The structural properties that SEM photograph of ZnO varistors surface studied by fractal mathematics program were investigated to verify the relations of electrical characteristics. The SEM photograph of ZnO varistors surface were changed by binary code and the grain shape of that were analyzed by fractal dimension. The void of ZnO varistors surface was found by fractal program. The relation between grain density and electrical properties depend on fractal dimension. The grain size in ZnO varistors surface was decreased by increasing of Sb$_2$O$_3$ addition. The spinel structure was formed by Sb$_2$O$_3$addition and it was depressed the ZnO grain formation. The grain size of ZnO by Sb$_2$O$_3$addition were from 5 to 10[${mu}{ extrm}{m}$]. Among of ZnO varistors, fractal dimension of ZnO4 was very high as a 1.764. The density of grain boundary in ZnO2 and ZnO3 varistors surface was 15[%] by formed spinal structure. The breakdown electric field of ZnO2 that fractal dimension has 1.752 was very high to be 8.5[kV/cm]. When the fractal dimensin was high, the grain shape of ZnO varistors was complex and the serial layers of ZnO grain was increased.