Average Damage is the damage dealt to the enemy. Average Damage should be adjusted to damage before reduction, see this section for steps
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Base Stat is the tested Stat of the monster before any additions (base atk, def, hp, etc).
Shown Stat is the tested Stat of the monster shown when holding the monster during selection for battles (Includes bonuses from runes).
Total Skill-up Bonus is the total bonus from skill-ups of the monster. Can be checked by clicking Skills on the monster in your inventory.
Other Bonus is the total of other bonuses that affect the tested Stat from leader skills or glory buildings which can be found in battle by clicking "settings > ally buff".