Generally you shouls treat such tests with gender as 'between subjects', i.e. independent, unpaired.

A agree with Jos. "Paired" means that you have meaingful, defined, unambigious pairs of values. It does not matter by what criterion this pairing is done, but it must be unambigiuous. As soon as you may choose what value "to pair" with another, then it is not paired, and a paired analysis is not adequate.

However, the data may have some correlation structure, meaning that there are subgroups of values that might be more similar to others in their sub group than to values in other sub groups. The experimental factor you are interested in is obviously and hopefully one that defines such sub groups. But there might be other (uncontrolled) factors that could also be used to divide the data into groups that were systematically different. If you don't consider this, this may be the reason for some variance in the data that you cannot explain, making it more difficult to "see" the systematic difference between the groups defined by your experimental factor.

Example:

you are interested in the effect of a treatment, and you have data from a group of people being either treated or untreated (controls). You want the systematic difference between these group means, and you may test that for instance with a t-test.

Now consider you have additional knowledge that gender* has a huge impact on the variable you are analyzing, possible as much or even more than the treatment. If you have data from female and male people, their values will vary a lot, not only depending on the treatment (you are intersted in) but also depending on thier gender. The variation introduced by the gender effect will result in a bad signal-to-noise ratio and larger p-values when testing the treatment effect.

This may be improved if you use a model that is able to consider both, the treatment and the gender effect. This would then be a two-way ANOVA model. And there can be an additional complication: it may be that the treatment effect itself differs between males and females. This is called an interaction of treatment and gender, and this can also be considered in such a model.

If the differences between genders are really not of interest, one may also chose a model that assumes that the gender means are also only random samples from a distribution with its own mean and variance - i.e., to model the gender effect as a random effect in mixed or hierarchical model. This is a bit weired for a factor like gender with levels male and female, but it may be more appropriate if gender is really coded as a social feature with levels like trans, two-spirit, cis, non-binary, fluid, neutral...

It is possible to account for several such covariables in one statistical model.

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*gender is a social feature (used as gender role, gender identity or gender expression), wheras sex is the biological property; which one is more appropriate to use depends on the field of study.

The paired data belong to a unit, while -the units of- the samples on unpaired data are quite separated (independent). Babak Jamshidi