  The interest in polar codes has been increasing significantly since their
adoption for use in the 5$^{\rm th}$ generation wireless systems standard.
Successive cancellation (SC) decoding algorithm has low implementation
complexity, but yields mediocre error-correction performance at the code
lengths of interest. SC-Flip algorithm improves the error-correction
performance of SC by identifying possibly erroneous decisions made by SC and
re-iterates after flipping one bit. It was recently shown that only a portion
of bit-channels are most likely to be in error. In this work, we investigate
the average log-likelihood ratio (LLR) values and their distribution related to
the erroneous bit-channels, and develop the Thresholded SC-Flip (TSCF) decoding
algorithm. We also replace the LLR selection and sorting of SC-Flip with a
comparator to reduce the implementation complexity. Simulation results
demonstrate that for practical code lengths and a wide range of rates, TSCF
shows negligible loss compared to the error-correction performance obtained
when all single-errors are corrected. At matching maximum iterations, TSCF has
an error-correction performance gain of up to $0.45$ dB compared with SC-Flip
decoding. At matching error-correction performance, the computational
complexity of TSCF is reduced by up to $40\%$ on average, and requires up to
$5\times$ lower maximum number of iterations.
