# Just Enough to Notice

Portrait of Ernst Heinrich Weber.

Ernst Heinrich Weber
Among the different fields of science, mathematics is the umbrella of them all. It amazes me how we can mathematically make sense of anything around us. Although it may take a quite a bit of time to fully understand the function of something, the possibility is there. Because of this ability, the problems both physically and psychologically can be deconstructed. This train of thought eventually led me to Ernst Heinrich Weber (1795 – 1878). His works and legacy proved to be phenomenal and led to many new studies in the field of psychology, physiology, and anatomy. His dedication towards his career gave him great recognition among the science community. He became known as one of the founders of psychophysics and experimental psychology. The implications of experimental psychology itself gave a new perspective for many psychologists in creating new research studies and finding new discoveries. Unquestionably Ernst Heinrich Weber was one of the most intuitive people during his time. Not only was he one of the pioneers of experimental psychology, he was able to produce to what we know as Weber’s Law.

Weber’s Law
Some of Ernst Weber’s most known works come from his studies on human physiological perception. During his research, he claimed that humans had the ability to distinguish the relative difference of two different items and not the absolute difference. We can essentially differentiate two different objects by using our sensory and perception to notice if one object is bigger, heavier, or louder than the other. This type of intuitive statistics is what Weber called Just-Noticeable Difference (JND). Just-Noticeable Difference is also known as difference limen, differential threshold, or least perceptible difference. Weber’s first experiments with this notion began by using weights and comparing them. He wanted to know the minimum amount of difference between two weights in order to tell them apart. What he discovered was that the most noticeable amount of weight was in the difference of 3% between the two. This discovery eventually led him to produce an equation that is now known as Weber’s Law.

Weber’s Law is shown as follows:
ΔR/R = k

Where R is the amount of existing stimulation present. (Delta) R is the difference of existing stimulation that gives us the amount that needs to be added for the Just-Noticeable Difference. Lastly k represents a constant that is different for every sense being measured. Something to note is that it has been found that in a range of high intensities, this law is proved to be invalid. However, one of Ernst Weber’s colleagues, Gustav Theador Fechner, was able to use Weber’s findings and was able to develop his own formula that measured all types of Just-Noticeable Difference senses.

Examples of Weber’s Law
To understand Weber’s Law a little more, there are a couple of examples that can be made.

Image: Roman Oleinik, via Wikimedia Commons.

Suppose there are 50 pennies and each penny weighs 2.5 grams (g). If we add the total weight, we will get 125g. If we take 3% of 125g to get our Just-Noticeable Difference, we will get about 3.75g. Therefore if we have two bags of pennies and put 50 (125g) in one and 52 (132.5 g) in another, we should be able to tell which bag is heavier, just by holding the two bags with your arms in the air.

A good visual example of this is through this link on YouTube:
https://youtu.be/wVhiezByMSU?t=1m45s

In the video, the person is using his stereo system in his car and adjusting the volume to show how Weber’s Law can be applied. He shows that it is difficult distinguish the difference when the volume is loud or quiet. Although he doesn’t show a precise way to measure and find the Just-Noticeable Difference, his presentation gives a clear idea of how Weber’s Law is applied.

Conclusion
With a little bit of curiosity, we can find hundreds of different types of mathematics being applied. Ernst Heinrich Weber is an example of using conventional math and applying into something at the time that may seem less mathematical. Because of his ability to combine both physics and psychology, he was able to create a new field of study and even a law named after him. As we continue to study mathematics and its history, we can see that all types of studies can be seen with a mathematical lens.