What averaging more scans actually buys

My own published research is in microwave spectroscopy, and anyone who has stared at a weak line knows the reflex: run it again and add the scans together. The reflex is right. What gets lost is the arithmetic, and the conditions under which that arithmetic holds.

The square-root law

A 2018 methods paper by Mikkelsen and colleagues states the textbook version for magnetic resonance spectroscopy: signal-to-noise scales with field strength, voxel volume and the square root of N, where N is the number of signal averages. The reason, in their words, is that the signals “are phase-coherent and noise is an inherently random process; therefore, signal will increase faster than noise does.”

Add N scans and the real signal, identical every time, adds up N-fold. Random noise partly cancels itself, so it grows only as √N. The ratio improves by √N. In practice that means:

• 4 times the scans doubles the signal-to-noise.
• 100 times the scans buys a factor of 10.
• Doubling an overnight run improves it by about 41 percent, not 100.

That last line is the one people forget when they book instrument time. Each further improvement costs more than the one before.

The rule assumes the noise plays fair

In 1965, Richard Ernst analyzed time averaging with “special emphasis on the correlation effects caused by non-white noise.” His abstract says plainly that “only in special cases” does signal-to-noise improve exactly as the square root of the number of scans. For one common kind of correlated noise he shows the deviation stays within a factor of √2; for others the law holds exactly. He also notes that a noise peak at a multiple of the scan repetition frequency matters only if it is narrower than that repetition frequency.

The practical reading: √N is a promise about random, uncorrelated noise. Drift, interference locked to your scan rate, or anything else that repeats from scan to scan does not average away the same way, because it adds up like signal. If your baseline wanders or a line keeps creeping, more scans can make an artifact look more convincing, not less.

When averaging blurs what you are measuring

Averaging assumes the sample is the same in every scan. A reaction is not. Flook and Lloyd-Jones (J. Org. Chem., 2024) describe the dilemma for in situ NMR reaction monitoring: stacking scans raises signal-to-noise but leaves fewer time points, so “improve one aspect … results in the detriment of the other.”

Their fix is to save every scan on its own and average afterward with a moving window, which lets you pick the trade-off after the fact. And they measured the cost of getting it wrong. For a fast reaction at 50 °C, monitored with 45 single scans, averaging over too wide a time window distorted the kinetics: the onset of what they call “overaveraging” was “evident when n = 16, and becomes substantial when n = 32,” worst where the curve bends most. An 8-scan window kept the kinetics faithful.

The general lesson travels well beyond NMR. Averaging trades time resolution for signal-to-noise. If the thing you are measuring changes within your averaging window, the clean-looking result is an average of different states.

When more signal stops buying a better answer

The Mikkelsen paper's real contribution is the next step. For GABA-edited spectroscopy in the brain, with simulations plus data from 18 participants across five brain regions, they confirmed that signal-to-noise kept rising as √N. But the number a study actually needs, the spread of results across a group of people, leveled off.

• In the simulations, group-level variability (coefficient of variation) was 12.8 percent at half the acquisition, 12.5 percent at the full acquisition, against a true value of 12.3 percent.
• In the human data, group variance “stabilizes at approximately 200 averages,” with no statistically significant difference between 200 and 300.
• The point of diminishing returns depended on where in the brain they measured.

Their conclusion is that GABA quantification is “typically not SNR-limited,” and that results reached statistical equivalence as early as about 7 minutes into a 10-minute scan. Once the instrument's noise is small next to the real differences between people (or samples, or batches), extra averaging polishes the part of the error that no longer dominates.

One caution on scope: this is one technique, one metabolite and one cohort, and the authors frame it as a design framework, not a universal number. The shape of the argument is what generalizes.

What to look for in a methods section

• N, stated. How many scans or averages, and was it the same for every sample?
• How signal-to-noise was defined. Mikkelsen's group divided the peak amplitude by twice the standard deviation of a noise region. Other labs use other conventions, so SNR numbers from different papers are often not directly comparable.
• Was the sample stable over the averaging time? For anything changing, ask how wide the averaging window was relative to how fast it changes.
• What dominates the error? If between-sample variation is the bigger term, a report of ever-higher SNR is answering the wrong question.

The signal

Averaging scans is real, cheap signal-to-noise, but it pays out as the square root: four times the scans for twice the quality. Ernst showed in 1965 that the square root is exact only when the noise is well behaved. Flook and Lloyd-Jones showed that averaging a changing sample blurs it. And Mikkelsen's group showed that once instrument noise stops being the biggest error, more averaging stops improving the answer you actually need.