I’m not sure that this is true in general, but there are still some factors we can talk about.
For one thing, female insects (and spiders, and other arthropods) are often bigger than males. Insects tend to maximize reproduction by producing a large number of young, and bigger females can make more eggs. This pushes females to be larger.
In mammals on the other hand, reproduction is usually maximized by producing fewer offspring and caring for them more effectively. There’s no pressure on females to grow huge to hold large clutches of eggs (or live babies more likely). On the other hand, male mammals often wind up competing with each other for access to females, and big males tend to win these competitions. That drives males to be bigger (although there are also many mammal species where there’s no size difference)
Aside from sheer size difference, another factor that makes female insects more likely to bite is that there are many insects that consume blood. As I noted before, female insects make a ton of eggs. This requires a lot of nutrients. Blood has a lot of nutrients. So many insect species go for blood. Males in many species (mosquitoes for example) don’t even go for blood at all because they don’t have to make eggs.
The situation is different in mammals. Most adult insects live brief lives and just need enough to mate and lay eggs. But mammals live a long time and have to maintain pretty expensive warm blooded metabolisms. Between the need to often maintain a larger body size, and the high baseline need for calories just to live a full life, there’s not such a huge difference in need of nutrients between male and female mammals, so, eg, female tigers aren’t the only ones that hunt meat.
Another notable difference is eusocial insects. Ants, bees, wasps, specifically. In these species, for several reasons but importantly because of a quirk in genetics, worker females are closely related to the queen’s offspring. These workers help defend the nest and the offspring and are thus more likely to sting and bite. Indeed, since the stinger is a modified egg laying apparatus, they are the only ones that _can_ sting. But all this is more of a quirk of one specific type of common insect, not something that’s generally true of all insects.
Of course, “protecting offspring” is a pretty common reason for mammals to attack as well, and female mammals with offspring under threat can be pretty aggressive too.
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