Fluorinated graphene has a tunable band gap that is useful in making flexible graphene electronics. But the carbon-fluorine (C-F) bonds in fluorinated graphene can be easily broken by increased temperature or electron beam irradiation. Here, we demonstrate that the stability of fluorinated graphene is mainly determined by its C-F configuration. The double-sided fluorinated graphene has a much stronger stability than the single-sided fluorinated graphene under the same irradiation dose. Density functional theory calculations show that the configuration of double-sided fluorinated graphene has a negative and low formation energy, indicating to be an energetically stable structure. On the contrary, the formation energy of single-sided fluorinated graphene is positive, leading to an unstable C-F bonding that is easily broken by the irradiation. Our findings make a new step towards a more stable and efficient design of graphene electronic devices.