  TThe Period-Luminosity (PL) relation for Cepheid variable stars in the Large
Magellanic Cloud (LMC) is crucial for distance measurements in astronomy. This
study analyzes the impact of using the median rather than the mean on the PL
relation's slope and zero point. It also examines the persistence of the break
at approximately 10 days and addresses specification issues in the PL relation
model. Using $VI$-band median and mean magnitudes from the OGLE-IV survey,
corrected for extinction, we fit the PL relation employing robust
$MM$-regression, which features a high breakdown point and robust standard
errors. Statistical tests and residual analysis are conducted to identify and
correct model deficiencies. Our findings indicate a significant change in the
PL relation for Cepheids with periods of 10 days or longer, regardless of
whether median or mean magnitudes are used. A bias in the zero point and slope
estimators is observed when using median magnitudes instead of mean magnitudes,
especially in the $V$-band. By identifying and correcting regression issues and
considering the period break, our estimators for slope and zero point are more
accurate for distance calculations. Comparative analysis of the models for each
band quantifies the bias introduced by using median magnitudes, highlighting
the importance of considering the Cepheids' period for accurate location
measure results, similar to those obtained using mean magnitudes.
