Yes, telling us what phone you have is important, since many smartphones use data in the background ALL THE TIME regardless of what apps you install, activate.  This is how these particular devices are designed to work, so its not a malfunction.  But many of these “background” services can be turned off, if you tell us what phone you are talking about.   On some phones, turning off data does nothing except deactivate it until the next time an app turns it back on.  Believe it or not.  But there are ways to “really” turn off the data connection.