The electrical contact resistance between two stainless steel balls was measured to study the electrical conduction of a metallic contact separated by a thin oxide layer. Through a statistical approach, the contact resistance was found to have a bimodal log-normal distribution. This result reflects conduction by tunneling and electrically induced metal bridge, which was explained by the simulation of a random circuit breaker model, inside the insulating layer. Based on the results of this study, we suggest an effective conduction model to explain macroscopic electrical contact systems with a nano- or microscopic filamentary conduction mechanism.