Sounds like Ordinary Magic (2012) by Caitlen Rubino-Bradway.
In Abby’s world, magic isn’t anything special: it’s a part of everyday life. So when Abby learns that she has zero magical abilities, she’s branded an “Ord”—ordinary, bad luck, and quite possibly a danger to society.
The outlook for kids like Abby isn’t bright. Many are cast out by their families, while others are sold to treasure hunters (ordinary kids are impervious to spells and enchantments). Luckily for Abby, her family enrolls her in a school that teaches ordinary kids how to get around in a magical world. But with treasure-hunting kidnappers and carnivorous goblins lurking around every corner, Abby’s biggest problem may not be learning how to be ordinary—it’s whether or not she’s going to survive the school year!
According to this review, the protagonist, Abby, undergoes a test all children take at a certain age, where they're assigned a numerical rating indicating their magical potential. Abby fails the first test, and is determined to have no magical potential. She subsequently goes to a school for ordinary children, which is protected by her older sister, Alexa:
Abby Hale leads a happy and normal life for a twelve year old – she goes to school every day and she’s lucky enough to have a large, loving family and plenty of friends. And, just like anyone else her age, Abby cannot wait to be Judged, because Judgement means she will finally be an Adult and that she’ll FINALLY be able to use magic (just like everyone else in the world). On the day of her Judgement, Abby is an excited mess of nerves and she wonders what rating she’ll receive – most kids get a Judgement of 5 or so (although Abby’s eldest sister, Alexa, got a nearly unheard-of 9 rating!). But Abby doesn’t even pass her first test, and she is Judged as having absolutely zero magical potential.
In other words, Abby is an Ord – that is, she’s “Ordinary” therefore impervious to magic and anathema to society.
You see, in Abby’s world, everything relies on magic, from the rooms in her family’s house to the shortcake that Abby’s mother conjures in her bakery. Ords are a danger because they can see through any spells and cannot be affected by magic, and reviled – they are treated as though their Ord-ness is a contagious disease (it’s not), and their basic human rights are stripped away. For instance Ords, especially Ord children, are often sold as slaves to traveling Adventurers (who find it useful to have someone impervious to magic on hand to walk through magical booby traps in the pursuit of treasure).
Abby soon learns all of this, as she’s kicked out of school, her supposed friends keep their distance, and a pair of brutal Adventurers show up at her family’s doorstep looking to purchase Abby for their next adventure. At least Abby has her family who stand by her and love her just as much as if she had been proclaimed an immensely powerful young mage. And just when all seems lost, Abby learns that her life is not without hope and opportunity – her eldest sister, Alexa, works a top secret job in Education for the kingdom, and it turns out she (and newly coronated King Stephen) has been a champion for Ords for years, protecting a school for Ordinary children in the heart of the kingdom’s capitol city. Soon enough, Abby is whisked away to Margaret Green School in Rothmere, where she learns that she is one of many Ords, where she makes new friends, and learns how to protect herself and use her Ordinariness to her advantage. Of course, danger abounds (what with redcaps and desperate Adventurers about), but with the help of her family and her friends, Abby may just be able to make it through her first year of school alive.
This preview of the book confirms that Abby's family uses a magic carpet to get around:
We climbed on, Mom putting me right between her and Dad, with Jeremy behind us, so they could both get to me quickly. The carpet had spells to keep everyone's balance, and there was a shield to prevent people from falling off. For extra insurance, Dad called up the rug fibers, twisting them into a rope around my waist. If I moved too much one way or another, it tightened up like a snake and jerked me back into my seat.
Dad murmured something, and the carpet lifted up. There's always a moment of freefall with magic carpets when your body goes up but your stomach stays down. It's my favourite part. The carpet picked up speed and my head started spinning as we kept going up and up and up and then -- everything was perfect. The wind was rushing all around, lifting us up until we were light as frosting and twice as weightless. The sky stretched out all around us, blue and cloudless. I could see part of one moon peeking out in front of us.
