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A wrist GPS solves the problem for you only if you actually use it, meaning you look at the data and interpret it. Otherwise it's just a nice watch on your wrist. The useful stuff is like the distance tracked (especially if you change routes), your average pace, and if you sync it with apps like Strava or Garmin Connect you can see trends over time. But fair warning: GPS isn't super accurate, especially in cities with lots of buildings or under trees, so if you always run in the same places the track could vary by 50-100 meters from run to run. It's not ideal if you want precise data.
What others don't point out is how important battery life is. If you're looking at an entry-level watch for 150-200 euros, maybe it lasts a couple days if you use it to track races, but if you run often and want to keep it on all the time for step counting, charging becomes a pain. Better models like the Garmin Epix or Apple Watch Ultra last longer, but they cost twice as much.
If you run 3-4 times a week and you really want to improve, yeah, it's worth it. You keep better track of yourself, you see if you're faster or slower than two months ago, and you have the data in black and white. But if it's just occasional or just for moving around, a smartphone with a free app like Strava does exactly the same job without spending a dime.
It really depends on what kind of runner you are honestly. If you run just for the pure pleasure of it and you don't care about the data, a GPS is like a waste of money, but if you're serious about improving your times or you want to track your routes, it's pretty useful because it gives you the precise distance and speed in real time, stuff that your phone might not be able to do well if you keep it in your pocket.
Most of the runners I know use it mainly so they don't get lost and to see how long they've been running, and it's convenient because you don't have to carry your phone around. But honestly, if you're just starting out now, you could also wait a bit and save up, it's not like having the watch is going to change your performance on its own.
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