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w-index (Wohlin)

Wohlin's w-index (Wohlin 2009) is similar to others that try to address the issue where not all citations are included in the h-index and that many different distributions of citations can have identical h-indices. Unlike the other indices, however, it does not start with h, and instead uses a somewhat complicated procedure of dividing papers into classes based on the number of citations. Rather than give publications more weight for every citation, this index gives more weight to citations as the publication moves from one class to the next. Publications with fewer than five citations are ignored (given a weight of 0). The first class represents publications with 5-9 citations; each subsequent class has a width double that of the previous class, thus the second class represents 10-19 citations, the third class 20-39 citations, etc. This structure was chosen (other classification schemes could be substituted) because citations curves are usually skewed with many publications with relatively smaller numbers of citations, and few publications with relative large numbers of citations.

Class
(c)
Citation Rangeln (lower limit)
(Tc)
Cumulative
Sum of Tc
(Vc)
00–40.00000.0000
15–91.60941.6094
210–192.30263.9120
320–392.99576.9078
440–793.688910.5966
580–1594.382014.9787
6160–3195.075220.0538
7320–6395.768325.8222
8640–12796.461532.2836
91280–25597.154639.4382
102560–51197.847847.2860
etc.

To calculate the metric, for each of the c' classes, one can count the number of publications within the cth class, Xc. Skewed distributions are often normalized using a logarithmic transform. Therefore, one calculates the natural logarithm of the lower limit of each class as Tc = ln(Bc) where Bc is the lower limit of the cth class. One can also calculate Vc as the cumulative sum of Tc for all classes from 1 to c. The w-index is then calculated as

$$w=\sum\limits_{c=1}^{c^\prime}{X_c V_c}$$

The w-index increases as a publication moves from one class to the next. Moving between larger classes gives more weight than moving between smaller classes. Because it considers citations more broadly, the w-index is more fine-grained than the h-index.

Example

Publications are ordered by number of citations, from highest to lowest.

Citations (Ci)592616111110433211100000

Class
(c)
Citation
Range
Count of Pubs
in Range
(Xc)
ln (lower limit)
(Tc)
Cumulative
Sum of Tc
(Vc)
XcVc
00–4120.00000.00000.0000
15–901.60941.60940.0000
210–1942.30263.912015.6481
320–3912.99576.90786.9078
440–7913.688910.596610.5966

The index is the sum of XcVc, thus w = 33.1525.

History

Yearw
19970.0000
19981.6094
19999.4335
200017.9507
200133.1525
200258.4809
200383.8094
2004125.2796
2005159.1489
2006211.2158
2007247.6346
2008313.0944
2009343.5218
2010377.8374
2011454.1407
2012497.7141
2013555.1268
2014576.7900
2015603.9748
2016646.1619
2017694.5873
2018717.6369
2019757.7446
2020789.7813
2021814.6635
2022844.1745
2023873.6855
2024916.0958
2025949.5189

References