The subject of effects of solar wind density (Np) on auroral electrojets has recently received increasing attention. For unbiased events in which only Np varied significantly but other solar wind parameters were relatively constant, we calculated rates of change of the auroral electrojets per unit change of Np (dAU/dNp and dAL/dNp). Here we report that the distribution of dAU/dNp and dAL/dNp binned by the Z component of interplanetary magnetic field (IMF) shows a butterfly pattern. The rates are found to be small for northward IMF. As the southward IMF becomes larger, the distribution bifurcates--one branch extending to positive rates and the other to negative rates. Although it has been commonly believed that the enhanced Np increases the auroral electrojets, the enhanced Np can also decrease the auroral electrojets with a 49% probability.