sum((x - mean(x))^4)/n)/((sum((x - mean(x))^2)/n)^2)

16.5 Exercise

• Calculate the following summary statistics of temperature and radiation of the weather data

  • Mean, Median, Quartiles
  • Range, IQR, Variance, SD, CV
  • Skewness, Kurtosis

• Write a function which will return all the above summary statistics of a vector

• Calculate the standard deviation using the formula and compare with the R function output