An evaluating method for determining deterioration of a lean NO.sub.x catalyst installed in an exhaust gas system of an internal combustion engine for reducing NO.sub.x under a lean air-to-fuel ratio condition, comprising the steps of determining oxygen concentration upstream and downstream of the catalyst with a sensor when the engine is operated under the lean air-to-fuel ratio condition, and
- evaluating said catalyst by comparing a difference of the determined oxygen concentration upstream and downstream of said catalyst via the sensor.