A Mann-Kendall Example ("By Hand")


    Say we have a time series with successive values:

    12.9 14.8 13.2 15.7 13.8 15.6 12.9 11.0 17.3 15.9 19.0 18.3 20.1 21.7

    The Mann-Kendall S  is computed by comparing each value to all later values  
    and recording a 1 if the later value is larger, a -1 if the later value
    is smaller, and a 0 if the later value is the same:  Here we get

    12.9   1  1  1  1  1  0  -1  1  1  1  1  1  1
    14.8     -1  1 -1  1 -1  -1  1  1  1  1  1  1
    13.2         1  1  1 -1  -1  1  1  1  1  1  1
    15.7           -1 -1 -1  -1  1  1  1  1  1  1
    13.8               1 -1  -1  1  1  1  1  1  1
    15.6                 -1  -1  1  1  1  1  1  1
    12.9                     -1  1  1  1  1  1  1
    11.0                         1  1  1  1  1  1
    17.3                           -1  1  1  1  1
    15.9                               1  1  1  1
    19.0                                 -1  1  1
    18.3                                     1  1
    20.1                                        1

    Counting up, there are 91 pairs, 18 -1's, 1 0, and 72 1's.  Hence 

                 S = -18 + 0 + 72 = 54

    The variance is computed like this (there is one pair of equal values):

                 V = (14)(13)(33)/18 - (2)(1)(9)/18

    Then the Mann-Kendall statistic is

                 Z = (S - 1)/sqrt(V) = 2.9058 (approximately)


    Interpretation:  We can see that there appears to be an upward trend
    in the data.  The Mann-Kendall test confirms this with a pretty 
    small p-value.  The probability of observing this large a value of Z
    if there were no trend would be about  p = .0018 (using the standard
    normal table).