textbf{Step 1: Electron-withdrawing nature.}
Electron-withdrawing groups (EWGs) are groups that pull electron density towards themselves through inductive or resonance effects. Among the given groups, we need to evaluate their ability to withdraw electrons.
- \( \text{NO}_2 \) (Nitro group): Strong electron-withdrawing group due to both inductive and resonance effects. It withdraws electrons through its electronegativity and the resonance structure it forms with the rest of the molecule. Hence, \( \text{NO}_2 \) is the strongest EWG.
- \( \text{CN} \) (Cyanide group): A strong electron-withdrawing group due to its triple bond with carbon and the electronegativity of nitrogen. It is also highly effective in withdrawing electron density through its inductive effect.
- \( \text{COOH} \) (Carboxyl group): The carboxyl group is an electron-withdrawing group, though weaker than \( \text{NO}_2 \) and \( \text{CN} \). It can withdraw electrons via the resonance effect, but it is not as strong as the nitro or cyanide group.
- \( I \) (Iodine): Although iodine is electronegative, it is a weak electron-withdrawing group compared to the others listed here. This is because iodine has a lone pair of electrons that can participate in resonance, providing some electron-donating character, though still an electron-withdrawing group overall.
Step 2: Final order.
Based on the electron-withdrawing ability, the decreasing order is: \[ \text{NO}_2>\text{CN}>\text{COOH}>I \] \[ \boxed{\text{NO}_2>\text{CN}>\text{COOH}>I} \]