Signal Detection (sensory decision) Theory
is a mathematical, theoretical system that recognizes that individuals are not merely passive receivers of stimuli. They are also engaged in the process of deciding whether they are confident enough to say "Yes, I detect that stimuli" when engaged in psychophysics experiments.

With two possible experimental trials (signal present or absent) and two possible participant responses ("yes" it is present or "no" it isn't there) there are four possible outcomes to each of many trials.
 

Signal
Response
Yes
No
Present
Hit
Miss
Absent
False Alarm
Correct Negative
 
Participants' responses on each trial are going to be consequences of both their perceptual sensitivity to the stimuli presented and their decision strategy or bias toward saying some thing is there or not when they are in doubt.

Calculating d' From a Single Outcome matrix

Data required for each point on an isosensitivity (ROC) curve requires hundreds of trials (to get accurate probabilities for Hits and False Alarms). With a few assumptions, d' can be calculated from a single outcome matrix using Signal Detection theory.

This method assumes that:
1. Noise is normally distributed. Presenting a signal on top of that noise, will therefore shift the amount of sensory activity to the right (higher), by an amount equal to that sensory systems sensitivity to that signal. The difference between the mean amount of sensory activity generated by the noise alone trials and the signal+noise trials will equal sensitivity (d') measured in z-score (standard deviation) units.
2. Participants adopt a criterion (b) for dealing with those values of sensory activity that could result from either noise alone or signal plus noise (the area where the noise and signal+noise distributions overlap). If the amount of sensory activity exceeds that amount, the participant will say the detected the signal, any amount less than that and they will say they did not detect the signal.

With these assumptions, the four cells of an outcome matrix can be represented as areas under the two normal distributions (for sensory activity experienced on noise alone trials and signal+noise trials).

d' = ZFA - ZHit

Tables for the z-score distribution or percent area under the normal curve typically present the z-score distances between the mean and the Criterion value (b). If you are using such a table, ZFA can be found by looking up the z-score associated with (50 - False Alarm %). If this number is positive, then the z-score to be put into the above formula will also be positive, if it is negative, the z-score value for the formula will also be negative. It is essential that the proper signs be used. A good way of checking would be to draw the distributions and the criterion and see the relationship between d' and the two z-scores. Similarly, to find ZHit, look up (50 - Hit %), again, the resulting sign will be the same as is used for the z-score in the formula.

E.g.,

Signal
Proportion of Responses
Yes
No
Present
.60
.40
Absent
.20
.80
 
d' = ZFA - ZHit
d' = Z (50-20) - Z (50-60) 

looking up the z-score associated with 50-20= 30% of the area under the normal curve, it is .842; for 50-60= -10% it is .253. Since 50 - 60 is a negative, -.253 is put into the formula to get:

d' =.842-(-.253) = .842+.253= 1.095