The Continuous Ranked Probability Score (CRPS) measures the integral square difference between the cumulative distribution function (cdf) of the forecast $\mathrm{F}_S \left( q \right)$, and the corresponding cdf of the observed variable $\mathrm{F}_Q \left( q \right)$,

\begin{equation} \mathrm{CRPS} = \int_{-\infty}^{\infty} \left\lbrace \mathrm{F}_S \left( q \right) - \mathrm{F}_Q \left( q \right) \right\rbrace \mathrm{d} q \textrm{.} \end{equation}

The mean CRPS comprises the CRPS averaged across $J$ pairs of forecasts and observations,

\begin{equation} \overline{\mathrm{CRPS}} = \frac{1}{J} \sum \limits_{j=1}^{J} \mathrm{CRPS}_j \textrm{.} \end{equation}

The Continuous Ranked Probability Skill Score (CRPSS) is a function of the ratio of the mean CRPS of the main prediction system, $\overline{\mathrm{CRPS}}$, and a reference system, $\mathrm{\overline{CRPS}_{\mathrm{ref}}}$,

\begin{eqnarray} \mathrm{CRPSS} &=& \frac{ \mathrm{\overline{CRPS}} - \mathrm{\overline{CRPS}}_\mathrm{ref} }{ \mathrm{\overline{CRPS}}_\mathrm{perfect} - \mathrm{\overline{CRPS}}_\mathrm{ref} } \\ &=& \frac{ \mathrm{\overline{CRPS}} - \mathrm{\overline{CRPS}}_\mathrm{ref} }{ 0 - \mathrm{\overline{CRPS}}_\mathrm{ref} } \nonumber \\ &=& \frac{ \mathrm{\overline{CRPS}}_\mathrm{ref} - \mathrm{\overline{CRPS}} }{ \mathrm{\overline{CRPS}}_\mathrm{ref} } \nonumber \\ &=& 1 - \frac{ \mathrm{\overline{CRPS}} }{ \mathrm{\overline{CRPS}}_\mathrm{ref} } \nonumber \end{eqnarray}