# What does “number of ties of extent m” mean?

I'm trying to figure out how variance of Kendall's tau statistic is determined in the presence of ties. From Yue et al. 2002:

The variance of $S$ is given by:

$E(S) = \frac{n(n-1)(2n+5)-\sum_{m=1}^{n}t_{m}m(m-1)(2m+5)}{18}$

where $t_m$ is the number of ties of extent $m$.

What does "of extent $m$" mean?

Yue, Sheng, et al. "The influence of autocorrelation on the ability to detect trend in hydrological series." Hydrological processes 16.9 (2002): 1807-1829.

• Please give a full reference so we have some chance of checking the context. "Yue et al" and a year isn't much to go on. Is this Yue, S., Pilon, P. & Cavadias, G. (2002), "Power of the Mann-Kendall and Spearman’s rho tests for detecting monotonic trends in hydrological series.", J. Hydrol. 259 , 254–271 or something else? – Glen_b -Reinstate Monica Jul 3 '18 at 2:44

Presumably $m$ represents the number of values that are tied at some particular number. For example, if your sample were sorted into increasing order and had $(1.9, 2.0, 2.0, 3.1, 3.5, 3.5, 3.5, 3.8, 4.3, 4.3, 5.0)$ there's a pair of tied values at each of $2.0$ and $4.3$ and a triple at $3.5$ (a pair of them means $m=2$, a triple means $m=3$) while $t_2$ represents the number of times you get $m=2$ across the whole sample (the number of values with ties where there are just two values tied). So here $t_2=2$ and $t_3=1$ (while $t_4=0$ because there were no values in our sample where 4 observations were all tied together).