Everyone has this issue, because once we become adults we all have our own level of ego when it comes to who is right. It's just part of dealing with other adults that there is a kind of political game at work behind the scenes, which, if you're good at it, allows you to communicate that someone is incorrect without insulting their confidence.
Unfortunately, people, within my experience, often decide to ignore that possibility and just say that the other side is wrong, or else that someone has to be wrong and someone has to be right. Neither of those statements ever helps. Look at how society is split down the middle on religious grounds - atheists are convinced that there is no God, and religious people are convinced that there is, or else that there are many gods, or that there are spirits, or magic, etc. All of this division is really, obviously, a result of ego. A result of our need to be correct over others, and to divide ourselves from those who's disagreement is an inconvenience.
While it is true that saying you're right when you have very powerful evidence saying that you are makes you objectively justified, only through empathy and subjective reasoning will you ever convince those who disagree. In other words, tolerance comes before charisma, and charisma before diplomacy, and diplomacy before decisions, if you want to convince people on this. Few will surrender their ego immediately except if the empirical evidence is on hand - if it is, then that is a powerful tool indeed.
The situation with your sister is one example. If you had seen this information on a computer with her present, you might have said, "huh, I didn't know that, listen to this, sis", and read the thing that she didn't believe out. She would have accepted it, I'd bet. This is a result of the lack of ego-threatening danger in the situation - instead of directly attempting to prove her incorrect or "wrong", you have given her new data that suggests she was misinformed. That situation means that her self-worth is not threatened. So try the new data approach next time you have such an issue, with your sister if you can, since with family it is possible to have the data nearby in the house.
Never expect the scientific approach to thinking from others. You will be disappointed, and in any case, you might be wrong yourself on occasion. Not everything can be solved with objective logic.
AkJay said:
This is a REAL argument I had with someone. I swear to your God that I wish I was lying:
(I live on the North-East Coast of the U.S.A)
Girl: "The Moon is closer to us than Florida"
Me: "No it isn't"
Girl: "Yes it is! I can see the Moon, it's there, and I cannot see Florida, so it's simple logic that the Moon is closer."
Me in the middle of face-palming: "The Moon is thousands of miles away, Florida is in our country, We can drive to Florida in a few hours or one day at the most. It took a spaceship with rocket propulsion 3 days, and trust me, those rockets are going a lot faster than 65 MPH."
Girl: "Whatever, you're full of shit."
EDIT: Another argument I just remembered.
Guy : "Where is the other side of the map?"
Me : "The map is flat, there is no other side."
Guy : "You're joking? right? The world is a sphere, any idiot knows that. So if the map is one side, where is the other side?"
This will be held in my mind for good as proof that, at this stage in history, America still has horrible general education for geography. Or, possibly none. Then again, it might be a child speaking in both of those situations. It wouldn't be a big deal if a three year old said it. An adult, well... I guess there's a chance they were playing with your mind, too.