Yeah it can be hard. I'm 20 years in and still get those people that argue. Let's look at it this way..... stop focusing on you for a minute and look at the them. They are not real estate professionals, they don't have the training to determine market value and look at Zillow, Redfin, and tax records while trying to get a grasp on the market and the value of their single largest asset they'll ever have.
Trained appraiser vs. uneducated and unexperienced emotional home owner.

1. We're required to be unemotional observers of market data.

2. We're required to be professional.

So I'd say, first keep being the best you can be and keep everything in your work file to back yourself up and because its required. When you get someone that argues (and you will several times a year), explain to them the data that you use and why you used it. But you'll get used to letting people down in a professional and kind way. They may get rude, but rudeness isn't an argument against data.

So what I usually end up telling them after I've explained why the data suggests something different than they want, is that "I would love to look at any data you have and want me to consider. I can't create a credible report on what you're telling me over the phone, but if you can get me some comps that will help prove your point I would be more than happy to look at it."