What does it mean when you lose inches and not weight?

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People always say it's because you're losing fat and gaining muscle. But people keep saying that's a myth. So what exactly is happening? I've been stuck trying to lose my last 20 pounds and i'm going to try and be EXTREMELY strict this week to see if my diet really has something to do with the loss (but i can't seem to find my mistake). Luckily i did find i actually lost a inch around my waist.

But what does that actually mean? What's happening with my body? Should i try to go less in calories? (I'm eating about 1,500 average?) Should i try to switch up my exercise? Go with heavier weights?