# How do we compare two numbers

Posted by Yuling Yao on Sep 15, 2021.I was reading an article on how the politician’s height can have a causal effect on electability. But then I realize we often have a different scale for comparing numbers when we know these numbers represent some physical objects.

Here are two examples:

- As per Google, Pete Buttigieg’s height is 5’8 and Gavin Newsom’s is 6’3, who are on the relatively short and tall end of the modern day politician’s height spectrum respectively. With these two numbers in mind, certainly 6’3 is much bigger than 5’8, right?
- In the 2020 U.S. Presential election, the Democratic share in TX was 47% and the Republican share was 52%. Hey, it was 47 and 52: what a tossup!

The point is that 6’3 / 5’8 = 190.5 cm / 172.7 cm = 1.10, and 52 / 47 = 1.11. These two sets of comparisons have the same multiplicative difference, but why do we automatically read that 6’3 $»$ 5’8, while 52 $\approx$ 47?

One explanation is some sort of anchor effect. We encounter this arbitrary anchor choice in data visualization too: when comparing two coefficients, what $y$-axis scale are we using? Here by looking at the multiplicative difference, we have implicitly included zero as the lower end of the $y$-axis. But an adult male politician’s heigh cannot be zero, so maybe implicitly we have a different lower end point, or the anchor, say 5’6, then the actual multiplicative difference we are reading in mind is (6’3 - 5’6) / (5’8-5’6) = 4.6.

Another explanation is that we have mapped the parameters into some decision theory. When a computer reads 6’3, it is just some 32-bit integer. But we are not computers after all. We automatically generate a decision theory, in which the integer 6’3 is mapped to a masculine man wearing a brooks brother suit and oxford shoes, while the number 52% is mapped to some annoying recounting and the reflection of 2000. None of such additional information is coded by the numbers as they are presented.