It seems fairly straightforward. You're not using your Wifi, you should just turn it off to save battery. However, this question isn't as straightforward as it might've been in years past, and while part of that has to do with software advancements, it still has a lot do with the the hardware of your phone and more importantly the trade-off between which radios you keep running.
Most of the time, you should leave your Wifi on. Here's why.
Wifi's not as battery hungry as cellular radios
Wifi uses your battery, it's true, but Wifi doesn't use nearly the battery that your cellular radios do. Wifi will instantly connect when in range, and while out of range, it (usually) sips a small amount of battery looking for a network. Let's also remember that turning off Wifi means that you'll be using your cellular data, and while some of you lucky ducks may have unlimited data, that's not the case for most users and if you turn off Wifi and forget to turn it back on, you're wasting valuable data in your cap that could've been just as easily obtained over Wifi.
If you're about to embark on a day-long trip and won't be around any Wifi for it, then yes, turning off Wifi will save your more battery, but there's no real need to shut it off if you're just going between one Wifi zone and another, such as between home and work or out to run some errands. If you're looking to conserve battery while out and about, disable Auto-Sync rather than Wifi — it'll help you avoid distracting G+ notifications, too.
Wi-Fi's not just for internet, it's for location
That said, if that day-long trip is going to be exploring a metroplex that's swimming in Wifi signals anyway, you might want to keep it on, even if you don't log into them. If you've ever gotten this pop-up while starting to navigate, you might wonder why your phone wants the Wifi rather than the GPS, and this is because of the Wifi Positioning System.
This is used in area where GPS signals may get confused or blocked out by other sources, and it's also a way to help better determine a user's position without relying solely on GPS, which can be a battery-hog while giving turn-by turn positioning and directions. Ever wonder how Google can help you navigate inside the mall? Wifi Positioning for the win.
Wi-Fi is usually free
While we will be the first to remind you — after Doctor Who — that you shouldn't use Wifi you don't recognize, we'd also like to remind you that Wifi is free to use and offered in a great and growing number of areas like malls, gyms, theme parks, and offices. Some carriers even have Wifi Hotspot networks for their customers to use, like AT&T, who will automatically log me into one if I'm in range of it. Some might argue that's intrusive, and at times I do agree with that, but those of us on capped plans should take advantage of free data where we can get it.
So keep your Wi-Fi on
Unless you're about to go racing off into the wild, Wifi-less wilderness for the day, turning off Wifi simply isn't as needed as it once was. Our phones last longer than they used to, and let's not forget that these phones are made to be used. If you turn off the Wifi and the mobile data, yeah, your phone's going to last longer. But you're also not going to use it as much.
Is there a reason for or against this we missed? Are you a lucky unlimited plan holder who doesn't have to worry about how many gigs you burn through before you find a hotspot? Or are you a mAh-pinching miser who turns off your WifI after each and every time you check traffic/weather/email? Hit us up with your Wifi strategy in the comments below.
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