the signal-to-noise ratio
A faint voice trying to reach you across a crowded, noisy room: whether you can make out the words depends not on how loud the voice is in absolute terms but on how it stands against the background din. The signal-to-noise ratio captures exactly that comparison, the size of the thing you want set against the size of the random fluctuations trying to bury it. It, not raw signal strength, decides whether a measurement is detectable.
The signal-to-noise ratio is the ratio of signal amplitude (or power) to noise amplitude (or power); as a power ratio it is often quoted in decibels, SNR_dB = 10 log10(P_signal / P_noise). In a counting experiment where the noise is the statistical fluctuation sqrt(B) of a background B, a signal of S counts has significance S / sqrt(B). Because the signal grows with the number of events N while the noise grows only as sqrt(N), that significance improves as sqrt(N) the longer you run. The same square-root gain appears when you average M repeated traces: coherent signal adds linearly while random noise adds in quadrature, so the SNR climbs by sqrt(M).
You meet it in every measurement: a lock-in amplifier dragging a tiny modulated signal out of a sea of noise, LIGO extracting a gravitational-wave chirp buried far below the instrument's noise floor, a resonance bump rising above a smooth background in a particle detector. The honest caveat is that boosting SNR by averaging works only against random, zero-mean noise. It does nothing against a systematic offset, and it fails for coherent (correlated) noise, which does not average away the way independent noise does, so a high SNR alone is no guarantee of an accurate result.
To see a spectral line twice as clearly, that is to double the signal-to-noise ratio when the noise is statistical, you must collect four times as many counts, since SNR grows only as the square root of the integration time.
SNR ~ sqrt(integration time): doubling clarity costs quadruple the data.
Averaging improves signal-to-noise only for random, uncorrelated, zero-mean noise, and only against that noise; a systematic bias survives any amount of averaging, so a high SNR is not by itself proof of an accurate measurement.