The deputies shall perform their labors under the direction and advice of the assessor. It shall be his duty to examine and revise the lists of property taken by his deputies to see that the assessment is equal and uniform throughout his county; and if in his opinion any property, real or personal, in any of such lists, is valued at more or less than its value, according to the rule prescribed by law, he shall correct the same by giving to such property its value according to such rule. If it should appear that any property liable to taxation has been omitted from any taxpayer's list, the assessor shall list and value the same.