Why won’t Outlook sign out after it’s choosen and you try to return to your home page?????

I uncheck the stupid and I mean stupid "stay signed in" everytime I use Outlook which once I sign out and try to go back to my homepage still displays all my emails. Since I use a public computer as I am at college why won't the sign out in outlook actually sign you out? I can't trust the page to sign me out so I have to restart the whole system every time and it is quickly coming to the point where I may close the account and just use Google for it actually signs me out of the account when I log off.