Signal Detection Theory Calculator
Calculate d-prime (d'), criterion c, and sensitivity from hit rate and false alarm rate.
Used in psychophysics, radiology, and machine learning to separate sensitivity from bias.
Signal detection theory (SDT) separates a person’s sensitivity from their response bias. It was developed in the 1950s for radar operators and became central to psychophysics, medical diagnosis, and machine learning performance evaluation.
The setup. On each trial, a signal is either present or absent. The observer responds “yes” or “no.” This produces four outcomes: hits (signal present, said yes), misses (present, said no), false alarms (absent, said yes), correct rejections (absent, said no).
d-prime (d’) measures sensitivity — the distance between the signal and noise distributions in units of their standard deviation:
d’ = z(hit rate) - z(false alarm rate)
where z is the inverse of the standard normal CDF (the z-score corresponding to that probability).
d’ = 0: the observer cannot distinguish signal from noise. d’ = 1: moderate sensitivity. d’ = 2: good sensitivity. d’ > 3: excellent sensitivity.
Criterion c measures response bias — how liberal or conservative the observer is:
c = -0.5 x (z(hit rate) + z(false alarm rate))
c = 0: unbiased (no preference for yes or no). c > 0: conservative (reluctant to say yes — misses more than false alarms). c < 0: liberal (eager to say yes — false alarms exceed misses).
A key insight: two observers can have the same hit rate yet completely different sensitivity if one is liberal and the other conservative. d’ removes this confound. This is why d’ appears in radiology (how well does this reader detect tumors?), airport security, and any classifier evaluation that needs more than accuracy.