I'm sure I could find the answer to this in a nanosecond by searching for it, but my guess it would be something to do with Rigor Mortis, or the peculiar stiffening of a corpse. If you were "scared stiff" you would be literally scared to death.
Phil
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"Scared Stiff" (1953).
Scared stiff
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"Scared stiff" is an idiom that refers to being completely frozen with fear. It's origin is not known, but it has been in use since at least 1900 a. d. .
Scared Stiff (1953)
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The dead body was already stiff when I found it there.