Signal Detection Theory Calculator

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).

Likelihood ratio beta (β), the third number this calculator returns, is the other way of expressing bias. Where c measures the criterion as a distance from the midpoint, β measures the height ratio of the two distributions at the point where the observer has placed their cutoff:

β = exp[(z(F)² − z(H)²) / 2]

β = 1: no bias, the criterion sits where signal and noise are equally likely. β > 1: conservative, the observer demands more evidence before saying yes. β < 1: liberal.

c and β always agree about the direction of the bias, because they are two descriptions of the same criterion placement. Most modern papers report c, since β is awkward to compare across observers with different sensitivity. β survives because it has a cleaner decision-theory reading: it is the odds ratio at which the observer switches their answer.

A worked example. A radiologist reads 200 scans. Of the 100 with a tumour present she flags 85, and of the 100 clean scans she flags 15.

• z(0.85) = 1.036, z(0.15) = −1.036
• d′ = 1.036 − (−1.036) = 2.07, good discrimination
• c = −0.5 × (1.036 + (−1.036)) = 0.00, no bias at all
• β = exp[((−1.036)² − 1.036²)/2] = exp(0) = 1.00

Now a second radiologist catches 95 of the tumours but also flags 30 of the clean scans. His hit rate looks far better. But z(0.95) = 1.645 and z(0.30) = −0.524, so d′ = 2.17 and c = −0.56. His sensitivity is barely different from hers. He is simply readier to say yes, which is what a negative c means. Judged on hit rate alone he looks like the stronger reader, and he is not.

That is the whole point of the method, and it is why d′ appears in radiology, in airport screening, in eyewitness identification research, and in any classifier evaluation where plain accuracy would reward a machine for guessing.