The frequency axis of this symbolic diagram would be logarithmically scaled.

The center frequency f₀ (resonant frequency) is the geometric mean between the lower cutoff frequency f₁ and the upper cutoff frequency f₂ of a frequency band. See also: Band-pass filter. f2 - f1 is called the bandwidth B. Bandwidth diagram made by User:Kieff using mIRC and Microsoft Paint An alternative version is here: Image:Bandwidth blue. ... This article is about resonance in physics. ... The geometric mean of a set of positive data is defined as the product of all the members of the set, raised to a power equal to the reciprocal of the number of members. ... In telecommunications, the term cutoff frequency fc has the following meanings: 1. ... The frequency axis of this symbolic diagram would be logarithmically scaled. ... // Analog For analog signals, bandwidth is the width, usually measured in hertz, of a frequency band f2 − f1. ...

Only if the bandwidth f₂ - f₁ is very small in comparison to the center frequency it is sometimes possible to use this arithmetic mean for calculations, but this is often calculated in mistake: // Analog For analog signals, bandwidth is the width, usually measured in hertz, of a frequency band f2 − f1. ... In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ...

At radio stations (medium wave) the bandwidth is often only 9 kHz. A transmitter, which has 1500 kHz, is transmitting from 1495.5 kHz to 1504.5 kHz.
The exact formula gives:

and the short formula gives in this case the very close result of:

The short calculated value is always too large. If the bandwidth is given by B = f₂ - f₁, the difference is:

.

But if for instance we are looking for the center frequency of the telephone audio band from 300 Hz to 3300 Hz, we get (3300 + 300) / 2 = "1800 Hz" for the short arithmetic mean calculation, but the root of 300 x 3300 = "995" Hz with the correct geometric mean formula. What a big difference!

That the geometric mean is not the arithmetic mean can be seen in a calculation program at the bottom in the external link. There one can compare the difference of both values.

External links

• Calculations and comparisons between the geometric mean and the arithmetic mean