The just noticeable difference (JND) is the smallest difference in something that can be perceived. For instance, if you show me two pieces of string that are very nearly the same length, and then another similar pair, and another, there’s a certain length difference that I will just be able to perceive, and any that are closer together than that I won’t be able to tell apart. Similarly for other senses: two audio tones have to have a certain amount of difference in their volume or frequency in order for me to tell that they were different rather than the same tone repeated. The size of the JND is dependent on methods: you can notice a smaller difference in lengths if you look at two pieces of string side-by-side rather than one on one day, and one on the other. It can also help if there’s a point of reference, such as a grid in the background. But nevertheless, there will be some small difference below which you will be unable to tell two things apart.
So the JND can vary quite a bit depending on the experimental procedures, but given a particular method, the JND scales with the starting size of what you’re looking at: JND ∝ dl/L. If you have double the length of string, the difference in length between two comparison pieces also has to double before you’ll notice that there has been a change. If you’re in a dark room with one candle lit, lighting a 2nd is very noticeable addition to the brightness. If you’re in a bright room with a thousand candle power light on, lighting a candle may not noticeably increase the brightness — and if you can just notice adding (or subtracting) one candle against a background of say 200, then you should be able to just notice a change of 1/200th of a candle against a background of one candle.
Let’s consider the case of hair. I cut mine every 50 days or so. It goes from about 0.3″ when freshly cut to about 1″ in that time, for a rate of growth of 0.014″ per day. After I cut my hair it takes about a week before I notice that it’s gotten longer. So the constant for the JND is:
0.014*7/0.3 = 0.33
If the starting length for hair was instead say, 12″, then the scaling indicates the JND would be 3.9″. That is, a girl with shoulder-length hair would have to cut off about 4″ in order to have a good expectation that — with a one day to the next observation — a boy would notice that indeed her hair had been cut. Getting a 2″ trim would fall well below the JND, and psychophysically, it would be highly unlikely for such a difference to be spontaneously noticed. Nay, nearly physiologically impossible for such a difference to be detected under such conditions.
Everyone’s JND constant will be different, and circumstances can vary (e.g., someone may consistently wear shirts with horizontal markings on them to serve as a guidepost, or an observer may have superhuman vision discrimination, or the hair may be pulled into a ponytail, making the judgment even more difficult).
But whatever the individual circumstances, don’t forget the pioneering psychophysics work of Weber when someone doesn’t notice your haircut — they may not have been able to!