Not really victim, it's still 24 hours long. It starts at midnight and ends just before the next day's midnight.
It's impossible to measure the instant between seconds, because it has zero length, and our minds can't conceive of it. So we measure the space between instants. The first second of a minute we label "0" and the 60th we call "59". (Just like a ruler/measuring stick starts with zero.) And a day goes from 12:00:00 a.m. to just before the next 12:00:00 a.m..
Like you said, 14 full days before the middle point, and 14 full days after. The middle happens after the 14th day and before the 15th day.
And now that I've confused anyone actually trying to think too deeply about this, the middle of a 31-day month is the 16th at noon. And the middle of a 30-day month is midnight of the 16th day.
