Joint Frequencies

• The absolute frequency hij = h(ai, bj) of the combination (ai, bj), i = 1, … , k, j = 1, … , m, corresponds to the number of pairs(xi, yj) of the original list that match (ai, bj)
• Joint distributions = values hij i = 1, … , k, j = 1, … , m
• the data on the joint absolute frequencies can be summarized in a contingency table:

x <- matrix(c(1,4,7,4,0,8,9,4,6,3,2,2,5,7,5,6), byrow = TRUE, nrow = 4)
addmargins(x)

                Sum
     1  4  7  4  16
     0  8  9  4  21
     6  3  2  2  13
     5  7  5  6  23
Sum 12 22 23 16  73

• the sum of the rows and columns are called the marginal absolute frequencies
• the same can be done with the relative frequencies, which can be calculated that way: fij = hij / n
• note that k is the number of rows and m is the number of columns

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