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A detailed mathematical justification of the indices can be found in Golosov, Grigorii V. "The Effective Number of Parties: A New Approach", Party Politics, Vol. 16, No. 2, March 2010, pp. 171-192 and Golosov, Grigorii V. "Party System Nationalization: The Problems of Measurement with an Application to Federal States", Party Politics, forthcoming; first published electronically on September 9, 2014 as doi:10.1177/1354068814549342. The formulas are provided in these publications as appropriate. However, I have found that some of the scholars find the mathematical notation insufficient for using the formulas with the standard software such as Excel. The example below is intended to help them with that. When providing the example, I presume that the reader is absolutely unsophisticated in mathematics. Please do not feel insulted if you are not. Then the formulas in the cited publications will suffice for you, and you can safely skip further reading. Consider a real-life distribution of the vote in national parliamentary elections (Liechtenstein 2005 is selected for illustrative simplicity)
Well, this is easy. We don’t need the territorial distribution of the vote here, so we take the national totals and (1) transform them into decimal shares by dividing by the overall number of votes. The formula in D1 is =B1/C1, and so on to D3. Then (2) we copy the value (NOT the formula) of D1 into E1 and drag it down. This value is a constant: the decimal share of the vote of the largest party. Smaller values must not be entered into E1. Then (3) we enter the formula, =D1/(D1+E1^2-D1^2), into F1 and drag it down. Note that the resulting value in F1, i.e. for the largest party, is always one. (Since Golosov's effective number of parties can be decomposed into individual-party components, the values F1, F2, and F3 are relative sizes of individual parties, dubbed the "effective size scores", ESS). Then (4) we use F4 to sum up (F1:F3).
This is a lot trickier. (1) We start with transforming the numbers of votes received by parties in territorial units into decimal shares. Since the procedure is simple and already described above, I skip this stage in the presentation below. (B2:D2) and (B3:D3) should sum up to 1 in E2 and E3. Then (2) Enter =SUMSQ(B2:B3) into B4 and drag it to D4. Then (3) Enter =SUM(B2:B3)^2 into B5 and drag it to D5. Then (4) Enter =1-(2-(B5/B4))/(2-1) into B6 and drag it to D6. IMPORTANT: the numeral 2 in this formula stands for the number of territorial units. In other words, this value is variable. If there were 10 territorial units, then the formula would be =1-(10-(B5/B4))/(10-1). (The values B6, C6, and D6 are individual nationalization scores for parties 1, 2, and 3, respectively). (5) Enter the decimal shares of the vote received by the parties nationally, established as for the effective number of parties, into B7, C7, and D7. (6) Enter =B6*B7 into B8 and drag it to D8. (7) Use E8 to sum up (B8:D8).
Note that in Russified Excel, the equivalents of SUMSQ and SUM are СУММКВ and СУММ, respectively. |