The Just -noticeable difference is the smallest detectable difference between a starting and secondary level of a particular sensory stimulus. I will mathematically prove this cannot exist because it creates a contradiction.
1. Your estimate of the insentities of two stimuli are equal if their intensities differ less than j.
2. Your estimate of the insentities of two stimuli are different if their intensities differ by j or more.
A, B and C are three stimuli.
The intensity of A is q
The intensity of B is q + (2/3)j
The insensity of C is q + (4/3)j
A and B differ by (q + (2/3)j) – q = (2/3)j which is less than j, so your estimates of their intensities must be equal. Let the value of your estimate = x.
B and C differ by (q + (4/3)j) – (q + (2/3)j) = (2/3)j which is less than j, your estimates of their intensities must be equal. Let the value of your estimate = y.
x must not equal y because they are the values of your estimates of A and C which differ more than j (assumption 2).
x must equal y because they are both the value of your estimates of the intensity of B.
As you can see, there is a contradiction. Therefore the assumptions that were made were not all correct, meaning that the just noticeable difference cannot exist since it is dependent on the assumptions.
Apple sells a brand of displays named “Retina Display” which they claim has a high enough pixel density that the human eye is unable to notice pixelation at a typical viewing distance. This wrong – there is no ultimate pixel density – you will always be able to improve how a display looks by increasing its pixel density.
In reality we don’t always estimate the same value for two stimuli of the same intensity. But this does nothing to harm the proof I just gave beacause all it means is that instead of using individual values, we use the average of an infinite number of estimates and everything holds.
You might tell me that you can’t differentiate between a line of length 1m and 1.000000001m, but you’d be wrong. You would be able to tell the difference, it would however take a very large number of trials. The probability that you were able to correctly pick the longer of the two lines approaches 0.5 as the difference in their lengths decreases.
What does this mean for everyday things? You should just evaluate stuff in terms of worth. Instead of thinknig “Spending another minute on working out will produce no noticeable difference so I won’t do it”, think “Spending another minute on working out will produce a very small difference and isn’t worth the time I put into it, so I won’t do it”.