Three new aspects regarding the analysis of Langmuir probe data are presented. First, we demonstrate that the numerical results of Laframboise for spherical probes can be parametrized easily for arbitrary ratios of the probe radius r(p) to the Debye length lambda(D). The ion current can be expressed in the form a(-X)(b), where a and b are parameters depending on r(p)/lambda(D), and X is the dimensionless probe voltage. This functional form is the same as the one for cylindrical probes reported previously, but the values of a and b are different. Second, we use numerical simulations to show that unless the plasma potential V-s is known, it is in general difficult to determine accurately the form of the ion current characteristic I-i(V-p), and thus the ion density N-i, from typical probe data. This is because I-i(V-p), N-i, and r(p)/lambda(D) are interdependent. Third, the simulations indicate that the apparent electron energy distribution is very sensitive to the exact form of I-i(V-p) and to the method by which I-i(V-p) is subtracted from the total probe current to obtain the electron current. A linear extrapolation of I-i(V-p) is often adequate for determining the electron temperature, but assuming a constant ion current leads to electron energy distributions that appear to have two components with different electron temperatures. Additional issues discussed include the consequences of a slightly collisional probe sheath and the importance of end-effect corrections with a cylindrical probe.