# Averaging decibels

The decibel (dB) is a logarithmic unit that indicates the ratio of a physical quantity (usually power or intensity) relative to a specified or implied reference level.

If I measure some physical quantity in decibels, then what is the preferred way to calculate the mean of the measured values? Is it enough to simply average them, or should I convert them back to linear scale, calculate the average, and convert it back to decibels (example)? When should I use which approach, and why?

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The "physically natural" quantity to average is the actual power, or energy, but it depends exponentially on the number of decibels. So if you were averaging the power or energy, the result would be pretty much equal to the power or energy of the largest (loudest) reading in decibels.

So even though it's physically less natural, you probably want to compute the average number of decibels itself. But as you said, it's abuot "preferences". Your question isn't a question about observables, it's about subjective choices, so there can't of course be any "only correct and objective" answer. For various applications, various averages may be more or less useful or representative.

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Thank you! Do you think that the preference can be guided by the distribution of the measured quantity? I mean, I would average on the scale (linear or decibel) on which the distribution of measured values is more close to the Gaussian distribution. –  kol Dec 7 '12 at 17:20
Dear kol, well, everything may matter. But the Gaussian distribution isn't a likely expectation here. It's only relevant if the number of decibels is almost exactly the same, within "one decibel or less". In that case, if someone tries to make the loudness constant, it doesn't matter whether you exponentiate or not. However, decibels are useful exactly because the power of sound in various situations differs by many orders of magnitudes, so the readings in decibels tend to be vastly variable, and the power differs hugely in various situations. –  Luboš Motl Dec 7 '12 at 17:48
Quite generally, you may have some noise whose number of decibels is variable. Sometimes an airplane takes off at the airport, and so on. So you will have something between 50 and 130 decibels at a point. This corresponds to a huge variability of power (energy per time) that differs by 8 orders of magnitude (80 decibels). It's so wide and the behavior of the sources as well as impacts are so nonlinear that there's absolutely no reason to expect any simple distribution in this interval that is "amazingly wide", using physical criteria. The Gaussian distribution is only OK for narrow, linearized –  Luboš Motl Dec 7 '12 at 17:51
situations so exactly when the decibels become useful as a description of the sound's volume, the reasons to expect the normal distribution evaporate. The same comment applies to any other thing that is described by a logarithmic scale, for example Richter scale for earthquakes or pH for acidity. In those cases, the actual physical quantity also exponentially depends on the reading and this choice is used exactly because the physical quantities (energy in earthquake, concentration of OH- ions) has no trouble to change by many many orders of magnitude. –  Luboš Motl Dec 7 '12 at 17:53
Thank you very much! –  kol Dec 7 '12 at 18:07

There are reasons more than "preference" for the averaging. You defined it that way usually because you can get more information from that, particular for those additive quantities.

Suppose you preform a set of measurement at a particular point in space, there are two cases: (a) get the averaged value (b) take the average for the intensity itself, and then converted to decibel.

If you have the quantity in situation (b), you can know how much average energy flux passing through that point. Also, you can know the total energy flowing through that point. This information cannot be obtained from the method (a).

Similar situation for the earthquake, if you take the average for the energy, you can know the total energy released by that particular point, which is important. However, you cannot obtain this information by simply taking the average of earthquake scale.

Sure, as pointed out by Lubos, if the variation is small, these two definitions are basically the same as the $\log$ (any) function is local linear, and you can now have additive quantity again.

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