Measuring

Is the circumference of the FINISHED hat the same size as the chart indicates or when it is placed on the head?

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In my experience, the measurements usually pertain to the piece as it lies by itself, not when it is stretched or worn. (If blocking is part of thefinishing process, that may be a different matter.) So if a beanie pattern tells me to work until the piece measures 7 inches, I can measure it while still on the needles (not take it off and try it on my head).
In hats, the pattern often indicates what size head the hat is designed to fit, but the actual piece might be slightly larger or smaller to allow for ease or stretching, depending on the fabric and how snug it should fit. (A hat made entirely in ribbing might measure 18 inches around, but fit a head that is 20 inches around because of the stretch of the fabric.)