A statistical model of short time-scale geomagnetic fluctuations is developed and used to evaluate how geomagnetic dynamics are influenced by different solar wind controlling parameters. The functional form of the probability distribution function (PDF) that describes extreme-value (greater than 4σ) minute-to-minute changes in the ground magnetic field (Δx) at magnetometer station Sodankyl¿ (geomagnetic latitude and longitude of <63.87,107.61>) is shown to be nearly independent of the variables solar wind (SW) forcing, local time (LT), and day of year (DOY). Instead of modifying the intrinsic dynamics, as characterized by the functional form of the PDF of Δx, these variables are shown either to amplify or reduce the absolute level of variability of the fluctuations: The primary difference in the PDF tail of Δx during weak and strong solar wind forcing is the standard deviation, σ; the functional form of the PDF = f<Δx/σ(DOY,LT,SW)> is nearly invariant. In a statistical interpretation, we conclude that differences in solar-generated conductivity, seasonal effects, strength of solar wind forcing and variability, and position of the magnetometer ground station in local time do not change the structure of the extreme-value dynamics, as characterized by the probability distribution of Δx, but they serve to amplify the intrinsic variability.