# How to test difference in variability with paired data?

I am dealing with what seems to be a very simple problem, but my statistical knowledge is now a bit rusty, and I can't think of any good solution. I tried to cast my problem in a simpler one, hope it still makes sense!

The problem

Let assume I have two machines, A and B, both generating a toy composed of two metal spheres of weight $w_1$ and $w_2$, respectively. For the toy to work properly there should be a small variability between $w_1$ and $w_2$ (they should be balanced).

Let assume I measured $n$ toys produced by the machine A, and let $a = \{a_1, \dots, a_n\}$ be the vector describing the weights measurements, where each $a_i \in a$ is composed by a pair $\{w_1, w_2\}$. Let assume I measured $m$ toys produced by the machine B, and let $b = \{b_1, \dots, b_n\}$ the vector describing the weights measurements, as before.

The question

I would like to test if the variability of weight between each sphere pairs generated by the machine A is larger than the one generated by the machine B, that is to test if the machine A generates toys that are less balanced than those generated by the machine B.

My bits

• If the measurements weren't to be taken in pairs, I would use a Levene's test to test if the variability of the spheres generated by A and the one generated by B are different.
• each pair of weights could be summarised using its absolute difference, thus creating a new vector $a' = \{a'_1, \dots, a'_n\}$ ($b'$), where $a'_i = |w_1 - w_2|$ for each $a'_i \in a'$. ANOVA can then be used to test the difference in mean between $a'$ and $b'$ -- the main idea being the fact that if toys generated by the machine A have larger variability, their mean difference should be larger as well. However, I am not sure if this makes any sense.
• each pair of weights could be summarised using its difference, thus creating a vector $a' = \{a'_1, \dots, a'_n\}$ ($b'$), where $a'_i = w_1 - w_2$ for each $a'_i \in a'$. The Levene's test can then be used to test the difference in variability (following the same intuition as before), but, again, I am not sure if this makes any sense.

Extra question:

Let's say I measured 30 pairs of toys generated by the machine A and 300 generated by the machine B. Is the proposed solution still valid? That is: is the solution valid if $n < m$? And if $n > m$?

Thank you very much for your help!