# Sensitivity index

The sensitivity index or d (pronounced 'dee-prime') is a statistic used in signal detection theory. It provides the separation between the means of the signal and the noise distributions, compared against the standard deviation of the signal or noise distribution. For normally distributed signal and noise with mean and standard deviations μS and σS, and μN and σN, respectively, d′ is defined as:[1][page needed]

${\displaystyle d'={\frac {\mu _{\mathrm {S} }-\mu _{\mathrm {N} }}{\sqrt {{\frac {1}{2}}\left(\sigma _{\mathrm {S} }^{2}+\sigma _{\mathrm {N} }^{2}\right)}}}}$

d′ assumes that the standard deviations for signal and noise are equal. d′ can be estimated from the observed hit rate and false-alarm rate, as follows:[1]:7

d′ = Z(hit rate) − Z(false alarm rate),

where function Z(p), p ∈ [0,1], is the inverse of the cumulative distribution function of the Gaussian distribution.

d′ can be related to the area under the receiver operating characteristic curve, or AUC, via[1]:63

${\displaystyle d'={\sqrt {2}}Z({\mbox{AUC}}).}$

d′ is a dimensionless statistic. A higher d′ indicates that the signal can be more readily detected.