  Extremely red quasars (ERQs) are an interesting sample of quasars in the
Baryon Oscillation Spectroscopic Sample (BOSS) in the redshift range of $2.0 -
3.4$ and have extreme red colours of $i-W3\ge4.6$. Core ERQs have strong CIV
emission lines with rest equivalent width of $\ge100$\AA. Many core ERQs also
have CIV line profiles with peculiar boxy shapes which distinguish them from
normal blue quasars. We show, using a combination of kernel density estimation
and local outlier factor analyses on a space of the $i-W3$ colour, CIV rest
equivalent width and line kurtosis, that core ERQs likely represent a
  separate population rather than a smooth transition between normal
  blue quasars and the quasars in the tail of the colour-REW distribution. We
apply our analyses to find new criteria for selecting ERQs in this 3D parameter
space. Our final selection produces $133$ quasars, which are \emph{three} times
more likely to have a visually verified CIV broad absorption line feature than
the previous core ERQ sample. We further show that our newly selected sample
are extreme objects in the intersection of the WISE AGN catalogue with the
MILLIQUAS quasar catalogue in the colour-colour space of ($W1-W2$, $W2-W3$).
This paper validates an improved selection method for red quasars which can be
applied to future datasets such as the quasar catalogue from the Dark Energy
Spectroscopic Instrument (DESI).
