Central tendency measures

note that in the R codes x is a vector of values. Vectors can be created with c() for examle:

c(1,2,3,4,5,6)

[1] 1 2 3 4 5 6

Mode

• A simple central tendency that can be used for all scale levels
• The mode indicates which expression occurs most frequently
• If the absolute (or relative) frequency distribution has a unique maximum, the mode is unique and is denoted by x(mod)
• In the bar chart, the mode is the expression with the highest column
• the mode does not always have to be unique
• calculation in R: Mode <- function(x) { ux <- unique(x) ux[which.max(tabulate(match(x, ux)))] } mode <- Mode(x)

Properties of the mode

• If only very few values of the list of observations match the mode is not very meaningful

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Arithmetic mean

• calculation in R: mean(x)
• for grouped data the arithmetic mean from the midpoints of the classes has to be calculated
  

Properties of the arithmetic mean

• the arithmetic mean is very sensitive to outliers
• This sensitivity to outliers can be undesirable (e.g. individual errors in the data lead to large changes in the arithmetic mean)

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Median

• for at least ordinally scaled characteristics
• One half of the values of the original list should be below the median and the other half above
• For odd 𝒏𝒏 the median is the middle value of the ordered data
• calculation in R: median(x)

Properties of the median

• robust against outliers
• At least 50% of the values in the data are less or equal to the median
• At least 50% of the values in the data are greater or equal to the median

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Quantiles

• represent central tendency measures that also already provide information on the dispersion of the data
• Let 0 < p < 1. For any value xp for which
  • Let 0 < p < 1. For any value xp for which
  • at least a proportion 1 − p is greater or equal than xp
• calculation in R: Quantiles 0.2, 0.25, 0.5, 0.75
  c(xordered[floor(n * 0.20) + 1], xordered[floor(n * 0.25) + 1], xordered[floor(n * 0.5) + 1], xordered[floor(n * 0.75) + 1])
• calculation of interquartile range
  (xordered[floor(n * 0.75) + 1]) - (xordered[floor(n * 0.25) + 1])

Properties of the Quantiles and the interquartile range

• robust against outliers
• do not contain any information about the left and right ends

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five point summary

• The five-point summary of the distribution of ametric variable consists of :
  x(min), 0.25 - Quantile, median, 0.75 - Quantile, x(max)
• visualization in R: boxplot(x)
• example:

x <- c(1,4,6,3,7,9,4,2,1,4)
boxplot(x)

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Overview

		Mode	Arithmetic mean	Median	Quantiles
Applicable for different scale levels	nominal	yes	no	no	no
	ordinal	yes	no	yes	yes
	cardinal	yes	yes	yes	yes
Robust against outliers		yes	no	yes	yes

Storage Rules

mean ~ median ~ mode	symmetric
mean > median > mode	left-skewed
mean < median < mode	right-skewed

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