The long-term accuracy of the grid frequency is very high. You'll remember the old ac clocks that used a synchronous motor would almost always keep perfect time. Over the short term you'll see some variation, but nothing like the one percent error your meter is showing.
Back in 2011, the power companies experimented with letting the frequency drift a bit and not correcting it over the long term. I googled and found a few old articles about the experiment, but nothing about the RESULTS. So I don't know if they decided go back to the old rigorous regulation or to allow frequency variation to continue.
I think the real problem with your meter is noise. It's not easy to make a cheap, accurate, broad-band frequency meter. (If you try comparing different cheap guitar tuners, you'll quickly find they are not at all the same.) A little noise is going to introduce some extra zero crossings that will really trash the measurement. You might try adding a low-pass RC filter to your meter and see if that improves the measurement. Also, is there a spec for your meter's accuracy? If the manufacturer doesn't publish the accuracy, it's fair to assume that one percent is about all you'll get.
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