Average of Fourier Transforms
This is the basics of spectral analysis using digital Fourier transform. Mountains of literature is written on the subject. 

In short, when you do a FFT over the entire data set 1-minute long, you will have fairly high spectral resolution of about 1/60 Hz, or 16.6 mHz (milliHz). But your confidence interval of the amplitude at each frequency line will be very low, you will have a very high ~100% noise level.

When you break your data into smaller blocks, so for each second-long data your spectral resolution will be 1 Hz, low. But, if you take an average of all 60 periodograms, your noise level will be SQRT(60) =~8 times smaller, so you can be much more confident if you see some spectral peaks in your spectrum. 

Note, that this works only on power spectrum estimates, Re^2 + Im^2.

So, with a limited data set, there is a trade-off between accuracy of estimates of spectral amplitudes, and frequency resolution of your power spectrum.