wrong end of the stick, the - Phrases/Idioms
The American Heritage® Dictionary of Idioms by Christine Ammer. Copyright © 2003, 1997 by The Christine Ammer 1992 Trust. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.
A misunderstanding or distortion, as in We ordered a “full quart” of rice, but the clerk got hold of the wrong end of the stick and sent us “four quarts” instead. This expression refers to a walking stick held upside down, which does not help a walker much. It originated in the 1400s as worse end of the staff and changed to the current wording only in the late 1800s. Also see short end of the stick.