Why isn't the data producing a constant when I divide the force at x distance by the force at 2x distance? Shouldn't all inverse relationships produce a constant if you divide by the same ratio?
Let me address both questions.
To answer the first question: because the relationship between force and distance is non-linear. The data you collected via your experiment shows an inverse relationship between force and distance (the greater the distance the lower the force and vice versa), which you already know but brings me to the other question.
To answer the second question: no, all inverse relationships don't produce a constant if you divide by the same ratio. The key word is relationship; there's a particular relationship called proportionality which is for linear relationships only, in this case, when you have a variable $a$ inversely proportional to another variable $b$ this means that
$$a=\frac{k}{b} \ \mbox{ or if you prefer } \ a*b=k $$
where $k$ is a constant. Notice that this is what you're thinking about, but a relationship is a more general term which you're confusing with this special case of relationship called proportionality. So take the case you have between force ($F$) and distance ($d$), because as you have proved clearly
$$F*d\neq k$$
In this case there isn't an inverse proportionally relationship, and you actually just say that there's an inverse relationship. Also when you are dividing between the forces (i.e. comparing the same variable) you are thinking about direct proportionality $\frac{F_1}{F_2}=k$ and this is also not true because force depends on distance and as you saw it does so in a non-linear manner.
A really easy way to check which is it for your data is to simply plot it in a graph, if it's proportional you'll get a line, if it isn't you'll get something else, like in your case:
Analysis Requested by post author
Is there no way I can show in my data analysis that at double the distance, force is decreased a certain factor?
Yes you can. If you plot your data with the best trend line (i.e. the mathematical model that better explains the data) you'll get this:
This means you can model the relationship of your data as
$$F=0.8621*d^{-1.55}$$
Or if you prefer
$$F=\frac{0.8621}{d^{1.55}}$$
Notice the equation shows the inverse relationship between $F$ and $d$ but also notice that it is a non-linear equation. Also, that $R^2$ factor there is an statistical coefficient to test correlation between the variables, the closer it is to $1$ the better the mathematical model to explain the data. So in this case you've got $R^2=0.9724$ pretty decent uh?
You also talked about linearization and how you gave a simple estimate for the model. A simple way to linearize your data would be to simply plot the relation between force ($F$) and inverse of distance ($1/d$) and you'll get this neat linear plot:
And notice that this model has a better score as $R^2=0.994$ as the mathematical expression
$$F=0.9423\frac{1}{d}-0.092$$
explains better the relationship between $F$ and $1/d$ (which could be estimated as directly proportional) than the other model explains the relationship between $F$ and $d$ (which are inversely related).
Pick the one you prefer and to estimate the double of the distance ($2d$) simply replace in the one you picked, here I'll do it in both.
Model 1 ($F$ vs $2d$)
$$F=\frac{0.8621}{(2d)^{1.55}}=\frac{0.8621}{2^{1.55}d^{1.55}}$$
Model 2 ($F$ vs $1/2d$)
$$F=0.9423\frac{1}{2d}-0.092$$
PS: my focus here was on the data, i.e., I'm assuming the experiment and measurements to be correct.
